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Geometric and Rlgebrctic
Topologicol Methods in
Quantum Mechanics

Geometric one) fllgebroic
Topologicol Methods in
Quantum Mechanics
Giovanni Giachetta & Luigi Mangiarotti
University of Camerino, Italy
Gennadi Sardanashvily
Moscow State University, Russia
fc World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
GEOMETRIC AND ALGEBRAIC TOPOLOGICAL METHODS IN
QUANTUM MECHANICS
Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means,
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Forphotocopying of material inthis volume, pleasepay acopying fee through the Copyright Clearance
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is not required from the publisher.
ISBN 981-256-129-3
Printed in Singapore.
ISBN 891-256-129-3

Preface
Contemporary quantum mechanics meets an explosion of different types
of quantization. Some of these quantization techniques (geometric quan-
tization, deformation quantization, BRST quantization, noncommutative
geometry, quantum groups, etc.) call into play advanced geometry and al-
gebraic topology. These techniques possess the following main peculiarities.
• Quantum theory deals with infinite-dimensional manifolds and fibre
bundles as a rule.
• Geometry in quantum theory speaks mainly the algebraic language of
rings, modules, sheaves and categories.
• Geometric and algebraic topological methods can lead to non-
equivalent quantizations of a classical system corresponding to different
values of topological invariants.
Geometry and topology are by no means the primary scope of our book,
but they provide the most effective contemporary schemes of quantization.
At the same time, we present in a compact way all the necessary up to date
mathematical tools to be used in studying quantum problems.
Our book addresses to a wide audience of theoreticians and mathe-
maticians, and aims to be a guide to advanced geometric and algebraic
topological methods in quantum theory. Leading the reader to these fron-
tiers, we hope to show that geometry and topology underlie many ideas
in modern quantum physics. The interested reader is referred to extensive
Bibliography spanning mostly the last decade. Many references we quote
are duplicated in E-print arXiv (http://xxx.lanl.gov).
With respect to mathematical prerequisites, the reader is expected to
be familiar with the basics of differential geometry of fibre bundles. For
the sake of convenience, a few relevant mathematical topics are compiled
in Appendixes.
V

Contents
Preface v
Introduction 1
1. Commutative geometry 17
1.1 Commutative algebra 17
1.2 Differential operators on modules and rings 23
1.3 Connections on modules and rings 27
1.4 Homology and cohomology of complexes 31
1.5 Homology and cohomology of groups and algebras . . . . 39
1.6 Differential calculus over a commutative ring 56
1.7 Sheaf cohomology 59
1.8 Local-ringed spaces 70
1.9 Algebraic varieties 85
2. Classical Hamiltonian systems 91
2.1 Geometry and cohomology of Poisson manifolds 91
2.2 Geometry and cohomology of symplectic foliations . . . . 110
2.3 Hamiltonian systems 115
2.4 Hamiltonian time-dependent mechanics 136
2.5 Constrained Hamiltonian systems 157
2.6 Geometry and cohomology of Kahler manifolds 172
2.7 Appendix. Poisson manifolds and groupoids 189
3. Algebraic quantization 195
3.1 GNS construction I. C*-algebras of quantum systems . . . 195
vii

viii Geometric and Algebraic Topological Methods in Quantum Mechanics
3.2 GNS construction II. Locally compact groups 209
3.3 Coherent states 217
3.4 GNS construction III. Groupoids 224
3.5 Example. Algebras of infinite qubit systems 229
3.6 GNS construction IV. Unbounded operators 234
3.7 Example. Infinite canonical commutation relations . . . . 238
3.8 Automorphisms of quantum systems 249
4. Geometry of algebraic quantization 257
4.1 Banach and Hilbert manifolds 257
4.2 Dequantization 271
4.3 Berezin's quantization 274
4.4 Hilbert and C*-algebra bundles 278
4.5 Connections on Hilbert and C*-algebra bundles 282
4.6 Example. Instantwise quantization 286
4.7 Example. Berry connection 290
5. Geometric quantization 295
5.1 Leafwize geometric quantization 295
5.2 Example. Quantum completely integrable systems . . . . 306
5.3 Quantization of time-dependent mechanics 312
5.4 Example. Non-adiabatic holonomy operators 324
5.5 Geometric quantization of constrained systems 332
5.6 Example. Quantum relativistic mechanics 335
5.7 Geometric quantization of holomorphic manifolds 342
6. Supergeometry 347
6.1 Graded tensor calculus 347
6.2 Graded differential calculus and connections 352
6.3 Geometry of graded manifolds 358
6.4 Lagrangian formalism on graded manifolds 366
6.5 Lagrangian supermechanics 382
6.6 Graded Poisson manifolds 385
6.7 Hamiltonian supermechanics 388
6.8 BRST complex of constrained systems 392
6.9 Appendix. Supermanifolds 401
6.10 Appendix. Graded principal bundles 423
6.11 Appendix. The Ne'eman-Quillen superconnection 426

Contents ix
7. Deformation quantization 433
7.1 Gerstenhaber's deformation of algebras 433
7.2 Star-product 444
7.3 Fedosov's deformation quantization 450
7.4 Kontsevich's deformation quantization 459
7.5 Deformation quantization and operads 472
7.6 Appendix. Monoidal categories and operads 475
8. Non-commutative geometry 483
8.1 Modules over C*-algebras 484
8.2 Non-commutative differential calculus 486
8.3 Differential operators in non-commutative geometry . . . . 492
8.4 Connections in non-commutative geometry 498
8.5 Connes' non-commutative geometry 503
8.6 Landsman's quantization via groupoids 507
8.7 Appendix. if-Theory of Banach algebras 509
8.8 Appendix. The Morita equivalence of C*-algebras 512
8.9 Appendix. Cyclic cohomology 514
8.10 Appendix. KK-Theory 518
9. Geometry of quantum groups 523
9.1 Quantum groups 523
9.2 Differential calculus over Hopf algebras 530
9.3 Quantum principal bundles 535
10. Appendixes 541
10.1 Categories 541
10.2 Hopf algebras 546
10.3 Groupoids and Lie algebroids 553
10.4 Algebraic Morita equivalence 565
10.5 Measures on non-compact spaces 569
10.6 Fibre bundles I. Geometry and connections 586
10.7 Fibre bundles II. Higher and infinite order jets 611
10.8 Fibre bundles III. Lagrangian formalism 618
10.9 Fibre bundles IV. Hamiltonian formalism 626
10.10 Fibre bundles V. Characteristic classes 633
10.11 /f-Theory of vector bundles 648
10.12 Elliptic complexes and the index theorem 650

x Geometric and Algebraic Topological Methods in Quantum Mechanics
Bibliography 661
Index 683

Introduction
Geometry of classical mechanics and field theory is mainly differential geom-
etry of finite-dimensional smooth manifolds, fibre bundles and Lie groups.
The key point why geometry plays a prominent role in classical field
theory lies in the fact that it enables one to deal with invariantly de-
fined objects. Gauge theory has shown clearly that this is a basic phys-
ical principle. At first, a pseudo-Riemannian metric has been identified
to a gravitational field in the framework of Einstein's General Relativ-
ity. In 60-70th, one has observed that connections on a principal bundle
provide the mathematical model of classical gauge potentials [120; 284;
442]. Furthermore, since the characteristic classes of principal bundles
are expressed in terms of the gauge strengths, one can also describe the
topological phenomena in classical gauge models [142]. Spontaneous sym-
metry breaking and Higgs fields have been explained in terms of reduced
G-structures [341]. A gravitational field seen as a pseudo-Riemannian met-
ric exemplifies such a Higgs field [230]. In a general setting, differential
geometry of smooth fibre bundles gives the adequate mathematical formu-
lation of classical field theory, where fields are represented by sections of
fibre bundles and their dynamics is phrased in terms of jet manifolds [169].
Autonomous classical mechanics speaks the geometric language of sym-
plectic and Poisson manifolds [l; 279; 426]. Non-relativistic time-dependent
mechanics can be formulated as a particular field theory on fibre bundles
over R [294].
At the same time, the standard mathematical language of quantum me-
chanics and perturbative field theory, except gravitation theory, has been
long far from geometry. In the last twenty years, the incremental develop-
ment of new physical ideas in quantum theory (including super- and BRST
symmetries, geometric and deformation quantization, topological field the-
1

2 Geometric and Algebraic Topological Methods in Quantum Mechanics
ory, anomalies, non-commutativity, strings and branes) has called into play
advanced geometric techniques, based on the deep interplay between alge-
bra, geometry and topology.
Let us briefly survey some peculiarities of geometric and algebraic topo-
logical methods in quantum mechanics.
Let us recall that, in the framework of algebraic quantization, one as-
sociates to a classical system a certain (e.g., von Neumann, C*-, canonical
commutation or anticommutation relation) algebra whose different repre-
sentations are studied. Quantization techniques under discussion introduce
something new. Namely, they can provide non-equivalent quantizations
of a classical system corresponding to different values of some topological
and differential invariants. For instance, a symplectic manifold X admits
a set of non-equivalent star-products indexed by elements of the cohomol-
ogy group i?2
(X)[[/i]] [206; 340]. Thus, one may associate to a classical
system different underlying quantum models. Of course, there is a ques-
tion whether this ambiguity is of physical or only mathematical nature.
Prom the mathematical viewpoint, one may propose that any quantization
should be a functor between classical and quantum categories (e.g., some
subcategory of Poisson manifolds on the classical side and a subcategory of
C*-algebras on the quantum side) [271]. From the physical point of view,
dequantization becomes important.
There are several examples of sui generis dequantizations. For in-
stance, Berezin's quantization [145] in fact is dequantization. One can
also think of well-known Gelfand's map as being dequantization of a
commutative C*-algebra A by the algebra of continuous complex func-
tions vanishing at infinity on the spectrum of A. This dequantiza-
tion has been generalized to non-commutative unital C*-algebras [105;
239]. The concept of the strict C*-algebraic deformation quantization
implies an appropriate dequantization when h —> 0 [269; 372]. In
Connes' non-commutative geometry, dequantization of the spectral triple
in the case of a commutative algebra C°°(X) is performed in order
to restart the original differential geometry of a spin manifold X [107;
368].
I.
Let us start with familiar differential geometry. There are the following
reasons why this geometry contributes to quantum theory.
(i) Most of the quantum models comes from quantization of the original

Introduction 3
classical systems and, therefore, inherits their differential geometric prop-
erties. First of all, this is the case of canonical quantization which replaces
the Poisson bracket {/, /'} of smooth functions with the bracket [/, /'] of
Hermitian operators in a Hilbert space such that Dirac's condition
[f,T] = -ih{fJ!
)
holds. Let us mention Berezin-Toeplitz quantization [47; 145; 365] and ge-
ometric quantization [141; 401; 426; 438] of symplectic, Poisson and Kahler
manifolds.
(ii) Many quantum systems are considered on a smooth manifold
equipped with some background geometry. As a consequence, quan-
tum operators are often represented by differential operators which act
in a pre-Hilbert space of smooth functions. A familiar example is
the Schrodinger equation. The Kontsevich deformation quantization is
based on the quasi-isomorphism of the differential graded Lie algebra
of multivector fields (endowed with the Schouten-Nijenhuis bracket and
the zero differential) to that of polydifferential operators (provided with
the Gerstenhaber bracket and the modified Hochschild differential) [219;
255].
(iii) In some quantum models, differential geometry is called into play
as a technical tool. For instance, a suitable [/(l)-principal connection is
used in order to construct the operators / in the framework of geomet-
ric quantization. Another example is Fedosov's deformation quantization
where a symplectic connection plays a similar role [149]. Let us note that
this application has stimulated the study of symplectic connections [165].
(iv) Geometric constructions in quantum models often generalize the
classical ones, and they are build in a similar way. For example, connections
on principal superbundles [21], graded principal bundles [405], and quantum
principal bundles [293] are defined by means of the corresponding one-
forms in the same manner as connections on smooth principal bundles with
structure finite-dimensional Lie groups.
II.
In quantum models, one deals with infinite-dimensional smooth Banach
and Hilbert manifolds and (locally trivial) Hilbert and C*-algebra bun-
dles. The definition of smooth Banach (and Hilbert) manifolds follows that
of finite-dimensional smooth manifolds in general, but infinite-dimensional

Banach manifolds are not locally compact, and they need not be para-
compact [273; 422]. In particular, a Banach manifold admits the differen-
tiable partition of unity if and only if its model space does. It is essential
that Hilbert manifolds (but not, e.g., nuclear manifolds) satisfy the inverse
function theorem and, therefore, locally trivial Hilbert bundles are denned.
However, they need not be bundles with a structure group.
(i) Infinite-dimensional Kahler manifolds provide an important exam-
ple of Hilbert manifolds [327]. In particular, the projective Hilbert space
of complex rays in a Hilbert space E is such a Kahler manifold. This
is the space the pure states of a C*-algebra A associated to the same
irreducible representation n of A in a Hilbert space E [129]. There-
fore, it plays a prominent role in many quantum models. For instance,
it has been suggested to consider a loop in the projective Hilbert space,
instead of a parameter space, in order to describe Berry's phase [7;
43]. We have already mentioned the dequantization procedure which repre-
sents a unital C*-algebra by a Poisson algebra of complex smooth functions
on a projective Hilbert space [105].
(ii) Sections of a Hilbert bundle over a smooth finite-dimensional mani-
fold X make up a particular locally trivial continuous field of Hilbert spaces
in [129]. Conversely, one can think of any locally trivial continuous field
of Hilbert spaces or C*-algebras as being the module of sections of a topo-
logical fibre bundle. Given a Hilbert space E, let B C B{E) be some
C*-algebra of bounded operators in E. The following fact reflects the non-
equivalence of Schrodinger and Heisenberg quantum pictures. There is the
obstruction to the existence of associated (topological) Hilbert and C*-
algebra bundles £ —
» X and B —
+ X with the typical fibres E and B,
respectively. Firstly, transition functions of £ define those of B, but the
latter need not be continuous, unless B is the algebra of compact operators
in E. Secondly, transition functions of B need not give rise to transition
functions of £. This obstruction is characterized by the Dixmier-Douady
class of B in the Cech cohomology group H3
(X, Z). There is the similar ob-
struction to the [/(l)-extension of structure groups of principal bundles [73;
86]. One also meets the Dixmier-Douady class as the obstruction to a bun-
dle gerbe being trivial [58; 87].
(iii) There is a problem of the definition of a connection on C*-algebra
bundles which comes from the fact that a C*-algebra (e.g., any commutative
C*-algebra) need not admit non-zero bounded derivations. An unbounded
derivation of a C*-algebra A obeying certain conditions is an infinitesimal
generator of a strongly (but not uniformly) continuous one-parameter group

Introduction 5
of automorphisms of A [62]. Therefore, one may introduce a connection on
a C*-algebra bundle in terms of parallel transport curves and operators,
but not their infinitesimal generators [15]. Moreover, a representation of
A does not imply necessarily a unitary representation of its strongly (not
uniformly) continuous one-parameter group of automorphisms. In contrast,
connections on a Hilbert bundle over a smooth manifold can be defined both
as particular first order differential operators on the module of its sections
[296] and a parallel displacement along paths lifted from the base [228].
(iv) Instantwise geometric quantization of time-dependent mechanics is
phrased in terms of Hilbert bundles over R [174; 401]. Holonomy operators
in a Hilbert bundle with a structure finite-dimensional Lie group are well
known to describe the non-Abelian geometric phase phenomena [44]. At
present, holonomy operators in Hilbert bundles attract special attention
in connection with quantum computation and control theory [159; 181;
349].
III.
Geometry in quantum systems speaks mainly the algebraic language of
rings, modules and sheaves due to the fact that the basic ingredients in the
differential calculus and differential geometry on smooth manifolds (except
non-linear differential operators) can be restarted in a pure algebraic way.
(i) Any smooth real manifold X is homeomorphic to the real spectrum of
the M-ring C°°(X) of smooth real functions on X provided with the Gelfand
topology [17; 233]. Furthermore, the sheaf Cg of germs of / G C°°(X) on
this topological space fixes a unique smooth manifold structure on X such
that it is the sheaf of smooth functions on X. The pair (X, C^) exemplifies
a local-ringed space. A sheaf SRon a topological space X is said to be a
local-ringed space if its stalk 9lx at each point x £ X is a local commutative
ring [414], One can associate to any commutative ring A the particular
local-ringed space, called an affine scheme, on the spectrum Spec .4. of A
endowed with the Zariski topology [421].
Furthermore, one can assign the following algebraic variety to any com-
mutative finitely generated /C-ring A over an algebraically closed field K.
Given a ring fC[x] of polynomials with coefficients in /C, let us consider the
epimorphism 4> : Kx —
» A defined by the equalities <fi(xi) = a*, where a*
are generating elements of A. Zeros of polynomials in Ker (p make up an
algebraic variety V whose coordinate ring /Cy is exactly A. The subvarieties
of V constitute the system of closed sets of the Zariski topology on V [397].

