![Introduction
This book proves a number of important theorems that are commonly
given in advanced books on Commutative Algebra without proof, owing
to the difficulty of the existing proofs. In short, we give homological
proofs of these results, but instead of the original ones involving simpli-
cial methods, we modify these to use only lower dimensional homology
modules, that we can introduce in an ad hoc way, thus avoiding sim-
plicial theory. This allows us to give complete and comparatively short
proofs of the important results we state below. We hope these notes can
serve as a complement to the existing literature.
These are some of the main results we prove in this book:
Theorem (I) Let (A, m, K) → (B, n, L) be a local homomorphism of
noetherian local rings. Then the following conditions are equivalent:
a) B is a formally smooth A-algebra for the n-adic topology
b) B is a flat A-module and the K-algebra B ⊗A K is geometrically
regular.
This result is due to Grothendieck [EGA 0IV, (19.7.1)]. His proof is
long, though it provides a lot of additional information. He uses this
result in proving Cohen’s theorems on the structure of complete noethe-
rian local rings. An alternative proof of (I) was given by M. André [An1],
based on André–Quillen homology theory; it thus uses simplicial meth-
ods, that are not necessarily familiar to all commutative algebraists. A
third proof was given by N. Radu [Ra2], making use of Cohen’s theorems
on complete noetherian local rings.
Theorem (II) Let A be a complete intersection ring and p a prime
ideal of A. Then the localization Ap is a complete intersection.
1](https://image.slidesharecdn.com/88749-250718140451-45c39b6c/85/Smoothness-Regularity-and-Complete-Intersection-1st-Edition-Javier-Majadas-14-320.jpg)
![2 Introduction
This result is due to L.L. Avramov [Av1]. Its proof uses differential
graded algebras as well as André–Quillen homology modules in dimen-
sions 3 and 4, the vanishing of which characterizes complete intersec-
tions.
Our proofs of these two results follow André and Avramov’s arguments
[An1], [Av1, Av2] respectively, but we make appropriate changes so as
to involve André–Quillen homology modules only in dimensions ≤ 2: up
to dimension 2 these homology modules are easy to construct following
Lichtenbaum and Schlessinger [LS].
Theorem (III) A regular homomorphism is a direct limit of smooth
homomorphisms of finite type (D. Popescu [Po1]–[Po3]).
We give here Popescu’s proof [Po1]–[Po3], [Sw]. An alternative proof
is due to Spivakovsky [Sp].
Theorem (IV) The module of differentials of a regular homomorphism
is flat.
This result follows immediately from (III). However, for many years
up to the appearance of Popescu’s result, the only known proof was that
by André, making essential use of André–Quillen homology modules in
all dimensions.
Theorem (V) If f : (A, m, K) → (B, n, L) is a local formally smooth
homomorphism of noetherian local rings and A is quasiexcellent, then f
is regular.
This result is due to André [An2]; we give here a proof more in the
style of the methods of this book, mainly following some papers of André,
A. Brezuleanu and N. Radu.
We now describe the contents of this book in brief. Chapter 1 intro-
duces homology modules in dimensions 0, 1 and 2. First, in Section 1.1
we give the definition of Lichtenbaum and Schlessinger [LS], which is
very concise, at least if we omit the proof that it is well defined. The
reader willing to take this on trust and to accept its properties (1.4) can
omit Sections (1.2–1.3) on first reading; there, instead of following [LS],
we construct the homology modules using differential graded resolutions.
This makes the definition somewhat longer, but simplifies the proof of
some properties. Moreover, differential graded resolutions are used in
an essential way in Chapter 4.](https://image.slidesharecdn.com/88749-250718140451-45c39b6c/85/Smoothness-Regularity-and-Complete-Intersection-1st-Edition-Javier-Majadas-15-320.jpg)
![Introduction 3
Chapter 2 studies formally smooth homomorphisms, and in partic-
ular proves Theorem (I). We follow mainly [An1], making appropriate
changes to avoid using homology modules in dimensions > 2. This part
was already written (in Spanish) in 1988.
Chapter 3 uses the results of Chapter 2 to deduce Cohen’s theorems
on complete noetherian local rings. We follow mainly [EGA 0IV] and
Bourbaki [Bo, Chapter 9].
In Chapter 4, we prove Theorem (II). After giving Gulliksen’s result
[GL] on the existence of minimal differential graded resolutions, we fol-
low Avramov [Av1] and [Av2], taking care to avoid homology modules
in dimension 3 and 4. As a by-product, we also give a proof of Kunz’s re-
sult characterizing regular local rings in positive characteristic in terms
of the Frobenius homomorphism.
Finally, Chapters 5 and 6 study regular homomorphisms, giving in
particular proofs of Theorems (III), (IV) and (V).
The prerequisites for reading this book are a basic course in com-
mutative algebra (Matsumura [Mt, Chapters 1–9] should be more than
sufficient) and the first definitions in homological algebra. Though in
places we use certain exact sequences deduced from spectral sequences,
we give direct proofs of these in the Appendix, thus avoiding the use of
spectral sequences.
Finally, we make the obvious remark that this book is not in any
way intended as a substitute for André’s simplicial homological methods
[An1] or the proofs given in [EGA 0IV], since either of these treatments
is more complete than ours. Rather, we hope that our book can serve as
an introduction and motivation to study these sources. We would also
like to mention that we have profited from reading the interesting book
by Brezuleanu, Dumitrescu and Radu [BDR] on topics similar to ours,
although they do not use homological methods.
We are grateful to T. Sánchet Giralda for interesting suggestions and
to the editor for contributing to improve the presentation of these notes.
Conventions. All rings are commutative, except that graded rings are
sometimes (strictly) anticommutative; the context should make it clear
in each case which is intended.](https://image.slidesharecdn.com/88749-250718140451-45c39b6c/85/Smoothness-Regularity-and-Complete-Intersection-1st-Edition-Javier-Majadas-16-320.jpg)
![1
Definition and first properties of
(co-)homology modules
In this chapter we define the Lichtenbaum–Schlessinger (co-)homology
modules Hn(A, B, M) and Hn
(A, B, M), for n = 0, 1, 2, associated to
a (commutative) algebra A → B and a B-module M, and we prove
their main properties [LS]. In Section 1.1 we give a simple definition
of Hn(A, B, M) and Hn
(A, B, M), but without justifying that they are
in fact well defined. To justify this definition, in Section 1.3 we give
another (now complete) definition, and prove that it agrees with that
of 1.1. We use differential graded algebras, introduced in Section 1.2. In
[LS] they are not used. However we prefer this (equivalent) approach,
since we also use differential graded algebras later in studying complete
intersections. More precisely, we use Gulliksen’s Theorem 4.1.7 on the
existence of minimal differential graded algebra resolutions in order to
prove Avramov’s Lemma 4.2.1. Section 1.4 establishes the main prop-
erties of these homology modules.
Note that these (co-)homology modules (defined only for n = 0, 1, 2)
agree with those defined by André and Quillen using simplicial methods
[An1, 15.12, 15.13].
1.1 First definition
Definition 1.1.1 Let A be a ring and B an A-algebra. Let e0 : R → B
be a surjective homomorphism of A-algebras, where R is a polynomial
A-algebra. Let I = ker e0 and
0 → U → F
j
−
−
→ I → 0
an exact sequence of R-modules with F free. Let φ:
2
F → F be the R-
module homomorphism defined by φ(x∧y) = j(x)y−j(y)x, where
2
F
4](https://image.slidesharecdn.com/88749-250718140451-45c39b6c/85/Smoothness-Regularity-and-Complete-Intersection-1st-Edition-Javier-Majadas-17-320.jpg)
Smoothness Regularity and Complete Intersection 1st Edition Javier Majadas
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- 5. Smoothness Regularity and Complete Intersection 1st Edition Javier Majadas Digital Instant Download Author(s): Javier Majadas, Antonio G. Rodicio ISBN(s): 9780521125727, 0521125723 Edition: 1 File Details: PDF, 1.25 MB Year: 2010 Language: english