Every affine variety V in turn yields the affine scheme Spec/Cy such that
there is one-to-one correspondence between the points of Spec/Cv and the
irreducible subvarieties of V. For instance, complex algebraic varieties have
a structure of complex analytic manifolds.
(ii) Given a (connected) compact topological space X and the ring
C°(X) of continuous complex functions on X, the well-known Serre-Swan
theorem [409] states that a C°(X)-module is finitely generated projective if
and only if it is isomorphic to the module of sections of some (topological)
vector bundle over X. Moreover, this isomorphism is a categorial equiv-
alence [237], and its variant takes place if X is locally compact [369]. If
X is a compact smooth manifold, there is the similar isomorphism of a
finitely generated projective C°°(X)-modules on X to the modules of sec-
tions of some smooth vector bundle over X [430], and this is also true if
X is not necessarily compact. A variant of the Serre-Swan theorem for
Hilbert modules over non-commutative C*-algebras holds [239].
(iii) Let K. be a commutative ring, A a commutative /C-ring, and P, Q
some ^-modules. The /C-linear Q-valued differential operators on P can
be defined [202; 233; 261]. The representative objects of the functors Q —
>
Diff S(P, Q) are the jet modules J"P of P. Using the first order jet module
Jl
P, one also restarts the notion of a connection on an ^-module P [260;
296]. Such a connection assigns to each derivation r £ dA of a /C-ring .4 a
first order P-valued differential operator VT on P obeying the Leibniz rule
Vr(ap)=r(o)p + aVT(p).
For instance, if P is a C°° (X)-module of sections of a smooth vector bundle
Y —
> X, we come to the familiar notions of a linear differential operator
on Y, the jets of sections of Y —
» X and a linear connection on Y —
»
X. Similarly, connections on local-ringed spaces are introduced [296]. In
supergeometry, connections on graded modules over a graded commutative
ring and graded local-ringed spaces are defined [2l].
In non-commutative geometry, different definitions of a differential op-
erator on modules over a non-commutative ring have been suggested [50;
136; 286]. Roughly speaking, the difficulty lies in the fact that, if d is a
derivation of a non-commutative /C-ring A, the product ad, a € A, need
not be so. There are also different definitions of a connection on modules
over a non-commutative ring [137; 267].
(iv) Let K. be a commutative ring, A a (commutative or non-
commutative) /C-ring, and Z(A) the center of A. Derivations of A make
up a Lie /C-algebra 5.4. Let us consider the Chevalley-Eilenberg com-

Introduction 7
plex of /C-multilinear morphisms of DA to A, seen as a DA-module [160;
426]. Its subcomplex O*($A, d) of Z(^-multilinear morphisms is a differ-
ential graded algebra, called the Chevalley-Eilenberg differential calculus
over A. It contains the minimal differential calculus O*A generated by
elements da, a € A. If ^4. is the R-ring C°°(X) of smooth real functions
on a smooth manifold X, the module QC°°(X) of its derivations is the Lie
algebra of vector fields on X and the Chevalley-Eilenberg differential cal-
culus over C°°(X) is exactly the algebra of exterior forms on a manifold X
where the Chevalley-Eilenberg coboundary operator d coincides with the
exterior differential, i.e., O*(X>C°°(X),d) is the familiar de Rham complex.
In a general setting, one therefore can think of elements of the Chevalley-
Eilenberg differential calculus Ok
(QA, d) over an algebra .4 as being differ-
ential forms over A. Similarly, the Chevalley-Eilenberg differential calculus
over a graded commutative ring is constructed [160].
IV.
As was mentioned above, homology and cohomology of spaces and al-
gebraic structures often play a role of sui generis hidden quantization pa-
rameters which can characterize non-equivalent quantizations.
(i) First of all, let us mention the abstract de Rham theorem [220] and,
as its corollary, the homomorphism
H*(X,Z)-+H*(X)
of the Cech cohomology of a smooth manifold X to the de Rham cohomol-
ogy of exterior forms on X. For instance, the Chern classes c* € H2l
(X, Z)
of a [/(n)-principal bundle P —» X are represented by the de Rham co-
homology classes of certain characteristic exterior forms V-2.%(FA) on X
expressed into the strength two-form FA of a principal connection A on
F - » I [142]. The Chern class c-i of a complex line bundle plays a promi-
nent role in many quantization schemes, e.g., geometric quantization.
The well-known index theorem establishes the equality of the index of
an elliptic operator on a fibre bundle to its topological index expressed in
terms of the characteristic forms of the Chern character, Todd and Euler
classes. Let us note that the classical index theorem deals with linear
elliptic operators on compact manifolds. They are Fredholm operators. In
order to generalize the index theorem to non-compact manifolds, one either
imposes conditions sufficient to force operators to be the Fredholm ones or

considers the operators which are no longer Fredholm, but their index can
be interpreted as a real number by some kind of averaging procedure [375].
(ii) Geometric quantization of a symplectic manifold (X, Cl) is af-
fected by the following ambiguity. Firstly, the equivalence classes of
admissible connections on a prequantization bundle (whose curvature
obeys the prequantization condition R = ifl) are indexed by the set
of homomorphisms of the homotopy group TT(X) of X to U(l) [257;
312]. Secondly, there are non-equivalent bundles of half-forms over X in
general and, consequently, the non-equivalent quantization bundles exist
[141]. This ambiguity leads to non-equivalent quantizations.
(iii) The cohomology analysis gives a rather complete picture of defor-
mation quantization of symplectic manifolds. Let K, be a commutative ring
and K.[[h] the ring of formal series in a real parameter h. Let us recall that,
given an associative (resp. Lie) algebra A over a commutative ring /C, its
Gerstenhaber deformation [166] is an associative (resp. Lie) /C[[/i]]-algebra
Ah such that Ah/hAh = A. The multiplication in Ah reads
oo
a*b = aob+ Y^ hr
Cr(a, b)
r=l
where o is the original associative (resp. Lie) product and Cr are 2-cochains
of the Hochschild (resp. Chevalley-Eilenberg) complex of A. The obstruc-
tion to the existence of a deformation of A lies in the third Hochschild (resp.
Chevalley-Eilenberg) cohomology group.
Let A = C°°(X) be the ring of complex smooth functions on a smooth
manifold X. One considers its associative deformations Ah where the
cochains Cr are bidifferential operators of finite order. The multidiffer-
ential cochains make up a subcomplex of the Hochschild complex of A, and
its cohomology equals the space of multi-vector fields on X [433]. If <C°° (X)
is provided with the standard Frechet topology of compact convergence for
all derivatives, one can consider its continuous deformation. The corre-
sponding subcomplex of the Hochschild complex of A is proved to have the
same cohomology as the differential one [332].
Let now X be a symplectic manifold, and let A = C°°(X) be the Poisson
algebra. Since the Poisson bracket is a bidifferential operator of order (1,1),
one has studied the similar deformations of A where the cochains Cr are
differential operators of order (1,1) with no constant term. The cohomology
of the corresponding subcomplex of the Chevalley-Eilenberg complex of A
equals the de Rham cohomology H*(X) of X [280], The equivalence classes

Introduction 9
of Poisson deformations of the Poisson bracket on a symplectic manifold X
are parameterized by i?2
(X)[[/t]]. A star-product on a Poisson manifold
is defined as an associative deformation of C°°(X) such that C (/,/') —
C (/',/) is the Poisson bracket. The existence of a star-product on an
arbitrary symplectic manifold has been proved in [125], and this is true for
any regular Poisson manifold [150; 302]. Moreover, any star-product on
a symplectic manifold is equivalent to Fedosov's one, and its equivalence
classes are parameterized by H2
(X)[[h}] [206; 340].
(iv) Let us also mention BRST cohomology, called into play in order
to describe constrained symplectic systems [156; 217; 259]. Let (Z, fl) be
a symplectic manifold endowed with a Hamiltonian action of a Lie group
G, J the corresponding momentum mapping of Z to the Lie coalgebra g*
of G, and T
V = J~1
(0) a regular constraint surface. The classical BRST
complex is defined as the bicomplex
Bn
'm
= AQ*®Ag®C°°(Z),
where the n- and m-gradings are the ghost and antighost degrees, respec-
tively. The differential
S . B*'* -> B*+h
*
is the coboundary operator of the Chevalley-Eilenberg cohomology of the
Lie algebra g of G with coefficients in the g-module g ® C°°(Z), while
d:B*'* -> B*'*'1
is the Koszul boundary operator. The algebra B is provided with the graded
Poisson bracket [,], and there exists an element 9 of B, called the BRST
charge, such that [0,0] = 0 and D = [Q,.] = 5 + d up to extra terms
of non-zero ghost number is the nilpotent classical BRST operator. The
BRST cohomology is defined as the cohomology of this classical BRST op-
erator. The BRST complex has been built for constrained Poisson systems
[245] and time-dependent Hamiltonian systems with Lagrangian constraints
[295] as an extension of the Koszul-Tate complex of constraints through in-
troduction of ghosts. Quantum BRST cohomology has been studied in the
framework of geometric [419] and deformation [49] quantization.
V.
Contemporary quantum models appeal to a number of new algebraic
structures and the associated geometric techniques.

(i) For instance, SUSY models deal with graded manifolds and differ-
ent types of supermanifolds, namely, H°°-, G°°-, GH°°-, G-supermanifolds
over (finite) Grassmann algebras, R°°- and .R-supermanifolds over Arens-
Michael algebras of Grassmann origin and the corresponding types of De-
Witt supermanifolds [21; 22; 69]. Their geometries are phrased in terms
of graded local-ringed spaces. Let us note that one usually considers su-
pervector bundles over G-supermanifolds. Firstly, the category of these
supervector bundles is equivalent to the category of locally free sheaves of
finite rank (in contrast, e.g., with Gff°°-supermanifolds). Secondly, deriva-
tions of the structure sheaf of a G-supermanifold constitute a locally free
sheaf (this is not the case, e.g., of G°°-supermanifolds). Moreover, this
sheaf is again a structure sheaf of some G-superbundle (in contrast with
graded manifolds). At the same time, most of the quantum models uses
graded manifolds. They are not supermanifolds, though there is the cor-
respondence between graded manifolds and DeWitt if00
-supermanifolds.
By virtue of the well-known Batchelor theorem, the structure ring of any
graded manifold with a body manifold Z is isomorphic to the graded ring
AE of sections of some exterior bundle AE* —> Z. In physical models,
this isomorphism holds fixed from the beginning as a rule and, in fact, by
geometry of a graded manifold is meant the geometry of the graded ring
AE- For instance, the familiar differential calculus in graded exterior forms
is the graded Chevalley-Eilenberg differential calculus over such a ring.
(ii) Non-commutative geometry is mainly developed as a generalization
of the calculus in commutative rings of smooth functions [107; 194; 267].
In a general setting, any non-commutative /C-ring A over a commutative
ring /C can be called into play. One can consider the above mentioned
Chevalley-Eilenberg differential calculus O*A over A, differential operators
and connections on A-modules (but not their jets). If the derivation K.-
module T)A is a finite projective module with respect to the center of A,
one can treat the triple (A, (3A, O*A) as a non-commutative space. For
instance, this is the case of the matrix geometry, where A is the algebra
of finite matrices, and of the quantum phase space, where A is a finite-
dimensional algebra of canonical commutation relations. Non-commutative
field theory also can be treated in this manner [133; 359], though the bracket
of space coordinates
[xfl
,x1/
} = ie^
in this theory is also restarted from Moyal's star-product xM
* xv
[99; 133].

Introduction 11
A different linear coordinate product
[x»,xl
'] = i<$'xx
comes from Connes' non-commutative geometry [195].
In Connes' non-commutative geometry, the more deep analogy to the
case of commutative smooth function rings leads to the notion of a spectral
triple (A, E, V) [107; 109]. It is given by an involutive subalgebra A C
B(E) of bounded operators on a Hilbert space E and an (unbounded) self-
adjoint operator V in E such that the resolvent (V - A)"1
, A € C E
is a compact operator and [D,A] C B(E). Furthermore, one assigns to
elements u> = aqdai • • • dak of the universal differential calculus (O*A, d)
over A the operators
7r(w) = a0[P,oi] •••[!>,ofc]
in E. This however fails to be a representation of the differential algebra
O*A because 7r(o>) = 0 does not imply ir(dw) = 0. The appropriate quotient
O*A 3 4> -> 1<P} € O*v
together with the differential d[u>] := [dw] overcomes this difficulty, though
7r([w]) is not an operator in E. Let us note that other variants of spectral
data, besides a spectral triple, are also discussed [158]. The algebra C°°{X)
and the Dirac operator P on a compact manifold X exemplifies Connes'
commutative geometry [108; 368]. Spectral triples have been studied for
non-commutative tori, the Moyal deformations of E", non-commutative
spheres 2-, 3- and 4-spheres [95; 110], and quantum Heisenberg manifolds
N.
(iii) Formalism of groupoids provides the above mentioned categorial
C*-algebraic deformation quantization of some class of Poisson manifolds
[270; 271]. A groupoid is a small category whose morphisms are invertible
[287; 367]. For instance, given an action of a group G on a set X on the
right, the product © = X x G is brought into the action groupoid where:
• a pair ((x,g), (x',g')) is composable if and only if x' — xg,
• the inversion (a;,^)"1
= (xg,g~x
),
• the partial multiplication (x,g)(xg,g') — (x,gg'),
• the range r{(x,g)) = (x, 1G),
• the domain l((x,g)) :— (xg, 1Q).
The unit space 0° = r(<8) = l(<5) of this groupoid is naturally identified
to X. Any group bundle Y —
> X (e.g., a vector bundle) is a groupoid

whose elements make up composable pairs if and only if they belong to
the same fibre, and whose unit space is the set of unit elements of fibres of
Y —> X. Let 21 —
> ©° be an Abelian group bundle over the unit space ©° of
a groupoid ©. The pair (©,21) together with a homomorphism © —
> Iso2l
is called the ©-module bundle. One can associate to any ©-module bundle
a cochain complex C*(©,21). Let 21 be a 6-module bundle in groups [/(I).
The key point is that, similarly to the case of a locally compact group [129],
one can associate a C*-algebra C*(0,u) to any locally compact groupoid
© provided with a Haar system by means of the choice of a two-cocycle
a G C2
(©,21) [367]. The algebras C*(<S,a) and C*(©,<T') are isomorphic
if a and a' are cohomology equivalent. If © is an r-discrete groupoid,
any measure A of total mass 1 on its unit space ©° induces a state of the
C*-algebraC*(©,a).
A Lie groupoid is a groupoid for which © and ©° are smooth manifolds,
the inversion and partial multiplication are smooth, while r and I are fibred
manifolds. Since a Lie groupoid admits a Haar system, one can assign to
it a C*-algebra C*(©). This assignment is functorial if certain classes of
morphisms of Lie groupoids and C*-algebras (isomorphism classes of regular
bibundles and those of Hilbert bimodules, respectively) are considered [270].
A Lie groupoid is called symplectic if it is a symplectic manifold (©, fi)
such that the multiplication relation
(x,y) -> (xy,x,y)
is a Lagrangian submanifold of the symplectic manifold
(© x ©x 6,flefi6fi)
[27]. A Poisson manifold P is called integrable if there exists a symplectic
groupoid ©(P) over P. It is unique up to an isomorphism. Integrable Pois-
son manifolds subject to a certain class morphisms (isomorphism classes
of regular dual pairs) make up a suitable category Poisson [270]. Since
the groupoid ©(P) is I- and /-simple connected, one considers the category
LG of Lie groupoids possessing this property. Any Lie groupoid yields an
associated Lie algebroid A(<S) which is the restriction to ©° of the vertical
tangent bundle of the fibration r : © -> ©° [287]. The key point is that,
similarly to the dual of a Lie algebra, the dual A*(<8) of A(<5) is a Pois-
son manifold. Then the assignment © i-> A*(&) is a functor from LG to
Poisson [271]. Let LPoisson denote its image. One can show that
T : L*(©) e-> C*(©)

Introduction 13
is a functor from the category LPoisson to the above mentioned category
of C*-algebras [271]. It is a desired functorial quantization. This functor is
equivariant under the Morita equivalence of Poisson manifolds in LG [444]
and that of C*-algebras [371]. Furthermore, the functorial quantization
A*(<&) I
—
> C*(6) is amplified into the above mentioned strict quantization
of C*(<&) by an appropriate continuous field of C*-algebras over R [27l].
Connes' tangent groupoid provides an example of such strict quantization
[269].
(iv) Hopf algebras and, in particular, quantum groups make a con-
tribution to many quantum theories [97; 249; 292; 293]. At the same
time, the development of differential calculus and differential geometry over
these algebras has met difficulties. Given a (complex or real) Hopf alge-
bra H = (H, m, A, e, 5), one introduces the first order differential calculus
(henceforth FODC) (fi1
,^) over H just as for a non-commutative alge-
bra. It is said to be left-covariant if Q1
possesses the structure of a left
iJ-comodule
A; : n1
-> H <
g
> 9}
such that
Ai{adb) = A(o)(Id ®d)A(b), a,beH,
[249]. By virtue of Woronowicz's theorem [440], left-covariant FODCs are
classified by right ideals
Tl— {x £ Kere : S(xi)dx2 = 0, x = y^x ®x2)
of H contained in the kernel of its counit e. The linear subspace
T - {X e H* : X(l) = 0, X(K) = 0}
of the dual H* is the quantum (enveloping) Lie algebra (quantum tangent
space [214]) associated to the left-covariant FODC (Ql
,d) (see [286] for
a general construction of the enveloping algebra for a non-commutative
FODC). A problem lies in the definition of vector fields as a sum a%
Ui
of invariant vector fields Ui{a) = aiXi{.a
2) [13] because they satisfy the
deformed Leibniz rule deduced from the formula
Xi(ab) = Xi(a)e(6) + $3/ij(a)x,-(&),
i
where {xi} is a basis for T and /y are complex linear forms on H. One can
model the vector fields obeying such a Leibniz rule by the so called Cartan

pairs [50]. These are elements u of the right H-du&l fi1
together with the
morphisms
u: H 3 a >->= u(da) £ H
which obey the relations
(bu)(a) — bu(a), u(ba) = u(b)a + (ub)(a).
Another problem of geometry of Hopf algebras is the notion of a quan-
tum principal bundle [75; 82; 293]. In the case of Lie groups, there are
two equivalent definitions of a smooth principal bundle, which is both
a set of trivial bundles glued together by means of transition functions
and a bundle provided with the canonical action of a structure group on
the right. In the case of quantum groups, these two notions of a princi-
pal bundle are not matched, unless the base is a smooth manifold [139;
355].
• The first definition of a quantum principal bundle repeats the classical
one and makes use of the notion of a trivial quantum bundle, a covering of
a quantum space (e.g., by a family of non-intersecting closed ideals), and its
reconstruction from local pieces [76] which however is not always possible
[81].
• The second definition of a quantum principal bundle is algebraic [74;
293]. Let H be a Hopf algebra and V a right K-comodule algebra with
respect to the coaction /3 : V —
» V <
g
> H. Let
M = {p£V : (3(p) =p®l}
be its invariant subalgebra. The triple (P,H,f3) is called a quantum prin-
cipal bundle if the map
ver : V ® V 9 (p ® q) ^ p/3{q) £ ? ® W
M M
is a linear isomorphisms. This condition, called the Hopf-Galois condition,
is a key point of this algebraic definition of a quantum principal bundle. By
some reasons, one can think of it as being a sui generis local trivialization.
(v) Finally, one of the main point of Tamarkin's proof of the for-
mality theorem in deformation quantization is that, for any algebra A
over a field of characteristic zero, its Hochschild cochain complex and
its Hochschild cohomology are algebras over the same operad [219; 411].
This observation has been the starting point of 'operad renaissance' [253;
297]. Monoidal categories provide numerous examples of algebras for

Introduction 15
operads. Furthermore, homotopy monoidal categories lead to the no-
tion of a homotopy monoidal algebra for an operad. In a general set-
ting, one considers homotopy algebras and weakened algebraic structures
where, e.g., a product operation is associative up to homotopy [276].
Their well-known examples are A^-spaces and Aoo-algebras [403]. At
the same time, the formality theorem is also applied to quantization
of several algebraic geometric structures such as algebraic varieties [255;
450].

Chapter 1
Commutative geometry
In comparison with classical mechanics and field theory phrased in terms of
smooth finite-dimensional manifolds, quantum theory speaks the algebraic
language adapted to describing systems of infinite degrees of freedom. Ge-
ometric techniques are involved in quantum theory due to the fact that the
differential calculus over an arbitrary ring can be denned. Their relation to
the familiar differential geometry of smooth manifolds is based on the fact
that any manifold can be characterized in full by a certain algebraic con-
struction and, furthermore, there is the categorial equivalence between the
vector bundles over a smooth manifold and the finite projective modules
over the ring of smooth real functions on this manifold.
1.1 Commutative algebra
In this Section, the relevant basics on modules over commutative algebras
is summarized [272; 288].
An algebra A is an additive group which is additionally provided with
distributive multiplication. All algebras throughout the book are associa-
tive, unless they are Lie algebras. A ring is a unital algebra, i.e., it contains
a unit element 1. Unless otherwise stated, we assume that 1 ^ 0 , i.e., a
ring does not reduce to the zero element. One says that A is a division
algebra if it has no a divisor of zero, i.e., ab = 0, a,b £ A, implies either
a — 0 or b = 0. Non-zero elements of a ring form a multiplicative monoid.
If this multiplicative monoid is a multiplicative group, one says that the
ring has a multiplicative inverse. A ring A has a multiplicative inverse if
and only if it is a division algebra. A field is a commutative ring whose
non-zero elements make up a multiplicative group.
A subset I of an algebra A is called a left (resp. right) ideal if it is a
17

subgroup of the additive group A and ab e l (resp. 6a € 1) for all a E A,
b e l If J is both a left and right ideal, it is called a two-sided ideal. An
ideal is a subalgebra, but a proper ideal (i.e., 1 ^ A) of a ring is not a
subring because it does not contain a unit element.
Let A be a commutative ring. Of course, its ideals are two-sided. Its
proper ideal is said to be maximal if it does not belong to another proper
ideal. A commutative ring A is called local if it has a unique maximal
ideal. This ideal consists of all non-invertible elements,of A. A proper two-
sided ideal I of a commutative ring is called prime if db £ 1 implies either
a £ J or b £ 1. Any maximal two-sided ideal is prime. Given a two-sided
ideal 1 c A, the additive factor group A/1 is an algebra, called the factor
algebra. If A is a ring, then A/1 is so. If J is a prime ideal, the factor ring
A/1 has no divisor of zero, and it is a field if J is a maximal ideal.
Remark 1.1.1. We will refer to the following particular construction in
the sequel. Let K be a commutative ring and S its multiplicative subset
which, by definition, is a monoid with respect to multiplication in K. Let
us say that two pairs (a,s) and (a',s'), a,a' £ /C, s,s' £ S, are equivalent
if there exists an element s" £ S such that
s"{s'a - so,') = 0.
We abbreviate with a/s the equivalence classes of (a, s). The set S~1
IC of
these equivalence classes is a ring with respect to the operations
s/a + s'/a' := (s'a + sa')/(ss'),
(a/s) • (a'/s') := (aa')/(ss').
There is a homomorphism
$s : K 3^ a/1 £ S^IC (1.1.1)
such that any element of $s(S) is invertible in S^1
^. If a ring K has no
divisor of zero and S does not contain a zero element, then $s (1.1.1) is
a monomorphism. In particular, if 5 is the set of non-zero elements of K-,
the ring S~1
fC is a field, called the field of quotients of the fraction field of
/C. If K. is field, its fraction field coincides with K. •
Given an algebra A, an additive group P is said to be a left (resp. right)
A-module if it is provided with distributive multiplication A x P —> P by
elements of A such that (ab)p = a(bp) (resp. (ab)p = b(ap)) for all a, b € A
and p £ P. If A is a ring, one additionally assumes that lp = p = pi for

Chapter 1 Commutative Geometry 19
all p € P. Left and right module structures are usually written by means
of left and right multiplications (a, p) H-» ap and (a, p) >
—
> pa, respectively.
If P is both a left module over an algebra A and a right module over
an algebra A', it is called an (A — .4')-bimodule (an .4-bimodule if A —
.4'). If A is a commutative algebra, an (.4 — .4)-bimodule P is said to be
commutative if ap = pa for all a € A and p £ P. Any left or right module
over a commutative algebra A can be brought into a commutative bimodule.
Therefore, unless otherwise stated, any module over a commutative algebra
A is called an .4-module (see Section 8.1).
A module over a field is called a vector space. If an algebra A is a
module over a ring K., it is said to be a IC-algebra. Any algebra can be seen
as a Z-algebra.
Remark 1.1.2. Any AC-algebra A can be extended to a unital algebra A
by the adjunction of the identity 1 to A. The algebra A, called the unital
extension of A, is defined as the direct sum of ^-modules K © A provided
with the multiplication
(Ai,ai)(A2,a2) = (AiA2,Aia2 + A2ai+aia2), Ai,A2£/C, a i , a 2 e A
Elements of A can be written as (A, a) = Al + a, A € /C, a G A.
Let us note that, if A is a unital algebra, the identity 1^ in A fails to
be that in A. In this case, the algebra A is isomorphic to the product of A
and the algebra K,(l — 1A)- D
In this Chapter (except Sections 1.5C), all associative algebras are as-
sumed to be commutative, unless they are graded.
The following are standard constructions of new modules from old ones.
• The direct sum Pi © P2 of ,4-modules Pi and P2 is the additive group
Pi x P2 provided with the .4-module structure
a(Pi,P2) = (api,ap2), Pi,2 G Pli2, a & A.
Let {Pi}ie/ be a set of modules. Their direct sum ©P* consists of elements
(..., pi,...) of the Cartesian product n Pi s u c n
that pi ^ 0 at most for a
finite number of indices i € I.
• The tensor product P ® Q of ^-modules P and Q is an additive group
which is generated by elements p® q, p € P, q 6 Q, obeying the relations
{p + p') ® q = P ® q + p' ® q, p ® (q + q') = p®q+p®q'',
pa (8) q = p <
g
> aq, p e P, q € Q, a € A,

(see Remark 10.4.1), and it is provided with the .4-module structure
a(p ® q) = (ap) ®q = p® (qa) = (p <
g
> q)a.
If the ring A is treated as an .4-module, the tensor product A ®^ Q is
canonically isomorphic to Q via the assignment
A ®A QBa®q<->aq£Q.
• Given a submodule Q of an .4-module P, the quotient P/Q of the
additive group P with respect to its subgroup Q is also provided with an
«4-module structure. It is called a factor module.
• The set Horn ,4(P, Q) of .4-linear morphisms of an .4-module P to an
.4-module Q is naturally an .4-module. The .4-module P* = Horn ^(P, A)
is called the dual of an .4-module P. There is a natural monomorphism
P-» P**.
An .4-module P is called free if it has a basis, i.e., a linearly indepen-
dent subset I C P spanning P such that each element of P has a unique
expression as a linear combination of elements of / with a finite number
of non-zero coefficients from an algebra A. Any vector space is free. Any
module is isomorphic to a quotient of a free module. A module is said to
be finitely generated (or of finite rank) if it is a quotient of a free module
with a finite basis.
One says that a module P is protective if it is a direct summand of a
free module, i.e., there exists a module Q such that P®Q is a free module.
A module P is projective if and only if P = pS where 5 is a free module
and p is a projector of S, i.e., p2
= p. If P is a projective module of finite
rank over a ring, then its dual P* is so, and P** is isomorphic to P.
THEOREM 1.1.1. Any projective module over a local ring is free. •
Now we focus on exact sequences, direct and inverse limits of modules
[288; 303].
A composition of module morphisms
P - U Q -^->T
is said to be exact at Q if Ker j = Im i. A composition of module morphisms
O^P -UQ -Ur^o (1.1.2)

is called a short exact sequence if it is exact at all the terms P, Q, and T.
This condition implies that: (i) i is a monomorphism, (ii) Ker j = Imi, and
(iii) j is an epimorphism onto the quotient T — Q/P.
THEOREM 1.1.2. Given an exact sequence of modules (1.1.2) and another
,4-module R, the sequence of modules
0->EomA{T,R) i^RomA{Q,R) ^Eom{P,R) (1.1.3)
is exact at the first and second terms, i.e., j * is a monomorphism, but i*
need not be an epimorphism. HI
One says that the exact sequence (1.1.2) is split if there exists a
monomorphism s :T —> Q such that j o s = IdT or, equivalently,
Q = i{P) ® s{T) ^P®T.
The exact sequence (1.1.2) is always split if T is a projective module.
A directed set 7 is a set with an order relation < which satisfies the
following three conditions:
(i) i < i, for all i € I;
(ii) if i < j and j < k, then i < k;
(iii) for any i,j € /, there exists k £ I such that i < k and j < k.
It may happen that i ^ j , but i < j and j < i simultaneously.
A family of modules {P{i^i (over the same algebra), indexed by a
directed set /, is called a direct system if, for any pair i < j , there exists a
morphism r* : Pi —
> Pj such that
r = I d Pi, r) or{ = ri, i<j< k.
A direct system of modules admits a direct limit. This is a module P^
together with morphisms r ^ : Pi —
> P^ such that r ^ = r£, o rj for all
i < j . The module P^ consists of elements of the direct sum ©Pj modulo
the identification of elements of Pi with their images in Pj for all i < j . An
example of a direct system is a direct sequence
Po —»Pi ^ • • • P / M . . . , J = N. (1.1.4)
It should be noted that direct limits also exist in the categories of commuta-
tive algebras and rings, but not in categories whose objects are non-Abelian
groups.