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- 8. LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at www.cambridge.org/mathematics 222 Advances in linear logic, J.-Y. GIRARD, Y. LAFONT & L. REGNIER (eds) 223 Analytic semigroups and semilinear initial boundary value problems, K. TAIRA 224 Computability, enumerability, unsolvability, S.B. COOPER, T.A. SLAMAN & S.S. WAINER (eds) 225 A mathematical introduction to string theory, S. ALBEVERIO et al 226 Novikov conjectures, index theorems and rigidity I, S.C. FERRY, A. RANICKI & J. ROSENBERG (eds) 227 Novikov conjectures, index theorems and rigidity II, S.C. FERRY, A. RANICKI & J. ROSENBERG (eds) 228 Ergodic theory of Zd-actions, M. POLLICOTT & K. SCHMIDT (eds) 229 Ergodicity for infinite dimensional systems, G. DA PRATO & J. ZABCZYK 230 Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS & E.V. FLYNN 231 Semigroup theory and its applications, K.H. HOFMANN & M.W. MISLOVE (eds) 232 The descriptive set theory of Polish group actions, H. BECKER & A.S. KECHRIS 233 Finite fields and applications, S. COHEN & H. NIEDERREITER (eds) 234 Introduction to subfactors, V. JONES & V.S. SUNDER 235 Number theory: Séminaire de théorie des nombres de Paris 1993–94, S. DAVID (ed) 236 The James forest, H. FETTER & B. GAMBOA DE BUEN 237 Sieve methods, exponential sums, and their applications in number theory, G.R.H. GREAVES et al (eds) 238 Representation theory and algebraic geometry, A. MARTSINKOVSKY & G. TODOROV (eds) 240 Stable groups, F.O. WAGNER 241 Surveys in combinatorics, 1997, R.A. BAILEY (ed) 242 Geometric Galois actions I, L. SCHNEPS & P. LOCHAK (eds) 243 Geometric Galois actions II, L. SCHNEPS & P. LOCHAK (eds) 244 Model theory of groups and automorphism groups, D.M. EVANS (ed) 245 Geometry, combinatorial designs and related structures, J.W.P. HIRSCHFELD et al (eds) 246 p-Automorphisms of finite p-groups, E.I. KHUKHRO 247 Analytic number theory, Y. MOTOHASHI (ed) 248 Tame topology and O-minimal structures, L. VAN DEN DRIES 249 The atlas of finite groups - Ten years on, R.T. CURTIS & R.A. WILSON (eds) 250 Characters and blocks of finite groups, G. NAVARRO 251 Gröbner bases and applications, B. BUCHBERGER & F. WINKLER (eds) 252 Geometry and cohomology in group theory, P.H. KROPHOLLER, G.A. NIBLO & R. STÖHR (eds) 253 The q-Schur algebra, S. DONKIN 254 Galois representations in arithmetic algebraic geometry, A.J. SCHOLL & R.L. TAYLOR (eds) 255 Symmetries and integrability of difference equations, P.A. CLARKSON & F.W. NIJHOFF (eds) 256 Aspects of Galois theory, H. VÖLKLEIN, J.G. THOMPSON, D. HARBATER & P. MÜLLER (eds) 257 An introduction to noncommutative differential geometry and its physical applications (2nd edition), J. MADORE 258 Sets and proofs, S.B. COOPER & J.K. TRUSS (eds) 259 Models and computability, S.B. COOPER & J. TRUSS (eds) 260 Groups St Andrews 1997 in Bath I, C.M. CAMPBELL et al (eds) 261 Groups St Andrews 1997 in Bath II, C.M. CAMPBELL et al (eds) 262 Analysis and logic, C.W. HENSON, J. IOVINO, A.S. KECHRIS & E. ODELL 263 Singularity theory, W. BRUCE & D. MOND (eds) 264 New trends in algebraic geometry, K. HULEK, F. CATANESE, C. PETERS & M. REID (eds) 265 Elliptic curves in cryptography, I. BLAKE, G. SEROUSSI & N. SMART 267 Surveys in combinatorics, 1999, J.D. LAMB & D.A. PREECE (eds) 268 Spectral asymptotics in the semi-classical limit, M. DIMASSI & J. SJÖSTRAND 269 Ergodic theory and topological dynamics of group actions on homogeneous spaces, M.B. BEKKA & M. MAYER 271 Singular perturbations of differential operators, S. ALBEVERIO & P. KURASOV 272 Character theory for the odd order theorem, T. PETERFALVI. Translated by R. SANDLING 273 Spectral theory and geometry, E.B. DAVIES & Y. SAFAROV (eds) 274 The Mandelbrot set, theme and variations, T. LEI (ed) 275 Descriptive set theory and dynamical systems, M. FOREMAN, A.S. KECHRIS, A. LOUVEAU & B. WEISS (eds) 276 Singularities of plane curves, E. CASAS-ALVERO 277 Computational and geometric aspects of modern algebra, M. ATKINSON et al (eds) 278 Global attractors in abstract parabolic problems, J.W. CHOLEWA & T. DLOTKO 279 Topics in symbolic dynamics and applications, F. BLANCHARD, A. MAASS & A. NOGUEIRA (eds) 280 Characters and automorphism groups of compact Riemann surfaces, T. BREUER 281 Explicit birational geometry of 3-folds, A. CORTI & M. REID (eds) 282 Auslander-Buchweitz approximations of equivariant modules, M. HASHIMOTO 283 Nonlinear elasticity, Y.B. FU & R.W. OGDEN (eds) 284 Foundations of computational mathematics, R. DEVORE, A. ISERLES & E. SÜLI (eds) 285 Rational points on curves over finite fields, H. NIEDERREITER & C. XING 286 Clifford algebras and spinors (2nd Edition), P. LOUNESTO 287 Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE, A.F. COSTA & E. MARTÍNEZ (eds) 288 Surveys in combinatorics, 2001, J.W.P. HIRSCHFELD (ed) 289 Aspects of Sobolev-type inequalities, L. SALOFF-COSTE 290 Quantum groups and Lie theory, A. PRESSLEY (ed) 291 Tits buildings and the model theory of groups, K. TENT (ed) 292 A quantum groups primer, S. MAJID 293 Second order partial differential equations in Hilbert spaces, G. DA PRATO & J. ZABCZYK 294 Introduction to operator space theory, G. PISIER 295 Geometry and integrability, L. MASON & Y. NUTKU (eds)
- 9. 296 Lectures on invariant theory, I. DOLGACHEV 297 The homotopy category of simply connected 4-manifolds, H.-J. BAUES 298 Higher operads, higher categories, T. LEINSTER (ed) 299 Kleinian groups and hyperbolic 3-manifolds, Y. KOMORI, V. MARKOVIC & C. SERIES (eds) 300 Introduction to Möbius differential geometry, U. HERTRICH-JEROMIN 301 Stable modules and the D(2)-problem, F.E.A. JOHNSON 302 Discrete and continuous nonlinear Schrödinger systems, M.J. ABLOWITZ, B. PRINARI & A.D. TRUBATCH 303 Number theory and algebraic geometry, M. REID & A. SKOROBOGATOV (eds) 304 Groups St Andrews 2001 in Oxford I, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) 305 Groups St Andrews 2001 in Oxford II, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) 306 Geometric mechanics and symmetry, J. MONTALDI & T. RATIU (eds) 307 Surveys in combinatorics 2003, C.D. WENSLEY (ed.) 308 Topology, geometry and quantum field theory, U.L. TILLMANN (ed) 309 Corings and comodules, T. BRZEZINSKI & R. WISBAUER 310 Topics in dynamics and ergodic theory, S. BEZUGLYI & S. KOLYADA (eds) 311 Groups: topological, combinatorial and arithmetic aspects, T.W. MÜLLER (ed) 312 Foundations of computational mathematics, Minneapolis 2002, F. CUCKER et al (eds) 313 Transcendental aspects of algebraic cycles, S. MÜLLER-STACH & C. PETERS (eds) 314 Spectral generalizations of line graphs, D. CVETKOVIĆ, P. ROWLINSON & S. SIMIĆ 315 Structured ring spectra, A. BAKER & B. RICHTER (eds) 316 Linear logic in computer science, T. EHRHARD, P. RUET, J.-Y. GIRARD & P. SCOTT (eds) 317 Advances in elliptic curve cryptography, I.F. BLAKE, G. SEROUSSI & N.P. SMART (eds) 318 Perturbation of the boundary in boundary-value problems of partial differential equations, D. HENRY 319 Double affine Hecke algebras, I. CHEREDNIK 320 L-functions and Galois representations, D. BURNS, K. BUZZARD & J. NEKOVÁŘ (eds) 321 Surveys in modern mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) 322 Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N.C. SNAITH (eds) 323 Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) 324 Singularities and computer algebra, C. LOSSEN & G. PFISTER (eds) 325 Lectures on the Ricci flow, P. TOPPING 326 Modular representations of finite groups of Lie type, J.E. HUMPHREYS 327 Surveys in combinatorics 2005, B.S. WEBB (ed) 328 Fundamentals of hyperbolic manifolds, R. CANARY, D. EPSTEIN & A. MARDEN (eds) 329 Spaces of Kleinian groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) 330 Noncommutative localization in algebra and topology, A. RANICKI (ed) 331 Foundations of computational mathematics, Santander 2005, L.M PARDO, A. PINKUS, E. SÜLI & M.J. TODD (eds) 332 Handbook of tilting theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) 333 Synthetic differential geometry (2nd Edition), A. KOCK 334 The Navier-Stokes equations, N. RILEY & P. DRAZIN 335 Lectures on the combinatorics of free probability, A. NICA & R. SPEICHER 336 Integral closure of ideals, rings, and modules, I. SWANSON & C. HUNEKE 337 Methods in Banach space theory, J.M.F. CASTILLO & W.B. JOHNSON (eds) 338 Surveys in geometry and number theory, N. YOUNG (ed) 339 Groups St Andrews 2005 I, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 340 Groups St Andrews 2005 II, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 341 Ranks of elliptic curves and random matrix theory, J.B. CONREY, D.W. FARMER, F. MEZZADRI & N.C. SNAITH (eds) 342 Elliptic cohomology, H.R. MILLER & D.C. RAVENEL (eds) 343 Algebraic cycles and motives I, J. NAGEL & C. PETERS (eds) 344 Algebraic cycles and motives II, J. NAGEL & C. PETERS (eds) 345 Algebraic and analytic geometry, A. NEEMAN 346 Surveys in combinatorics 2007, A. HILTON & J. TALBOT (eds) 347 Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds) 348 Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds) 349 Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 350 Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 351 Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH 352 Number theory and polynomials, J. MCKEE & C. SMYTH (eds) 353 Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) 354 Groups and analysis, K. TENT (ed) 355 Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI 356 Elliptic curves and big Galois representations, D. DELBOURGO 357 Algebraic theory of differential equations, M.A.H. MACCALLUM & A.V. MIKHAILOV (eds) 358 Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. LEARY (eds) 359 Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCÍA-PRADA & S. RAMANAN (eds) 360 Zariski geometries, B. ZILBER 361 Words: Notes on verbal width in groups, D. SEGAL 362 Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERÓN & R. ZUAZUA 363 Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds) 364 Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds) 365 Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds) 366 Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds) 367 Random matrices: High dimensional phenomena, G. BLOWER 368 Geometry of Riemann surfaces, F.P. GARDINER, G. GONZÁLEZ-DIEZ & C. KOUROUNIOTIS (eds) 369 Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ 370 Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH 371 Conformal fractals, F. PRZYTYCKI & M. URBAŃSKI 372 Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds) 373 Smoothness, regularity and complete intersection, J. MAJADAS & A.G. RODICIO 374 Geometric analysis of hyperbolic differential equations, S. ALINHAC
- 10. LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES: 373 Smoothness, Regularity and Complete Intersection JAVIER MAJADAS ANTONIO G. RODICIO Universidad de Santiago de Compostela, Spain
- 11. cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521125727 © J. Majadas and A. G. Rodicio 2010 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2010 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN 978-0-521-12572-7 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 12. Contents Introduction page 1 1 Definition and first properties of (co-)homology modules 4 1.1 First definition 4 1.2 Differential graded algebras 5 1.3 Second definition 11 1.4 Main properties 17 2 Formally smooth homomorphisms 22 2.1 Infinitesimal extensions 23 2.2 Formally smooth algebras 26 2.3 Jacobian criteria 29 2.4 Field extensions 34 2.5 Geometric regularity 39 2.6 Formally smooth local homomorphisms of noetherian rings 43 2.7 Appendix: The Mac Lane separability criterion 46 3 Structure of complete noetherian local rings 47 3.1 Cohen rings 47 3.2 Cohen’s structure theorems 52 4 Complete intersections 55 4.1 Minimal DG resolutions 56 4.2 The main lemma 60 4.3 Complete intersections 62 4.4 Appendix: Kunz’s theorem on regular local rings in characteristic p 64 v