THEOREM 1.1.3. Direct limits commute with direct sums and tensor prod-
ucts of modules. Namely, let {Pi} and {Qi} be two direct systems of
modules over the same algebra which are indexed by the same directed set
/, and let P^ and Q^ be their direct limits. Then the direct limits of
the direct systems {Pi © Qi} and {Pt <
g
> Qi} are P^ © Qoo and Poo ® Qoo,
respectively. •
A morphism of a direct system {Pi,rl
j}i to a direct system {Qi>,pl
j,}i>
consists of an order preserving map / : / — » / ' and morphisms Fj : Pj —
>
Qf(i) which obey the compatibility conditions
0S
^)oFi = Fior).
If PQQ and Qoo are limits of these direct systems, there exists a unique
morphism F^ : P ^ —
> Qoo such that
p£>oFi=F00ori0.
Moreover, direct limits preserve monomorphisms and epimorphisms. To be
precise, if all Ft : Pi —
> Q/(») axe monomorphisms or epimorphisms, so is
$00 : Poo —
* Qoo- As a consequence, the following holds.
THEOREM 1.1.4. Let short exact sequences
0-^Pi ^Qi ^Ti^O (1.1.5)
for all i £ I define a short exact sequence of direct systems of modules
{P,}/, {Qi}i, and {Tj}/ which are indexed by the same directed set /.
Then there exists a short exact sequence of their direct limits
O^Poo ^ Q o o ^ T o o ^ O . (1.1.6)
•
In particular, the direct limit of factor modules Qi/Pi is the factor
module Qoo/Poo- By virtue of Theorem 1.1.3, if all the exact sequences
(1.1.5) are split, the exact sequence (1.1.6) is well.
Example 1.1.3. Let P be an ^-module. We denote P®k
=®P. Let us
consider the direct system of ^-modules with respect to monomorphisms
A -^(A®P) — > - - - ( ^ © P © - - - © P 0 f e
) —>••• •

Its direct limit
®P = A® P <$>••• ®P®k
®--- (1.1.7)
is an N-graded ,4-algebra with respect to the tensor product <
g
>
. It is called
the tensor algebra of a module P. Its quotient with respect to the ideal
generated by elements p<g>p'+p'' ®p, p,p' e P, is an N-graded commutative
algebra, called the exterior algebra of a module P. •
We restrict our consideration of inverse systems of modules to inverse
sequences
P° «— P1
< p* *£1... . (1.1.8)
Its inductive limit (the inverse limit) is a module P°° together with mor-
phisms 7if° : P°° -> Pi
such that 7if° = irj o itf for all i < j . It consists
of elements (... ,p*,...), pl
€ Pl
, of the Cartesian product f] Pl
such that
p1
= K{ (p3
) for all i < j .
THEOREM 1.1.5. Inductive limits preserve monomorphisms, but not epi-
morphisms. If a sequence
Q-^Pi
^Q{
-51*T*, i e N ,
of inverse systems of modules {P1
}, {Q1
} and {T1
} is exact, so is the
sequence of the inductive limits
poo /f»oo
0_>p°° *-^>Q°° ?—>T°°.
n
In contrast with direct limits, the inductive ones exist in the category
of groups which are not necessarily commutative.
Example 1.1.4. Let {Pi} be a direct sequence of modules. Given another
module Q, the modules Hom(Pj,<3) make up an inverse system such that
its inductive limit is isomorphic to Horn (Poo, Q)- •
1.2 Differential operators on modules and rings
This Section addresses the notion of a (linear) differential operator on a
module over a commutative ring [202; 233; 26l].

Let K be a commutative ring and .A a commutative /C-ring. Let P and Q
be Amodules. The /C-module Homjc(P, Q) oi /C-module homomorphisms
$ : P —
> Q can be endowed with the two different Amodule structures
(afc)(p) := a$(p), ($ • o)(p) := $(ap), o £ i , p £ ? . (1.2.1)
For the sake of convenience, we will refer to the second one as the .A*-module
structure. Let us put
<Sa$ := a$ - $ • a, a £ A. (1-2.2)
DEFINITION 1.2.1. An element A <
£ Hom;c(-P, Q) is called a Q-valued
differential operator of order s on P if
$ao°---°5
aA = °
for any tuple of s + 1 elements ao,..., as of A •
The set Diff S(P, Q) of these operators inherits the A- and .A*-module
structures (1.2.1). Of course, an s-order differential operator is also of
(s + l)-order.
In particular, zero order differential operators obey the condition
<5aA(p) = oA(p) - A(ap) = 0, a € A, p € P,
and, consequently, they coincide with ,4-module morphisms P —> Q. A first
order differential operator A satisfies the condition
5boSaA(p) = baA(p)-bA(ap)-aA(bp) + A{abp)=0, a,b G A (1.2.3)
The following fact reduces the study of Q-valued differential operators
on an Amodule P to that of Q-valued differential operators on the ring A
PROPOSITION 1.2.2. Let us consider the Amodule morphism
/is:Diff,(A<3)->Q, MA) = A(1). (1-2-4)
Any Q-valued s-order differential operator A G Diff s (P, Q) on P uniquely
factorizes
A : P -^Diff S(AQ) - ^ Q (1-2-5)
through the morphism hs (1.2.4) and some homomorphism
fA :P-Diffs (AQ), (Up)(a) = A(op), a e A (1-2-6)

of the Amodule P to the .A*-module Diff s(.4,<3) [261]. The assignment
A H-> fA defines the isomorphism
Diffs(P,Q) = EomA_A.(P,Ditt8{A,Q)). (1.2.7)
n
Let P = A. Any zero order Q-valued differential operator A on A is
defined by its value A(l). Then there is an isomorphism Diff o(-4, Q) = Q
via the association
Q9QH4 A,eDiffo(AQ),
where A, is given by the equality A,(l) = q. A first order Q-valued
differential operator A on A fulfils the condition
A(ab) = bA(a) + aA(b) - 6aA(l), a, b e A.
It is called a Q-valued derivation of A if A(l) = 0, i.e., the Leibniz rule
A(ab) = A(a)b + aA(b), a,b € A, (1.2.8)
holds. One obtains at once that any first order differential operator on A
falls into the sum
A(a) = aA(l) + [A(a) - aA(l)]
of the zero order differential operator aA(l) and the derivation A(a) —
aA(l). If d is a derivation of A, then ad is well for any a € A- Hence,
derivations of A constitute an .A-module V(A, Q), called the derivation mod-
ule. There is the ,4-module decomposition
Difi1(A,Q) = Q®V(A,Q). (1.2.9)
Remark 1.2.1. Let us recall that, given a (non-commutative) /C-algebra
A and an .4-bimodule Q, by a Q-valued derivation of A is meant a K-
module morphism u : A —
> Q which obeys the Leibniz rule
u(ab) =u(a)b + au(b), a,b G A. (1.2.10)
It should be emphasized that this derivation rule differs from that (6.2.3)
of graded derivations. A Q-valued derivation u of A is called inner if there
exists an element q £ Q such that u(a) = qa — aq. O

If Q = A, the derivation module $A of A is also a Lie algebra over the
ring K. with respect to the Lie bracket
[ti,!i'] = iioti'-u'oti, u, u'£ A. (1.2.11)
Accordingly, the decomposition (1.2.9) takes the form
DiSi(A) = A®dA. (1.2.12)
An s-order differential operator on a module P is represented by a zero
order differential operator on the module of s-order jets of P as follows.
Given an .A-module P, let us consider the tensor product A ®JC P of
^-modules A and P. We put
5b
(a®p) := {ba)®p-a®(bp), p € P, a,b £ A. (1.2.13)
Let us denote by /xfc+1
the submodule of A®K. P generated by elements of
the type
S*0
O-.-OSbk
(a®p).
The k-order jet module Jk
{P) of a module P is defined as the quotient of
the ^-module A <8>JC P by /ife+1
. We denote its elements c ®k p.
In particular, the first order jet module JX
{P) consists of elements c®p
modulo the relations
Sa
o 5b
(l ®ip) = ab®xp-b <
g
>
i (op) - a ®i (bp) + 1 ®i (abp) = 0. (1.2.14)
The /C-module Jk
(P) is endowed with the A- and .A*-module structures
b(a ®k p) := ba ®k p, b» (a®kP) •= a®k(bp). (1.2.15)
There exists the module morphism
Jk
:P3p^l®kPe Jk
(P) (1.2.16)
of the .A-module P to the ,4*-module Jk
(P) such that Jk
{P), seen as an
,4-module, is generated by elements Jk
p, p 6 P.
Due to the natural monomorphisms /j,r
—
> fis
for all r > s, there are
.4-inodule epimorphisms of jet modules
7rj+1
: Ji+1
(P) -• J*(P). (1-2.17)
In particular,
Trl:J1
(P)Ba®ip^ap£P. (1.2.18)

The above mentioned relation between differential operators on modules
and jets of modules is stated by the following theorem [261].
THEOREM 1.2.3. Any Q-valued differential operator A of order k on an
,4-module P factorizes uniquely
A : P A j f e
( P ) —+Q
through the morphism Jk
(1.2.16) and some ,4-module homomorphism fA
:
Jk
(P) -> Q. •
The proof is based on the fact that the morphism Jk
(1.2.16) is a fc-order
i7fc
(P)-valued differential operator on P. Let us denote
J: P3pi->l®p£A®P.
Then, for any f e Horn .4(.4 ® P,Q), we obtain
6btfoJ)(p) = f(6b
(l®p)).
The correspondence A >
—
> fA
defines an .A-module isomorphism
DiBa(P,Q)= HornA(J'(P),Q). (1.2.19)
1.3 Connections on modules and rings
We employ the jets of modules in previous Section in order to introduce
connections on modules and commutative rings [296].
Let us consider the jet modules Js
= JS
{A) of the ring A itself. In
particular, the first order jet module J1
consists of the elements a <S> b,
a,b £ A, subject to the relations
ab <
g
>
! 1 - b <
g
>
i a - a <S>i b + 1 ®i (ab) = 0. (1.3.1)
The A- and ^.'-module structures (1.2.15) on J1
read
c(a <g>! b) := (ca) <g>i 6, c • (a <
E
>
i b) := a ®i (cb) = (a ®i b)c.
Besides the monomorphism
J1
: A^a^->l®la£Jl
(.2.1(^), there is the .4-module monomorphism
ii : A 3 a h^> a ®! 1 £ Jx
.

With these monomorphisms, we have the canonical A-module splitting
J1=ii(A)®O1, (1.3.2)
oJx(6) = a <
8
>
i b = ab ®x 1 + a(l ®i b - b <
g
>
i 1),
where the Amodule O1 is generated by the elements 1 ®i & — &(g>i 1 for all
6 € A Let us consider the corresponding Amodule epimorphism
/ i 1 : J 1 3 l ® i 6 i - + l i g i 1 & - f c ® 1 l G C 1 (1.3.3)
and the composition
d1 = h1oJ1:Aab^l®lb-b®1l€O1, (1.3.4)
which is a /C-module morphism. This is a C^-valued derivation of the Jt-ring
A which obeys the Leibniz rule
d1 (ab) = 1 ®i ab - ab ®i 1 + a ®i b - a <
8
>
! b = ad^ + ( d 1 ^ .
It follows from the relation (1.3.1) that adx6 = (d1b)a for all a, 6 € A Thus,
seen as an ,4-module, O1 is generated by the elements dla for all o e A
Let O1* = Horn .4 ( 0 .4) be the dual of the A-module O1. In view of
the splittings (1.2.12) and (1.3.2), the isomorphism (1.2.19) reduces to the
duality relation
DA=OU, (1.3.5)
QA 9 u <
-
> 4>u e O1*, (f>u(d1a) := u(a), a € A (1.3.6)
In a more direct way (see Proposition 8.2.1 below), the isomorphism (1.3.5)
is derived from the facts that C1 is generated by elements dla, a E A, and
that <f>(dla) is a derivation of A for any <f> £ O1*. However, the morphism
oi _
> o i . * = 0^»
need not be an isomorphism.
Let us define the modules Ok, k = 2,..., as the exterior products of the
Amodule O1. There are the higher degree generalizations
hk :ji(Ok-1)^Ok,
dk = hk o J1 : Ok-1 -+ Ok (1.3.7)
of the morphisms (1.3.3) and (1.3.4). The operators (1.3.7) are nilpotent,
i.e., dk o dk~l = 0. They form the cochain complex
0-»/C -^A ^ O 1 ^...Qkd<^... . (1.3.8)

Let us return to the first order jet module J1
(P) of an .A-module P. It
is isomorphic to the tensor product
J1
(P)=J1
®P, (a®1bp)*->{a®1b)®p. (1.3.9)
Then the isomorphism (1.3.2) leads to the splitting
JP) = {A®OX
)®P = {A®P)® {O1
® P), (1.3.10)
a (g>i bp <-» (ab + ad}{b)) ®p.
Applying the epimorphism ?r^ (1.2.18) to this splitting, one obtains the
short exact sequence of A- and A' -modules
0 —>O1
®P-» JP) ^ P —>0, (1.3.11)
(a ®! b — ab ®i 1) ® p —> (c (2)! 1 + a <
g
>
i b — ab ®i 1) <
g
> p —
» cp.
This exact sequence is canonically split by the ^.'-module morphism
P3 ap^ ®ap = a®p + dl
(a)®pe Jl
{P)-
However, it need not be split by an ,4-module morphism, unless P is a
projective .4-module.
DEFINITION 1.3.1. A connection on an ^4-module P is defined as an A-
module morphism
T:P^JP), T(ap) = aT(p), (1.3.12)
which splits the exact sequence (1.3.11) or, equivalently, the exact sequence
0-*O1
<8>P-+(A®O1
)®P-*P-^0. (1.3.13)
•
If a splitting T (1.3.12) exists, it reads
J1
p = T(p) + V(p), (1.3.14)
where V is the complementary morphism
V : P -> O1
® P, V(p) = l ® i p - r ( p ) . (1.3.15)
Though this complementary morphism in fact is a covariant differential
on the module P, it is traditionally called a connection on a module. It
satisfies the Leibniz rule
V(ap) =d1
a<8>p + aV(p), (1.3.16)

i.e., V is an (O1
® P)-valued first order differential operator on P. Thus,
we come to the equivalent definition of a connection [260].
DEFINITION 1.3.2. A connection on an ^.-module P is a /C-module mor-
phism V (1.3.15) which obeys the Leibniz rule (1.3.16). •
The morphism V (1.3.15) can be extended naturally to the morphism
V : O1
® P -> O2
<
8
> P.
Then we have the morphism
iJ = V2
:P^O2
®P, (1.3.17)
called the curvature of the connection V on a module P.
In view of the isomorphism (1.3.5), any connection in Definition 1.3.2
determines a connection in the following sense.
DEFINITION 1.3.3. A connection on an A-module P is an ^-module mor-
phism
ViA3ut-^VueDifii(P,P) (1.3.18)
such that the first order differential operators Vu obey the Leibniz rule
Vu(ap) = u(a)p + aVu(p), a € A, p £ P. (1.3.19)
•
Definitions 1.3.2 and 1.3.3 are equivalent if O1
= DA*.
The curvature of the connection (1.3.18) is defined as a zero order dif-
ferential operator
R{u, u') = [V«, Vu.) - V[UiU/] (1.3.20)
on the module P for all u, u' € DA.
Let P be a commutative .A-ring and DP the derivation module of P as
a AT-ring. Definition 1.3.3 is modified as follows.
DEFINITION 1.3.4. A connection on an ,4-ring P is an .4-module morphism
5 i 9 u H V u £ DP, (1.3.21)
which is a connection on P as an ^-module, i.e., obeys the Leinbniz rule
(1.3.19). •