- 13. vi Contents 5 Regular homomorphisms: Popescu’s theorem 67 5.1 The Jacobian ideal 68 5.2 The main lemmas 74 5.3 Statement of the theorem 83 5.4 The separable case 87 5.5 Positive characteristic 91 5.6 The module of differentials of a regular homomorphism 108 6 Localization of formal smoothness 109 6.1 Preliminary reductions 109 6.2 Some results on vanishing of homology 115 6.3 Noetherian property of the relative Frobenius 117 6.4 End of the proof of localization of formal smoothness 120 6.5 Appendix: Power series 121 Appendix: Some exact sequences 126 Bibliography 130 Index 134
- 14. Introduction This book proves a number of important theorems that are commonly given in advanced books on Commutative Algebra without proof, owing to the difficulty of the existing proofs. In short, we give homological proofs of these results, but instead of the original ones involving simpli- cial methods, we modify these to use only lower dimensional homology modules, that we can introduce in an ad hoc way, thus avoiding sim- plicial theory. This allows us to give complete and comparatively short proofs of the important results we state below. We hope these notes can serve as a complement to the existing literature. These are some of the main results we prove in this book: Theorem (I) Let (A, m, K) → (B, n, L) be a local homomorphism of noetherian local rings. Then the following conditions are equivalent: a) B is a formally smooth A-algebra for the n-adic topology b) B is a flat A-module and the K-algebra B ⊗A K is geometrically regular. This result is due to Grothendieck [EGA 0IV, (19.7.1)]. His proof is long, though it provides a lot of additional information. He uses this result in proving Cohen’s theorems on the structure of complete noethe- rian local rings. An alternative proof of (I) was given by M. André [An1], based on André–Quillen homology theory; it thus uses simplicial meth- ods, that are not necessarily familiar to all commutative algebraists. A third proof was given by N. Radu [Ra2], making use of Cohen’s theorems on complete noetherian local rings. Theorem (II) Let A be a complete intersection ring and p a prime ideal of A. Then the localization Ap is a complete intersection. 1
- 15. 2 Introduction This result is due to L.L. Avramov [Av1]. Its proof uses differential graded algebras as well as André–Quillen homology modules in dimen- sions 3 and 4, the vanishing of which characterizes complete intersec- tions. Our proofs of these two results follow André and Avramov’s arguments [An1], [Av1, Av2] respectively, but we make appropriate changes so as to involve André–Quillen homology modules only in dimensions ≤ 2: up to dimension 2 these homology modules are easy to construct following Lichtenbaum and Schlessinger [LS]. Theorem (III) A regular homomorphism is a direct limit of smooth homomorphisms of finite type (D. Popescu [Po1]–[Po3]). We give here Popescu’s proof [Po1]–[Po3], [Sw]. An alternative proof is due to Spivakovsky [Sp]. Theorem (IV) The module of differentials of a regular homomorphism is flat. This result follows immediately from (III). However, for many years up to the appearance of Popescu’s result, the only known proof was that by André, making essential use of André–Quillen homology modules in all dimensions. Theorem (V) If f : (A, m, K) → (B, n, L) is a local formally smooth homomorphism of noetherian local rings and A is quasiexcellent, then f is regular. This result is due to André [An2]; we give here a proof more in the style of the methods of this book, mainly following some papers of André, A. Brezuleanu and N. Radu. We now describe the contents of this book in brief. Chapter 1 intro- duces homology modules in dimensions 0, 1 and 2. First, in Section 1.1 we give the definition of Lichtenbaum and Schlessinger [LS], which is very concise, at least if we omit the proof that it is well defined. The reader willing to take this on trust and to accept its properties (1.4) can omit Sections (1.2–1.3) on first reading; there, instead of following [LS], we construct the homology modules using differential graded resolutions. This makes the definition somewhat longer, but simplifies the proof of some properties. Moreover, differential graded resolutions are used in an essential way in Chapter 4.