Two such connections Vu and V^ differ from each other in a derivation
of the .A-ring P, i.e., which vanishes on A C P. The curvature of the
connection (1.3.21) is given by the formula (1.3.20).
1.4 Homology and cohomology of complexes
This Section summarizes the relevant basics on homology and cohomology
of complexes of modules over a commutative ring [288; 303].
Let K. be a commutative ring. A sequence
0^B0 £-2?! <^...Bp
a
£±I... (1.4.1)
of /C-modules Bp and homomorphisms dp is said to be a chain complex if
dp o dp+i = 0, p G N,
i.e., Im9p+i C Ker9p. The homomorphisms dp are called boundary opera-
tors. Elements of the module Bp are said to be p-chains, while elements of
its submodules Kerdp C Bp and imdp+i c Ker9p are called p-cycles and
p-boundaries, respectively. The p-th homology group of the chain complex
B* (1.4.1) is defined as the factor module
Hp(B,) = Keidp/lmdlH.i.
It is a /C-module. In particular, we have
HQ{B.) = BQ/Imdi.
A chain complex (1.4.1) is exact at a term Bp if HP(B*) = 0. It is an
exact sequence if all homology groups are trivial. A chain complex (1.4.1)
is called acyclic if its homology groups Hp>o are trivial. A chain complex
5» is acyclic if there exists a homotopy operator h. This is defined as a set
of module morphisms
hp:Bp^Bp+1, peN,
such that
hp_! odp + dp+l o hp = IdBp, p&N+.
It follows that, if dpbp = 0, then bp = dp+i(hpbp), and Hp>0(Bt,) = 0.

A chain complex (1.4.1) is said to be a chain resolution of a module B
if it is acyclic and H0(B*) = B. This complex defines the exact sequence
0 <—B <— Bo &-Bx &-...Bp
9
^---. (1.4.2)
Any module B admits a chain resolution. Indeed, B is a quotient QO/BQ of
some free module Qo, where BQ is also a quotient Q/B of a free module
Qi, and so on.
The following are the standard constructions of new chain complexes
from old ones.
• Given chain complexes (£», d*) and (B^d*), their direct sum B1t®B'il
is a chain complex of modules
(B.®B'Jp = Bp®B'p
with respect to the boundary operators
df(bp + b'p):=dpbp + dpb'p.
• Given a subcomplex (C*,d*) of a chain complex (B*,d*), the factor
complex B+/C* is defined as the chain complex of factor modules Bp/Cp
provided with the boundary operators
dp[bp] := [dpbp],
where bp] € Bp/Cp denotes the coset of an element bp.
• Given chain complexes (B*,9») and (B»,9«), their tensor product
-B* <
8
> B!, is the chain complex of modules
( B , ® B't)p = © Bk <
g
> B'r
k+r=p
with respect to the boundary operators
9® (Bfc ® B'r) := (dkbk) ® b'r + (-l)k
bk ® (d'rbr)-
A chain morphism of chain complexes
7 : £* -> SI (1.4.3)
is denned as a family of degree-preserving /C-module homomorphisms
7 p : i?p -> s ; , p e N ,
which commute with the boundary operators, i.e.,
9
P + I ° 7 P + I =lp°dp+i.

It follows that if bp £ Bp is a cycle or a boundary, then 7P(6P) G B'p is
well. Therefore, the chain morphism of complexes (1.4.3) yields the induced
homomorphism of their homology groups
[7], : H.(B.) -» H.(B't), [7]([6]) := [7(6)], (1-4.4)
where [b] denotes the homology class of 6 € B».
Let 7,7' : £» —
> B't be two different chain morphisms of the same
chain complexes. By a chain homotopy h is meant a family of /C-module
homomorphisms
hp:Bp^ B'p+1, peN,
of degree +1 such that
d'p+i °hP + hp-i o dp = 7P - 7p.
If a chain homotopy exists, the chain morphisms 7 and 7' are called ho-
motopic. The difference 7 — 7' of homotopic chain morphisms sends cycles
onto boundaries, i.e., these morphisms induce the same homomorphisms
[7], and [7']* (1.4.4) of homology groups. In particular, a chain morphism
7 (1.4.3) is said to be a homotopy equivalence if there exists a chain mor-
phism £ : B'% —
» B* such that the compositions £°7 and 70£ are homotopic
to the identity morphisms of the chain complexes B* and B'^, respectively.
Chain complexes connected by a homotopy equivalence are called homo-
topic. Their homology groups are isomorphic.
Let us consider a short sequence of chain complexes
0 -» C, - ^ B. -$-> F. -> 0, (1.4.5)

represented by the commutative diagram
0 0
I I
dp
• • • <— C p - i < — Cp <
7P -1 I 7 P I
dB
• • • < -Dp-1 < Op < • • •
CP-I I CP I
••• < i'p-l * £p *
I I
0 0
It is called an exact sequence if all columns of this diagram are exact se-
quences of modules, i.e., 7 is a chain monomorphism and £ is a chain
epimorphism onto the factor complex F* = 5*/C*. One says that the
exact sequence (1.4.5) is split if there exists a set s* of degree-preserving
monomorphisms sp : Fp —> Bp such that
CpOSp = ldFp, d£ = Cp-i o dp3
o sp, peN+.
Then
Bp^Cp® Fp
for all p € N. A splitting s* is called a chain splitting if it is a chain
morphism, i.e.,
sp_! o d% = d^ o sp
or, equivalently, if 5 , is isomorphic to the direct sum of chain complexes
C* © Ft. Then we have
H,(B.) = H.(C.)®H,(F,).
THEOREM 1.4.1. The short exact sequence of chain complexes (1.4.5) yields
the long exact sequence of their homology groups
0 <—H0(F*) ^H0(B*) t^ffo (C.) ^^(F*) &-••• (1.4.6)
«—ffp(F.) &^HP{B.) [
^HP(C*)T
^HP+1(F*) < ,

where [7]* and [£]„ axe the induced homomorphisms (1.4.4), and r» is called
the joint homomorphism. O
Let us consider a direct sequence of chain complexes
Bl —> B — • • • B* % Bi+1
—»•••. (1.4.7)
Theorem 1.1.4 leads to the following important result [303].
THEOREM 1.4.2. The direct sequence (1.4.7) admits a direct limit 5J°
which is a chain complex, whose homology groups H*(B^) are the direct
limit of the direct system of homology groups
H,(B°) —>ff.(Bj) —»• • • H*(B*)h
^>]
H.(B!t+1
) — . . . .
D
This statement is also true for a direct system of chain complexes in-
dexed by an arbitrary directed set. However, the similar assertion for an
inverse system of chain complexes fails because the inductive limit of epi-
morphisms need not be an epimorphism.
Let us turn to cochain complexes. A sequence
0->B° -^B1
^•••Bp
-£*••• (1.4.8)
of modules Bp
and their homomorphisms 5P
is said to be a cochain complex
(henceforth, simply, a complex) if
p+i o <
J
P = 0, p £ N,
i.e., Im<P C Ker<5p+1
. The homomorphisms 6P
are called coboundary oper-
ators. For the sake of convenience, let us denote B~r
= 0 and 5~l
: 0 -+ B°.
Elements of the module Bp
are said to be p-cochains, while elements of its
submodules Ker<5p
C Bp
and Im6p
~1
C Ker<5p
are called p-cocycles and
p-coboundaries, respectively. The p-th cohomology group of the complex B*
(1.4.8) is the factor module
Hp
(B*)=KeiSp
/lm5p
-1
.
It is a ^-module. In particular, H°(B*) = Ker5°.
A complex (1.4.8) is exact at a term Bp
if HP
(B*) = 0. It is an exact
sequence if all cohomology groups are trivial.

Example 1.4.1. Given a chain complex JB» (1.4.1), let Cp
= B* be the K.-
duals of Bp. Let us define the /C-module homomorphisms Sp
: Cp
—
> Cp + 1
as
5p
cp
:= (f o 9p + 1 : Bp+1 -*K, tf> e Cp
. (1.4.9)
It is readily observed that Sp+1
o8p
= 0. Then {Cp
, 5P
} is the dual complex
of the chain complex £?». Let us note that, if the chain complex 5» is exact,
the dual complex need not be so (see Theorem 1.1.2). •
A complex (B*, 6*) is called acyclic if its cohomology groups HP>O
(B*)
are trivial. It is acyclic if there exists a homotopy operator h, defined as a
set of module morphisms
hP+i . BP+I ^ B P t p e N (
such that
hp+1
o6p
+ Sp
-1
ohp
= IdBp
, peN+.
Indeed, if 5p
bp
= 0, then bp
= 5p
-l
{ip
bp
), and HP>O
(B*) = 0.
A complex (B*, 6*) is said to be a resolution of a module B if it is acyclic
and H°{B*) = B.
The following are the standard constructions of new complexes from old
ones.
• Given complexes (B^, 5*) and (B^S^), their direct sum B± © J3| is a
complex of modules
(B*x ® B*)p
= Bp
® BP
with respect to the coboundary operators
6l(bp
+ bp
2):=5p
bp
+ 5P
bP
.
• Given a subcomplex (C*, 5*) of a complex (B*,S*), the factor complex
B*/C* is defined as a complex of factor modules Bp
/Cp
provided with the
coboundary operators
5P[bP] : = [<W],
where [bP] G Bp
/Cp
denotes the coset of the element bp
.
• Given complexes (5J, <5J) and (B^S^), their tensor product B <
g
> B%
is a complex of modules
(Bl ® B^)p
= © £f ® 5£
fc+r=p

with respect to the coboundary operators
a|(fl? ® Br
2) := (tf bf) ® &£ + (-l)fc
&? ® (*$&$).
A cochain morphism of complexes
7 : 5j* -> B2 (1.4.10)
is defined as a family of degree-preserving homomorphisms
Y : B -> Bp
, p e N ,
which commute with the coboundary operators, i.e.,
^O7" = y + 1
o ^ , peN.
It follows that if V e £?p
is a cocycle or a coboundary, then 7p
(bp
) £ Bp
is so. Therefore, the cochain morphism of complexes (1.4.10) yields an
induced homomorphism of their cohomology groups
[7]* : E*{B) - H*{B*2). (1.4.11)
Let 7,7' : B —> B2 be two different cochain morphisms. By their
cochain homotopy h is meant a family of homomorphisms
hP:Bl^Bl~ peN+,
such that
$P-I ohP + hp+1
6p
= 7P
- ip
.
If a cochain homotopy exists, the cochain morphisms 7 and 7' are called
homotopic. Homotopic cochain morphisms 7 and 7' induce the same ho-
momorphisms [7]* and [7']* (1.4.11) of cohomology groups. One says that
the cochain morphism 7 (1.4.10) is a homotopy equivalence if there exists
a cochain morphism £ : B2 —* B such that the compositions C ° 7 and
7 o £ axe homotopic to the identity morphisms of the complexes B* and B2,
respectively. Complexes connected by a homotopy equivalence are called
homotopic, and their cohomology groups are isomorphic.
Let us consider a short sequence of complexes
0 -^ C* ^UB* -^-> F* -> 0, (1.4.12)

represented by the commutative diagram
0 0
..._ U J._...
Tp I 7p+l I
. . . y BP
- ^ BP+1
> • • •
CP I Cp+i I
... » pp ifU pp+i > . . .
I I
0 0
It is said to be exact if all columns of this diagram are exact, i.e., 7 is a
cochain monomorphism and £ is a cochain epimorphism onto the quotient
F* = B*/C*.
The following assertions are similar to Theorems 1.4.1 and 1.4.2.
THEOREM 1.4.3. The short exact sequence of complexes (1.4.12) yields the
long exact sequence of their cohomology groups
0->H°(C*) l
^H°(B*) [
^H°(F*) ^HC*) — > • • • (1.4.13)
—>HP
(C*) ^ > F P
( B * ) 1
^>HP
(F*) ^HV+1
{C*) —••••.
•
THEOREM 1.4.4. A direct sequence of complexes
B* _»BJ —»• • • J3J7
^B*k+1 -*..• (1.4.14)
admits a direct limit B^ which is a complex whose cohomology ,ff*(.B£o)
is a direct limit of the direct sequence of cohomology groups
H*(B*0) -*H*{Bl) -^...H*(B*k)b
^>]
H*(B*k+1) — . . . .
This statement is also true for a direct system of complexes indexed by an
arbitrary directed set. •

1.5 Homology and cohomology of groups and algebras
Subsections: A. Homology and cohomology of groups, 39; B. The Koszul
complex, 44; C. Hochschild cohomology, 49; D. Chevalley-Eilenberg coho-
mology, 53.
We briefly sketch homology and cohomology of some algebraic systems
needed in the sequel. These are homology and cohomology of groups,
homology of the Koszul complex, Hochschild cohomology, Chevalley-
Eilenberg cohomology.
A. Homology and cohomology of groups
Homology and cohomology of groups demonstrate the standard tech-
niques of constructing homology and cohomology of algebraic systems [288].
Given a set Z, one can introduce a chain complex as follows. Let Zk be a
A;+l
free Z-module whose basis is the Cartesian product x Z. In particular, Zo
is a free Z-module whose basis is Z. Let us define Z-linear homomorphisms
d0 : Zo 3mi(zl
0) i-> Ylmi eZ
' m
» £ z
. (1.5.1)
i
3k+1 : Zk+1 -+ Z*, ke N,
fe+1
dk+i(z0, • • •, z/M-i) = ^2(~'i-y(zo,---,Zj,...,zk+i), (1.5.2)
j=o
where the caret "denotes omission. It is readily observed that dkodk+ = 0
for all k € N. Thus, we obtain the chain complex
0 <— L <— Zo <— Z <— • • • Zjt < • , (1.5.3)
called the standard chain complex of a set Z. The chain complex (1.5.3)
admits the homotopy operator
h z ( l ) = (z), h z ( z 0 , . . . , z k ) = ( z , z 0 , . . . , z k ) , (1.5.4)
where z is some fixed element of Z. Consequently, the chain complex (1.5.3)
is exact at all the terms Zk, k E N. Moreover, this complex is also exact
at the term Z since the boundary operator do (1.5.1) is an epimorphism.
Hence, it provides a resolution of a ring Z by free Z-modules.
Let Z = G be a group and G* the chain complex
O^z<^-Go ^ - d < Gp$±i--- (1.5.5)

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Montreville's family had introduced her; and she now seldom
deigned to name an untitled acquaintance—while Crofts hung on her
long narratives with affected admiration; and the two elder of her
three daughters, who were all in training to be beauties, aped their
mother in vanity and impertinence.
The eldest Miss Ashwood, now about fourteen, was an insupportable
torment to Emmeline, as she had taken it into her head to form,
with her, a sentimental friendship. She had learned all the cant of
sentiment from novels; and her mama's lovers had extremely edified
her in teaching her to express it. She talked perpetually of delicate
embarrassments and exquisite sensibilities, and had probably a lover,
as she extremely wanted a confidant; a post which Emmeline with
some difficulty declined.—Of 'the sweet novels' she had read, she
just understood as much as made her long to become the heroine of
such an history herself, and she wanted somebody to listen to her
hopes of being so. But Emmeline shrunk from her advances, and
repaid her fondness with general and cool civility; tho' Mrs.
Ashwood, who loved rather to listen to Crofts than to attend to her
daughters, continually promoted the intimacy, in hopes that she
would take them off her own hands, and allow them to be the
companions of her walks.
This, Emmeline was obliged studiously to evade, as such
companions would entirely have prevented her seeing Lady Adelina;
and by repeated excuses she not only irritated the curiosity of Mrs.
Ashwood and Miss Galton, but gave the former an additional cause
of dislike to that which she had already conceived; inasmuch as she
was younger, handsomer, and more admired than herself.
Emmeline received frequent letters from Delamere, as warm and
passionate as his personal professions. He told her, that as his
mother's health was greatly amended, he intended soon to visit
those parts of France with which he was yet unacquainted; and
should pass some time in the Northern Provinces, from whence he
entreated her to allow him to come only for a few days to England to

see her—an indulgence which he said would enable him to bear with
more tranquillity the remaining months of his exile.
Tho' now accustomed to consider him as her husband, Emmeline
resolutely refused to consent to this breach of his engagement to his
father. She had lately seen in her friends, Mrs. Stafford and Lady
Adelina, two melancholy instances of the frequent unhappiness of
very early marriages; and she had no inclination to hazard her own
happiness in hopes of proving an exception. She wished, therefore,
rather to delay her union with Delamere two or three years; but to
him she never dared hint at such a delay. A clandestine interview it
was, however, in her power to decline; and she answered his
request by entreating him not to think of such a journey; and
represented to him that he could not expect Lord Montreville would
finally adhere to his promises, if he himself was careless of fulfilling
the conditions on which his Lordship had insisted. Having thus, as
she supposed, prevented Delamere from offending his father, and
without any immediate uneasiness on her own account, she gave up
her mind to the solicitude she could not help feeling for Lady
Adelina. This occupied almost all her time when she was alone; and
gave her, when in company, an air of absence and reserve.
Tho' Mrs. Ashwood so much encouraged the attention of James
Crofts, she had not forgotten Fitz-Edward, whom she had vainly
sought at Lady Montreville's, in hopes of renewing an acquaintance
which had in it's commencement offered her so much satisfaction.
Fitz-Edward had been amused with her absurdity at the moment,
but had never thought of her afterwards; nor would he then have
bestowed so much time on a woman to him entirely indifferent, had
not he been thrown in her way by his desire to befriend Delamere
with Emmeline, on one of those days when Lady Adelina insisted on
his leaving her, to avoid the appearance of his passing with her all
his time. Happy in successful love, his gaiety then knew no bounds;
and his agreeable flattery, his lively conversation, his fashionable
manners, and his handsome person, had not since been absent from
the memory of Mrs. Ashwood. His being sometimes at the house he
had borrowed of Delamere, near Woodfield, was one of the principal

inducements to her to go thither. She indulged sanguine hopes of
securing such a conquest; and evaded giving to Crofts a positive
answer, till she had made another essay on the heart of the Colonel.
He came, however, so seldom to Woodfield, that Mrs. Stafford had
seen him there only once since her meeting Lady Adelina; and then
he appeared to be under encreased dejection, for which she knew
now, how to account.
Emmeline had given Mrs. Stafford so indifferent an account of Lady
Adelina one evening, that she determined the next morning to see
her. She therefore went immediately after breakfast, on pretence of
visiting a poor family who had applied to her for assistance; when as
Mrs. Ashwood, Miss Galton and Emmeline, were sitting together,
Colonel Fitz-Edward was announced.
He came down to Tylehurst only the evening before; and not
knowing there was company at Woodfield, rode over to pass an hour
with the two friends, to whom he had frequently been tempted to
communicate the source of his melancholy.
Whether it was owing to the consciousness of Lady Adelina's
mournful story that arose in the mind of Emmeline, or whether
seeing Fitz-Edward again in company with Mrs. Ashwood renewed
the memory of what had befallen her when they last met, she
blushed deeply the moment she beheld him, and arose from her
chair in confusion; then sat down and took out her work, which she
had hastily put up; and trying to recover herself, grew still more
confused, and trembled and blushed again.
Mrs. Ashwood was in the mean time overwhelming Fitz-Edward with
compliments and kind looks, which he answered with the distant
civility of a slight acquaintance; and taking a chair close to
Emmeline, enquired if she was not well?
She answered that she was perfectly well; and attempted to
introduce general conversation. But Fitz-Edward was attentive only
to her; and Mrs. Ashwood, extremely piqued at his distant manner,

meditated an excuse to get Emmeline out of the room, in hopes of
obtaining more notice.
Fitz-Edward, however, having talked apart with Miss Mowbray a
short time, arose and took leave, having by his manner convinced
Mrs. Ashwood of what she reluctantly believed, that some later
attachment had obliterated the impression she had made at their
first interview.
'I never saw such a figure in my life,' cried she, 'as Mr. Fitz-Edward.
Mercy on me!—he is grown so thin, and so sallow!'
'And so stupid,'interrupted Miss Galton. 'He is in love I fancy.'
Emmeline blushed again; and Mrs. Ashwood casting a malicious look
at her, said—'Oh! yes—he doubtless is in love. To men of his gay
turn you know it makes no difference, whether a person be actually
married or engaged.'
Emmeline, uncertain of the meaning of this sarcasm, and unwilling
to be provoked to make a tart reply, which she felt herself ready to
do, put up her work and left the room.
While she went in search of Mrs. Stafford, to enquire after Lady
Adelina, and to relate the conversation that had passed between her
and Fitz-Edward, Mrs. Ashwood and Miss Galton were indulging their
natural malignity. Tho' well apprized of Emmeline's engagement to
Delamere, yet they hesitated not to impute her confusion, and Fitz-
Edward's behaviour, to a passion between them. They believed, that
while her elopement with Delamere had beyond retreat entangled
her with him, and while his fortune and future title tempted her to
marry him, her heart was in possession of Fitz-Edward; and that
Delamere was the dupe of his mistress and his friend.
This idea, which could not have occurred to a woman who was not
herself capable of all the perfidy it implied, grew immediately
familiar with the imagination of Mrs. Ashwood, and embittered the
sense of her own disappointment.

Miss Galton, who hated Emmeline more if possible than Mrs.
Ashwood, irritated her suspicions by remarks of her own. She
observed 'that it was very extraordinary Miss Mowbray should walk
out so early in a morning, and so studiously avoid taking any body
with her—and that unless she had appointments to which she
desired no witness, it was very singular she should chuse to ramble
about by herself.'
From these observations, and her evident confusion on seeing him,
they concluded that she had daily assignations with Fitz-Edward.
They agreed, that it would be no more than common justice to
inform Mr. Delamere of their discovery; and this they determined to
do as soon as they had certain proofs to produce, with which they
concluded a very little trouble and attention would furnish them.
James Crofts, whose success was now indisputable, since of the
handsome Colonel there were no hopes, was let into the secret of
their suspicions; and readily undertook to assist in detecting the
intrigue, for which he assured them he had particular talents. While,
therefore, Mrs. Ashwood, Miss Galton, and James Crofts, were
preparing to undermine the peace and character of the innocent,
ingenuous Emmeline, she and Mrs. Stafford were meditating how to
be useful to the unhappy Lady Adelina. They became every day
more interested and more apprehensive for the fate of that devoted
young woman, whose health seemed to be such as made it very
improbable she should survive the birth of her child. Her spirits, too,
were so depressed, that they could not prevail on her to think of her
own safety, or to allow them to make any overtures to her family;
but, in calm and hopeless languor, she seemed resigned to the
horrors of her destiny, and determined to die unlamented and
unknown.
Her elder brother, Lord Westhaven, had returned from abroad almost
immediately after her concealment. His enquiries on his first arrival
in England had only informed him of the embarrassment of
Trelawny's affairs, and the inconvenience to which his sister had
consequently been exposed; and that after staying some time in

England, to settle things as well as she could, she had disappeared,
and every body believed was gone to her husband. His Lordship's
acquaintance and marriage with Augusta Delamere, almost
immediately succeeded; but while it was depending, he was
astonished to hear from Lord and Lady Clancarryl that Lady Adelina
had never written to them before her departure. He went in search
of Fitz-Edward; but could never meet him at home or obtain from his
servants any direction where to find him. Fitz-Edward, indeed,
purposely avoided him, and had left no address at his lodgings in
town, or at Tylehurst.
Lord Westhaven then wrote to Trelawny, but obtained no answer;
and growing daily more alarmed at the uncertainty he was in about
Lady Adelina, he determined to go, as soon as he was married, to
Switzerland; being persuaded that tho' some accident had prevented
his receiving her letters, she had found an asylum there, amongst
his mother's relations.
Fitz-Edward, with anxiety even more poignant, had sought her with
as little success. After the morning when she discharged her
lodgings, and left them in an hackney coach with her maid, he could
never, with all his unwearied researches, discover any traces of her.
He knew she was not gone to Trelawny; and dreading every thing
from her determined sorrow, he passed his whole time between
painful and fruitless conjectures, and the tormenting apprehension
of hearing of some fatal event. Incessantly reproaching himself for
being the betrayer of his trust, and the ruin of a lovely and amiable
woman, he gave himself up to regret and despondence. The gay
Fitz-Edward, so lately the envy and admiration of the fashionable
world, was lost to society, his friends, and himself.
He passed much of his time at Tylehurst; because he could there
indulge, without interruption, his melancholy reflections, and only
saw Mrs. Stafford and Emmeline, in whose soft and sensible
conversation he found a transient alleviation of his sorrow—sorrow
which now grew too severe to be longer concealed, and which he
resolved to take the earliest opportunity of acknowledging, in hopes

of engaging the pity of his fair friends—perhaps their assistance in
discovering the unhappy fugitive who caused it.
From Lady Adelina, they had most carefully concealed, that his
residence was so near the obscure abode she had chosen. Fatal as
he had been to her peace, and conscientiously as she had abstained
from naming him after their first conversation, they knew that she
still fondly loved him, and that her fears for his safety had assisted
her sense of rectitude when she determined to tear herself from
him. But were she again to meet him, they feared she would either
relapse into her former fatal affection, or conquer it by an effort,
which in her precarious state of health might prove immediately
fatal.
The request which Fitz-Edward had made to Emmeline, that he
might be allowed to see her and Mrs. Stafford together, without any
other person being present, they both wished to evade; dreading
least they should by their countenances betray the knowledge they
had of his unhappy story, and the interest they took in it's
catastrophe.
They hoped, therefore, to escape hearing his confession till Lady
Adelina should be removed—and to remove her became
indispensibly necessary, as Emmeline was convinced she was
watched in her visits to the cottage.
Twice she had met James Crofts within half a quarter of a mile of
the cottage; and at another time discovered, just as she was about
to enter it, that the Miss Ashwoods had followed her almost to the
door; which she therefore forbore to enter. These circumstances
made both her and Mrs. Stafford solicitous to have Lady Adelina
placed in greater security; and, added to Emmeline's uneasiness for
her, was the unpleasant situation in which she found herself.
Observed with malicious vigilance by Mrs. Ashwood, James Crofts,
Miss Galton, and the two Misses, she felt as awkward as if she really
had some secret of her own to hide; and with all the purity and even
heroism of virtue, learned the uneasy sensation which ever attends

mystery and concealment. The hours which used to pass tranquilly
and rationally with Mrs. Stafford, were now dedicated to people
whose conversation made her no amends; and if she retired to her
own room, it failed not to excite sneers and suspicions. She saw Mrs.
Stafford struggling with dejection which she had no power to
dissipate or relieve, and obliged to enter into frequent parties of
what is called pleasure, tho' to her it gave only fatigue and disgust,
to gratify Mrs. Ashwood, who hated all society but a crowd. James
Crofts, indeed, helped to keep her in good humour by his excessive
adulation; and chiefly by assuring her, that by any man of the least
taste, the baby face of Emmeline could be considered only as a foil
to her more mature charms, and that her fine dark eyes eclipsed all
the eyes in the world. He protested too against Emmeline for
affecting knowledge—'It is,' said he, 'a maxim of my father's—and
my father is no bad judge—that for a woman to affect literature is
the most horrid of all absurdities; and for a woman to know any
thing of business, is detestable!'
Mrs. Ashwood laid by her dictionary, determined for the future to
spell her own way without it.
Besides the powerful intervention of flattery, James Crofts had
another not less successful method of winning the lady's favour. He
told her that his brother, who had long cherished a passion in which
he was at length likely to be disappointed, was in that case
determined never to marry; that he was in an ill state of health; and
if he died without posterity, the estate and title of his father would
descend to himself.
The elder Crofts, very desirous of seeing a brother established who
might otherwise be burthensome or inconvenient to him, suggested
this finesse; and secured it's belief by writing frequent and
melancholy accounts of his own ill health—an artifice by which he
promoted at once his brother's views and his own. He affected the
valetudinarian so happily, and complained so much of the ill effect
that constant application to business had on his constitution, that
nobody doubted of the reality of his sickness. He took care that Miss

Delamere should receive an account of it, which he knew she would
consider as the consequence of his despairing love; and when he
had interested her vanity and of course her compassion, he
contrived to obtain leave of absence for three months from the
duties of his office, in order to go abroad for the recovery of his
health. He hastened to Barege; and soon found means to re-
establish himself in the favour of Miss Delamere; from which,
absence, and large draughts of flattery dispensed with French
adroitness, had a little displaced him. This stratagem put his brother
James on so fair a footing with the widow, that he thought her
fortune would be secured before she could discover it to be only a
stratagem, and that her lover was still likely to continue a younger
brother.
James Crofts seeing the necessity of dispatch, became so
importunate, that Mrs. Ashwood, despairing of Fitz-Edward, and
believing she might not again meet with a man so near a title, for
which she had a violent inclination, was prevailed on to promise she
would make him happy as soon as she returned to her own house.
It was now the end of June; and Lady Adelina, whose situation grew
very critical, had at length yielded to the entreaties of her two
friends, and agreed to go wherever they thought she could obtain
assistance and concealment in the approaching hour.
Mrs. Stafford and Emmeline, after long and frequent reflections and
consultations on the subject, concluded that no situation would be
so proper as Bath. In a place resorted to by all sorts of people, less
enquiry is excited than in a provincial town, where strangers are
objects of curiosity to it's idle inhabitants. To Bath, therefore, it was
determined Lady Adelina should go. But when the time of her
journey, and her arrangements there, came to be discussed, she
expressed so much terror least she should be known, so much
anguish at leaving those to whose tender pity she was so greatly
indebted, and such melancholy conviction that she should not
survive, that the sensible heart of Emmeline could not behold
without sharing her agonies; nor was Mrs. Stafford less affected.

When they returned home after this interview, Emmeline was
pursued by the image of the poor unhappy Adelina. But to give, to
the wretched, only barren sympathy, was not in her nature, where
more effectual relief was in her power. She thought, that if by her
presence she could alleviate the anguish, and soothe the sorrows of
the fair mourner, perhaps save her character and her life, and be the
means of restoring her to her family, she should perform an action
gratifying to her own heart, and acceptable to heaven. The more she
reflected on it, the more anxious she became to execute it—and she
at length named it to Mrs. Stafford.
Mrs. Stafford, tho' aware of the numberless objections which might
have been made to such a plan, could not resolve strenuously to
oppose it. She felt infinite compassion for Lady Adelina; but could
herself do little to assist her, as her time was not her own and her
absence must have been accounted for: but Emmeline was liable to
no restraint; and would not only be meritoriously employed in
befriending the unhappy, but would escape from the society at
Woodfield, which became every day more disagreeable to her. These
considerations, particularly the benevolent one of saving an unhappy
young woman, over-balanced, in the mind of Mrs. Stafford, the
objection that might be made to her accompanying a person under
the unfortunate and discreditable circumstances of Lady Adelina;
and her heart, too expansive to be closed by the cold hand of
prudery against the sighs of weakness or misfortune, assured her
that she was right. She knew that Emmeline was of a character to
pity, but not to imitate, the erroneous conduct of her friend; and she
believed that the reputation of Lady Adelina Trelawny might be
rescued from reproach, without communicating any part of it's
blemish to the spotless purity of Emmeline Mowbray.
CHAPTER II

As soon as Emmeline had persuaded herself of the propriety of this
plan and obtained Mrs. Stafford's concurrence, she hinted her
intentions to Lady Adelina; who received the intimation with such
transports of gratitude and delight, that Emmeline, confirmed in her
resolution, no longer suffered a doubt of it's propriety to arise; and,
with the participation of Mrs. Stafford only, prepared for her journey,
which was to take place in ten days.
Mrs. Stafford also employed a person on whom she could rely, to
receive the money due to Lady Adelina from her husband's estate.
But of this her Ladyship demanded only half, leaving the rest for
Trelawny. The attorney in whose hands Trelawny's affairs were
placed by Lord Westhaven, was extremely anxious to discover, from
the person employed by Mrs. Stafford, from whence he obtained the
order signed by Lady Adelina; and obliged him to attend several
days before he would pay it, in hopes, by persuasions or artful
questions, to draw the secret from him. He met, at the attorney's
chambers, an officer who had made of him the same enquiry, and
had followed him home, and since frequently importuned him—
intelligence, which convinced Mrs. Stafford that Lady Adelina must
soon be discovered, (as they concluded the officer was Fitz-Edward,)
and made both her and Emmeline hasten the day of her departure.
About a quarter of a mile from Woodfield, and at the extremity of
the lawn which surrounded it, was a copse in which the accumulated
waters of a trout stream formed a beautiful tho' not extensive piece
of water, shaded on every side by a natural wood. Mrs. Stafford, who
had particular pleasure in the place, had planted flowering shrubs
and caused walks to be cut through it; and on the edge of the water
built a seat of reeds and thatch, which was furnished with a table
and a few garden chairs. Thither Emmeline repaired whenever she
could disengage herself from company. Solitude was to her always a
luxury; and particularly desirable now, when her anxiety for Lady
Adelina, and preparations for their approaching departure, made her
wish to avoid the malicious observations of Mrs. Ashwood, the
forward intrusion of her daughters, and the inquisitive civilities of
James Crofts. She had now only one day to remain at Woodfield,

before that fixed for their setting out; and being altogether unwilling
to encounter the fatigue of such an engagement so immediately
previous to her journey, she declined being of the party to dine at
the house of a neighbouring gentleman; who, on the occasion of his
son's coming of age, was to give a ball and fête champêtre to a very
large company.
Mrs. Ashwood, seeing Emmeline averse, took it into her head to
press her extremely to go with them; and finding she still refused,
said—'it was monstrous rude, and that she was sure no young
person would decline partaking such an entertainment if she had not
some very particular reason.'
Emmeline, teized and provoked out of her usual calmness, answered
—'That whatever might be her reasons, she was fortunately
accountable to nobody for them.'
Mrs. Ashwood, provoked in her turn, made some very rude replies,
which Emmeline, not to irritate her farther, left the room without
answering; and as soon as the carriages drove from the door, she
dined alone, and then desiring one of the servants to carry her harp
into the summer-house in the copse, she walked thither with her
music books, and soon lost the little chagrin which Mrs. Ashwood's
ill-breeding had given her.
Fitz-Edward, who arrived in the country the preceding evening, after
another fruitless search for Lady Adelina, walked over to Woodfield,
in hopes, as it was early in the afternoon, that he might obtain, in
the course of it, some conversation with Mrs. Stafford and
Emmeline. On arriving, he met the servant who had attended
Emmeline to the copse, and was by him directed thither. As he
approached the seat, he heard her singing a plaintive air, which
seemed in unison with his heart. She started at the sight of him—
Mrs. Ashwood's suspicions immediately occurred to her, and at the
same moment the real motive which had made him seek this
interview. She blushed, and looked uneasy; but the innocence and
integrity of her heart presently restored her composure, and when

Fitz-Edward asked if she would allow him half an hour of her time,
she answered—'certainly.'
He sat down by her, dejectedly and in silence. She was about to put
aside her harp, but he desired her to repeat the air she was singing.
'It is sweetly soothing,' said he, 'and reminds me of happier days
when I first heard it; while you sing it, I may perhaps acquire
resolution to tell you what may oblige you to discard me from your
acquaintance. It does indeed require resolution to hazard such a
misfortune.'
Emmeline, not knowing how to answer, immediately began the air.
The thoughts which agitated her bosom while she sung, made her
voice yet more tender and pathetic. She saw the eyes of Fitz-Edward
fill with tears; and as soon as she ceased he said—
'Tell me, Miss Mowbray—what does the man deserve, who being
entrusted with the confidence of a young and beautiful woman—
beautiful, even as Emmeline herself, and as highly accomplished—
has betrayed the sacred trust; and has been the occasion—oh God!
—of what misery may I not have been the occasion!
'Pardon me,' continued he—'I am afraid my despair frightens you—I
will endeavour to command myself.'
Emmeline found she could not escape hearing the story, and
endeavoured not to betray by her countenance that she already
knew it.
Fitz-Edward went on—
'When first I knew you, I was a decided libertine. Yourself and Mrs.
Stafford, lovely as I thought you both, would have been equally the
object of my designs, if Delamere's passion for you, and the
reserved conduct of Mrs. Stafford, had not made me doubt
succeeding with either. But for your charming friend my heart long
retained it's partiality; nor would it ever have felt for her that pure
and disinterested friendship which is now in regard to her it's only

sentiment, had not the object of my present regret and anguish
been thrown in my way.
'To you, Miss Mowbray, I scruple not to speak of this beloved and
lamented woman; tho' her name is sacred with me, and has never
yet been mentioned united with dishonour.
'The connection between our families first introduced me to her
acquaintance. In her person she was exquisitely lovely, and her
manners were as enchanting as her form. The sprightly gaiety of
unsuspecting inexperience, was, I thought, sometimes checked by
an involuntary sentiment of regret at the sacrifice she had made, by
marrying a man every way unworthy of her; except by that fortune
to which she was indifferent, and of which he was hastening to
divest himself.
'I had never seen Mr. Trelawny; and knew him for some time only
from report. But when he came to Lough Carryl, my pity for her,
encreased in proportion to the envy and indignation with which I
beheld the insensible and intemperate husband—incapable of feeling
for her, any other sentiment, than what she might equally have
inspired in the lowest of mankind.
'Her unaffected simplicity; her gentle confidence in my protection
during a voyage in which her ill-assorted mate left her entirely to my
care; made me rather consider her as my sister than as an object of
seduction. I resolved to be the guardian rather than the betrayer of
her honour—and I long kept my resolution.'
Fitz-Edward then proceeded to relate the circumstances that
attended the ruin of Trelawny's fortune; and that Lady Adelina was
left to struggle with innumerable difficulties, unassisted but by
himself, to whom Lord Clancarryl had delegated the task of treating
with Trelawny's sister and creditors.
'Her gratitude,' continued he, 'for the little assistance I was able to
give her, was boundless; and as pity had already taught me to love
her with more ardour than her beauty only, captivating as it is,
would have inspired; gratitude led her too easily into tender

sentiments for me. I am not a presuming coxcomb; but she was
infinitely too artless to conceal her partiality; and neither her
misfortunes, or her being the sister of my friend Godolphin,
protected her against the libertinism of my principles.'
He went on to relate the deep melancholy that seized Lady Adelina;
and his own terror and remorse when he found her one morning
gone from her lodgings, where she had left no direction; and from
her proceeding it was evident she designed to conceal herself from
his enquiries.
'God knows,' pursued he, 'what is now become of her!—perhaps,
when most in need of tenderness and attention, she is thrown
destitute and friendless among strangers, and will perish in
indigence and obscurity. Unused to encounter the slightest hardship,
her delicate frame, and still more sensible mind, will sink under
those to which her situation will expose her—perhaps I shall be
doubly a murderer!'
He stopped, from inability to proceed—Emmeline, in tears, continued
silent.
Struggling to conquer his emotion and recover his voice, Fitz-Edward
at length continued—
'While I was suffering all the misery which my apprehension for her
fate inflicted, her younger brother, William Godolphin, returned from
the West Indies, where he has been three years stationed. I was the
first person he visited in town; but I was not at my lodgings there.
Before I returned from Tylehurst, he had informed himself of all the
circumstances of Trelawny's embarrassments, and his sister's
absence. He found letters from Lord Westhaven, and from my
brother, Lord Clancarryl; who knowing he would about that time
return to England, conjured him to assist in the attempt of
discovering Lady Adelina; of whose motives for concealing herself
from her family they were entirely ignorant, while it filled them with
uneasiness and astonishment. As soon as I went back to London,
Godolphin, of whose arrival I was ignorant, came to me. He

embraced me, and thanked me for my friendship and attention to his
unfortunate Adelina—I think if he had held his sword to my heart it
would have hurt me less!
'He implored me to help his search after his lost sister, and again
said how greatly he was obliged to me—while I, conscious how little
I deserved his gratitude, felt like a coward and an assassin, and
shrunk from the manly confidence of my friend.
'Since our first meeting, I have seen him several times, and ever
with new anguish. I have loved Godolphin from my earliest
remembrance; and have known him from a boy to have the best
heart and the noblest spirit under heaven. Equally incapable of
deserving or bearing dishonour, Godolphin will behold me with
contempt; which tho' I deserve, I cannot endure. He must call me to
an account; and the hope of perishing by his hand is the only one I
now cherish. Yet unable to shock him by divulging the fatal secret, I
have hitherto concealed it, and my concealment he must impute to
motives base, infamous, and pusillanimous. I can bear such
reflections no longer—I will go to town to-morrow, explain his
sister's situation to him, and let him take the only reparation I can
now make him.'
Emmeline, shuddering at this resolution, could not conceal how
greatly it affected her.
'Generous and lovely Miss Mowbray! pardon me for having thus
moved your gentle nature; and allow me, since I see you pity me, to
request of you and Mrs. Stafford a favour which will probably be the
last trouble the unhappy Fitz-Edward will give you.
'It may happen that Lady Adelina may hereafter be discovered—tho'
I know not how to hope it. But if your generous pity should interest
you in the fate of that unhappy, forlorn young woman, your's and
Mrs. Stafford's protection might yet perhaps save her; and such
interposition would be worthy of hearts like yours. As the event of a
meeting between me and Godolphin is uncertain, shall I entreat you,
my lovely friend, to take charge of this paper. It contains a will, by

which the child of Lady Adelina will be entitled to all I die possessed
of. It is enough, if the unfortunate infant survives, to place it above
indigence. Lord Clancarryl will not dispute the disposition of my
fortune; and to your care, and that of Mrs. Stafford, I have left it in
trust, and I have entreated you to befriend the poor little one, who
will probably be an orphan—but desolate and abandoned it will not
be, if it's innocence and unhappiness interest you to grant my
request. Delamere will not object to your goodness being so
exerted; and you will not teach it, generous, gentle as you are! to
hold in abhorrence the memory of it's father. This is all I can now do.
Farewell! dearest Miss Mowbray!—Heaven give you happiness, ma
douce amie! Farewell!'
These last words, in which Fitz-Edward repeated the name by which
he was accustomed to address Emmeline, quite overcame her. He
was hastening away, while, hardly able to speak, she yet made an
effort to stop him. The interview he was about to seek was what
Lady Adelina so greatly dreaded. Yet Emmeline dared not urge to
him how fatal it would be to her; she knew not what to say, least he
should discover the secret with which she was entrusted; but in
breathless agitation caught his hand as he turned to leave her,
crying—
'Hear me, Fitz-Edward! One moment hear me! Do not go to meet
Captain Godolphin. I conjure, I implore you do not!'
She found it impossible to proceed. Her eyes were still eagerly fixed
on his face; she still held his hand; while he, supposing her extreme
emotion arose from the compassionate tenderness of her nature,
found the steadiness of his despair softened by the soothing voice of
pity, and throwing himself on his knees, he laid his head on one of
the chairs, and wept like a woman.
Emmeline, who now hoped to persuade him not to execute the
resolution he had formed, said—'I will take the paper you have given
me, Fitz-Edward, and will most religiously fulfil all your request in it
to the utmost extent of my power. But in return for my giving you
this promise, I must insist'——

At this moment James Crofts stood before them.
Emmeline, shocked and amazed at his appearance, roused Fitz-
Edward by a sudden exclamation.
He started up, and said fiercely to Crofts—'Well, Sir!—have you any
commands here?'
'Commands, Sir,' answered Crofts, somewhat alarmed by the tone in
which this question was put—'I have no commands to be sure Sir—
but, but, I came Sir, just to enquire after Miss Mowbray. I did not
mean to intrude.'
'Then, Sir,' returned the Colonel, 'I beg you will leave us.'
'Oh! certainly, Sir,' cried Crofts, trying to regain his courage and
assume an air of raillery—'certainly—I would not for the world
interrupt you. My business indeed is not at all material—only a
compliment to Miss Mowbray—your's,' added he sneeringly, 'is, I see,
of more consequence.'
'Look ye, Mr. Crofts,' sharply answered Fitz-Edward—'You are to
make no impertinent comments. Miss Mowbray is mistress of her
actions. She is in my particular protection on behalf of my friend
Delamere, and I shall consider the slightest failure of respect to her
as an insult to me. Sir, if you have nothing more to say you will be so
good as to leave us.'
There was something so hostile in the manner in which Fitz-Edward
delivered this speech, that James Crofts, more at home in the
cabinet than the field, thought he might as well avoid another
injunction to depart; and quietly submit to the present, rather than
provoke farther resentment from the formidable soldier. He
therefore, looking most cadaverously, made one of his jerking bows,
and said, with something he intended for a smile—
'Well, well, good folks, I'll leave you to your tête a tête, and hasten
back to my engagement. Every body regrets Miss Mowbray's
absence from the ball; and the partner that was provided for her is
ready to hang himself.'

An impatient look, darted from Fitz-Edward, stopped farther effusion
of impertinence, and he only added—'Servant! servant!' and walked
away.
Fitz-Edward, then turning towards Emmeline, saw her pale and faint.
'Why, my dear Miss Mowbray, do you suffer this man's folly to affect
you? Your looks really terrify me!'
'Oh! he was sent on purpose,' cried Emmeline.—'Mrs. Ashwood has
lately often hinted to me, that whatever are my engagements to
Delamere I was much more partial to you. She has watched me for
some time; and now, on my refusing to accompany them to the ball,
concluded I had an appointment, and sent Crofts back to see.'
'If I thought so,' sternly answered Fitz-Edward, 'I would instantly
overtake him, and I believe I could oblige him to secresy.'
'No, for heaven's sake don't!' said Emmeline—'for heaven's sake do
not think of it! I care not what they conjecture—leave them to their
malice—Crofts is not worth your anger. But Fitz-Edward, let us return
to what we were talking of. Will you promise me to delay going to
London—to delay seeing Mr. Godolphin until—in short, will you give
me your honour to remain at Tylehurst a week, without taking any
measures to inform Godolphin of what you have told me. I will, at
the end of that time, either release you from your promise, or give
you unanswerable reasons why you should relinquish the design of
meeting him at all.'
Fitz-Edward, however amazed at the earnestness she expressed to
obtain this promise, gave it. He had no suspicion of Emmeline's
having any knowledge of Lady Adelina; and accounted for the deep
interest she seemed to take in preventing an interview, by
recollecting the universal tenderness and humanity of her character.
He assured her he would not leave Tylehurst 'till the expiration of the
time she had named. He conjured her not to suffer any impertinence
from Crofts on the subject of their being seen together, but to awe
him into silence by resentment. Emmeline now desired him to leave
her. But she still seemed under such an hurry of spirits, that he

insisted on being allowed to attend her to the door of the house,
where, renewing his thanks for the compassionate attention she had
afforded him, and entreating her to compose herself, he left her.
Emmeline intending to go to her own room, went first into the
drawing room to deposit her music book. She had hardly done so,
when she heard a man's step, and turning, beheld Crofts open the
door, which he immediately shut after him.
'I thought, Sir,' said Emmeline, 'you had been gone back to your
company.'
'No, not yet, my fair Emmeline. I wanted first to beg your pardon for
having disturbed so snug a party. Ah! sly little prude—who would
think that you, who always seem so cold and so cruel, made an
excuse only to stay at home to meet Fitz-Edward? But it is not fair,
little dear, that all your kindness should be for him, while you will
scarce give any other body a civil look. Now I have met with you I
swear I'll have a kiss too.'
Emmeline, terrified to death at his approaching her with this speech,
flew to the bell, which she rang with so much violence that the rope
broke from the crank.
'Now,' cried Crofts, 'if nobody hears, you are more than ever in my
power.'
'Heaven forbid!' shrieked Emmeline, in an agony of fear. 'Let me go,
Mr. Crofts, this moment.'
She would have rushed towards the door but he stood with his arms
extended before it.
'You did not run thus—you did not scream thus, when Fitz-Edward,
the fortunate Fitz-Edward, was on his knees before you. Then, you
could weep and sigh too, and look so sweetly on him. But come—
you see I know so much that it will be your interest, little dear, to
make me your friend.'
'Rather let me apply to fiends and furies for friendship! hateful,
detestable wretch! by what right do you insult and detain me?'

'Oh! these theatricals are really very sublime!' cried he, seizing both
her hands, which he violently grasped.
She shrieked aloud, and fruitlessly struggled to break from him,
when the footsteps of somebody near the door obliged him to let
her go. She darted instantly away, and in the hall met one of the
maids.
'Lord, Miss,' cried the servant, 'did you ring? I've been all over the
house to see what bell it was.'
Emmeline, without answering, flew to her own room. The maid
followed her: but desirous of being left alone, she assured the girl
that nothing was the matter; that she was merely tired by a long
walk; and desiring a glass of water, tried to compose and recollect
herself; while Crofts unobserved returned to the house where the
fête was given time enough to dress and dance with Mrs. Ashwood.
It was at her desire, that immediately after dinner Crofts had left the
company under pretence of executing a commission with which she
easily furnished him; but his real orders were to discover the motives
of Emmeline's refusal to be of the party. This he executed beyond
his expectation. It was no longer to be doubted that very good
intelligence subsisted between Emmeline and Fitz-Edward, since he
had been found on his knees before her; while she, earnestly yet
kindly speaking, hung over him with tears in her eyes. Knowing that
Emmeline was absolutely engaged to Delamere, he was persuaded
that Fitz-Edward was master of her heart; and that the tears and
emotion to which he had been witness, were occasioned by the
impossibility of her giving him her hand. He knew Fitz-Edward's
character too well to suppose he could be insensible of the lady's
kindness; and possessing himself a mind gross and depraved, he did
not hesitate to believe all the ill his own base and illiberal spirit
suggested.
Tho', interested hypocrite as he was, he made every other passion
subservient to the gratification of his avarice, Crofts had not coldly
beheld the youth and beauty of Emmeline; he had, however,

carefully forborne to shew that he admired her, and would probably
never have betrayed what must ruin him for ever with Mrs.
Ashwood, had not the conviction of her partiality to Fitz-Edward
inspired him with the infamous hope of frightening her into some
kindness for himself, by threatening to betray her stolen interview
with her supposed lover.
The scorn and horror with which Emmeline repulsed him served only
to mortify his self love, and provoke his hatred towards her and the
man whom he believed she favoured; and with the inveterate and
cowardly malignity of which his heart was particularly susceptible, he
determined to do all in his power to ruin them both.
CHAPTER III
Such was the horror and detestation which Emmeline felt for Crofts,
that she could not bear the thoughts of seeing him again. But as she
feared Mrs. Stafford might resent his behaviour, and by that means
embroil herself with the vain and insolent Mrs. Ashwood, with whom
she knew Stafford was obliged to keep on a fair footing, she
determined to say as little as she could of his impertinence to Mrs.
Stafford, but to withdraw from the house without again exposing
herself to meet him. As soon as she saw her the next morning, she
related all that had passed between Fitz-Edward and herself; and
after a long consultation they agreed that to prevent his seeing
Godolphin was absolutely necessary; and that no other means of
doing so offered, but Mrs. Stafford's relating to him the real
circumstances and situation of Lady Adelina, as soon as she could be
removed from her present abode and precautions taken to prevent
his discovering her. This, Mrs. Stafford undertook to do immediately
after their departure. It was to take place on the next day; and
Emmeline, with the concurrence of her friend, determined that she
would take no leave of the party at Woodfield: for tho' the

appearance of mystery was extremely disagreeable and distressing
to Emmeline, she knew that notice of her intentions would excite
enquiries and awaken curiosity very difficult to satisfy; and that it
was extremely probable James Crofts might be employed to watch
her, and by that means render abortive all her endeavours to
preserve the unhappy Lady Adelina.
Relying therefore on the generosity and innocence of her intentions,
she chose rather to leave her own actions open to censure which
they did not deserve, than to risk an investigation which might be
fatal to the interest of her poor friend. She took nothing with her,
Mrs. Stafford undertaking every necessary arrangement about her
cloaths—and having at night taken a tender leave of this beloved
and valuable woman, and promised to write to her constantly and to
return as soon as the destiny of Lady Adelina should be decided,
they parted.
And Emmeline, arising before the dawn of the following morning, set
out alone to Woodbury Forest—a precaution absolutely necessary, to
evade the inquisitive watchfulness of James Crofts. She stole softly
down stairs, before even the servants were stirring, and opening the
door cautiously, felt some degree of terror at being obliged to
undertake so long a walk alone at such an hour. But innocence gave
her courage, and friendly zeal lent her strength. As she walked on,
her fears subsided. She saw the sun rise above the horizon, and her
apprehensions were at an end.
As no carriage could approach within three quarters of a mile of the
house where Lady Adelina was concealed, they were obliged to walk
to the road where Mrs. Stafford had directed a post chaise to wait
for them, which she had hired at a distant town, where it was
unlikely any enquiry would be made.
Long disuse, as she had hardly ever left the cottage from the
moment of her entering it, and the extreme weakness to which she
was reduced, made Emmeline greatly fear that Lady Adelina would
never be able to reach the place. With her assistance, and that of
her Ladyship's woman, slowly and faintly she walked thither; and

Emmeline saw her happily placed in the chaise. Every thing had
been before settled as to the conveyance of the servant and
baggage, and to engage the secresy of the woman with whom she
had dwelt, by making her silence sufficiently advantageous; and as
they hoped that no traces were left by which they might be followed,
the spirits of the fair travellers seemed somewhat to improve as they
proceeded on their journey.—Emmeline felt her heart elated with the
consciousness of doing good; and from the tender affection and
assistance of such a friend, which could be considered only as the
benevolence of heaven itself, Lady Adelina drew a favourable omen,
and dared entertain a faint hope that her penitence had been
accepted.
They arrived without any accident at Bath, the following day; and
Emmeline, leaving Lady Adelina at the inn, went out immediately to
secure lodgings in a retired part of the town. As soon as it was dark,
Lady Adelina removed thither in a chair; and was announced by
Emmeline to be the wife of a Swiss officer, to be herself of
Switzerland, and to bear the name of Mrs. St. Laure—while she
herself, as she was very little known, continued to pass by her own
name in the few transactions which in their very private way of living
required her name to be repeated.
When Mrs. Ashwood found that Emmeline had left Woodfield
clandestinely and alone, and that Mrs. Stafford evaded giving any
account whither she was gone, by saying coldly that she was gone
to visit a friend in Surrey whom she formerly knew in Wales, all the
suspicions she had herself harboured, and Miss Galton encouraged,
seemed confirmed. James Crofts had related, not without
exaggerations, what he had been witness to in the copse; and it was
no longer doubted but that she was gone with Fitz-Edward, which at
once accounted for her departure and the sudden and mysterious
manner in which it was accomplished. James Crofts had suspicions
that his behaviour had hastened it; but he failed not to confirm Mrs.
Ashwood in her prepossession that her entanglement with Fitz-
Edward was now at a period when it could be no longer concealed—
intelligence which was to be conveyed to Delamere.

The elder Crofts, who had been some time with Lady Montreville and
her daughter, had named Delamere from time to time in his letters
to his brother. The last, mentioned that he was now with his mother
and sister, who were at Nice, and who purposed returning to
England in about three months. Crofts represented Delamere as still
devoted to Emmeline; and as existing only in the hope of being no
longer opposed in his intention of marrying her in March, when the
year which he had promised his father to wait expired; but that Lady
Montreville, as time wore away, grew more averse to the match, and
more desirous of some event which might break it off. Crofts gave
his brother a very favourable account of his progress with Miss
Delamere; and hinted that if he could be fortunate enough to put an
end to Delamere's intended connection, it would so greatly conciliate
the favour of Lady Montreville, that he dared hope she would no
longer oppose his union with her daughter: and when once they
were married, and the prejudices of the mother to an inferior
alliance conquered, he had very little doubt of Lord Montreville's
forgiveness, and of soon regaining his countenance and friendship.
This account from his brother added another motive to those which
already influenced the malignant and illiberal mind of James Crofts
to injure the lovely orphan, and he determined to give all his
assistance to Mrs. Ashwood in the cruel project of depriving her at
once of her character and her lover. In a consultation which he held
on this subject with his promised bride and Miss Galton, the ladies
agreed that it was perfectly shocking that such a fine young man as
Mr. Delamere should be attached to a woman so little sensible of his
value as Emmeline; that it had long been evident she was to him
indifferent, and it was now too clear that she was partial to another;
and that therefore it would be a meritorious action to acquaint him
of her intimacy with Fitz-Edward; and it could not be doubted but his
knowledge of it would, high spirited as he was, cure him effectually
of his ill-placed passion, and restore the tranquillity of his
respectable family. Hiding thus the inveterate envy and malice of
their hearts under this hypocritical pretence, they next considered
how to give the information which was so meritorious. Anonymous

letters were expedients to which Miss Galton had before had
recourse, and to an anonymous letter they determined to commit
the secret of Emmeline's infidelity—while James Crofts, in his letters
to his brother, was to corroborate the intelligence it contained, by
relating as mere matter of news what had actually and evidently
happened, Emmeline's sudden departure from Woodfield.
Delamere, when he saw his mother out of danger at Barege, had
returned to the neighbourhood of Paris, where he had lingered some
time, in hopes that Emmeline would accede to his request of being
allowed to cross the channel for a few days; but her answer, in
which she strongly urged the hazard he would incur of giving his
father a pretence to withdraw his promise, by violating his own, had
obliged him, tho' with infinite reluctance, to give up the scheme; and
being quite indifferent where he was, if he was still at a distance
from her, he had yielded to the solicitations of Lady Montreville, and
rejoined her at Nice. There, he now remained; while every thing in
England seemed to contribute to assist the designs of those who
wished to disengage him from his passion for Emmeline.
The day after Emmeline's departure with Lady Adelina, Fitz-Edward
went to Woodfield; and hearing that Miss Mowbray had suddenly left
it, was thrown into the utmost astonishment—astonishment which
Mrs. Ashwood and Miss Galton observed to each other was the finest
piece of acting they had ever seen.
The whole party were together when he was introduced—a
circumstance Mrs. Stafford would willingly have avoided, as it was
absolutely necessary for her to speak to him alone; and determined
to do so, whatever construction the malignity of her sister-in-law
might put upon it, she said—
'I have long promised you, Colonel, a sight of the two pieces of
drawing which Miss Mowbray and I have finished as companions.
They are now framed; and if you will come with me into my
dressing-room you shall see them.'

As the rest of the company had frequently seen these drawings,
there was no pretence for their following Mrs. Stafford; who,
accompanied by the Colonel, went to her dressing room.
A conference thus evidently sought by Mrs. Stafford, excited the
eager and painful curiosity of the party in the parlour.
'Now would I give the world,' cried Mrs. Ashwood, 'to know what is
going forward.'
'Is it not possible to listen?' enquired Crofts, equal to any meanness
that might gratify the malevolence of another or his own.
'Yes,' replied Mrs. Ashwood, 'if one could get into the closet next the
dressing-room without being perceived, which can only be done by
passing thro' the nursery. If indeed the nursery maids and children
are out, it is easy enough.'
'They are out, mama, I assure you,' cried Miss Ashwood, 'for I saw
them myself go across the lawn since I've been at breakfast. Do,
pray let us go and listen—I long of all things to know what my aunt
Stafford can have to say to that sly-looking Colonel.'
'No, no, child,' said her mother, 'I shall not send you, indeed—but
Crofts, do you think we should be able to make it out?'
'Egad,' answered he, 'I'll try—for depend upon it the mischief will
out. It will be rare, to have such a pretty tale to tell Mr. Delamere of
his demure-looking little dear.—I'll venture.'
Mrs. Ashwood then shewing him the way, he went on tip toe up
stairs, and concealing himself in a light closet which was divided
from the dressing room only by lath and plaister, he lent an attentive
ear to the dialogue that was passing.
It happened, however, that the window near which Mrs. Stafford and
Fitz-Edward were sitting was exactly opposite to that side of the
room to which Crofts' hiding-place communicated; and tho' the room
was not large, yet the distance, the partition, and the low voice in
which both parties spoke, made it impossible for him to distinguish
more than broken sentences. From Mrs. Stafford he heard—'Could

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