- 16. Introduction 3 Chapter 2 studies formally smooth homomorphisms, and in partic- ular proves Theorem (I). We follow mainly [An1], making appropriate changes to avoid using homology modules in dimensions > 2. This part was already written (in Spanish) in 1988. Chapter 3 uses the results of Chapter 2 to deduce Cohen’s theorems on complete noetherian local rings. We follow mainly [EGA 0IV] and Bourbaki [Bo, Chapter 9]. In Chapter 4, we prove Theorem (II). After giving Gulliksen’s result [GL] on the existence of minimal differential graded resolutions, we fol- low Avramov [Av1] and [Av2], taking care to avoid homology modules in dimension 3 and 4. As a by-product, we also give a proof of Kunz’s re- sult characterizing regular local rings in positive characteristic in terms of the Frobenius homomorphism. Finally, Chapters 5 and 6 study regular homomorphisms, giving in particular proofs of Theorems (III), (IV) and (V). The prerequisites for reading this book are a basic course in com- mutative algebra (Matsumura [Mt, Chapters 1–9] should be more than sufficient) and the first definitions in homological algebra. Though in places we use certain exact sequences deduced from spectral sequences, we give direct proofs of these in the Appendix, thus avoiding the use of spectral sequences. Finally, we make the obvious remark that this book is not in any way intended as a substitute for André’s simplicial homological methods [An1] or the proofs given in [EGA 0IV], since either of these treatments is more complete than ours. Rather, we hope that our book can serve as an introduction and motivation to study these sources. We would also like to mention that we have profited from reading the interesting book by Brezuleanu, Dumitrescu and Radu [BDR] on topics similar to ours, although they do not use homological methods. We are grateful to T. Sánchet Giralda for interesting suggestions and to the editor for contributing to improve the presentation of these notes. Conventions. All rings are commutative, except that graded rings are sometimes (strictly) anticommutative; the context should make it clear in each case which is intended.
- 17. 1 Definition and first properties of (co-)homology modules In this chapter we define the Lichtenbaum–Schlessinger (co-)homology modules Hn(A, B, M) and Hn (A, B, M), for n = 0, 1, 2, associated to a (commutative) algebra A → B and a B-module M, and we prove their main properties [LS]. In Section 1.1 we give a simple definition of Hn(A, B, M) and Hn (A, B, M), but without justifying that they are in fact well defined. To justify this definition, in Section 1.3 we give another (now complete) definition, and prove that it agrees with that of 1.1. We use differential graded algebras, introduced in Section 1.2. In [LS] they are not used. However we prefer this (equivalent) approach, since we also use differential graded algebras later in studying complete intersections. More precisely, we use Gulliksen’s Theorem 4.1.7 on the existence of minimal differential graded algebra resolutions in order to prove Avramov’s Lemma 4.2.1. Section 1.4 establishes the main prop- erties of these homology modules. Note that these (co-)homology modules (defined only for n = 0, 1, 2) agree with those defined by André and Quillen using simplicial methods [An1, 15.12, 15.13]. 1.1 First definition Definition 1.1.1 Let A be a ring and B an A-algebra. Let e0 : R → B be a surjective homomorphism of A-algebras, where R is a polynomial A-algebra. Let I = ker e0 and 0 → U → F j − − → I → 0 an exact sequence of R-modules with F free. Let φ: 2 F → F be the R- module homomorphism defined by φ(x∧y) = j(x)y−j(y)x, where 2 F 4
- 18. 1.2 Differential graded algebras 5 is the second exterior power of the R-module F. Let U0 = im(φ) ⊂ U. We have IU ⊂ U0, and so U/U0 is a B-module. We have a complex of B-modules U/U0 → F/U0 ⊗R B = F/IF → ΩR|A ⊗R B (concentrated in degrees 2, 1 and 0), where the first homomorphism is induced by the injection U → F, and the second is the composite F/IF → I/I2 → ΩR|A ⊗R B, where the first map is induced by j, and the second by the canonical derivation d: R → ΩR|A (here ΩR|A is the module of Kähler differentials). We denote any such complex by LB|A, and define for a B-module M Hn(A, B, M) = Hn(LB|A ⊗B M) for n = 0, 1, 2, Hn (A, B, M) = Hn (HomB(LB|A, M)) for n = 0, 1, 2. In Section 1.3 we show that this definition does not depend on the choices of R and F. 1.2 Differential graded algebras Definition 1.2.1 Let A be a ring. A differential graded A-algebra (R, d) (DG A-algebra in what follows) is an (associative) graded A-algebra with unit R = n≥0 Rn, strictly anticommutative, i.e., satisfying xy = (−1)pq yx for x ∈ Rp, y ∈ Rq and x2 = 0 for x ∈ R2n+1, and having a differential d = (dn : Rn → Rn−1) of degree −1; that is, d is R0-linear, d2 = 0 and d(xy) = d(x)y + (−1)p xd(y) for x ∈ Rp, y ∈ R. Clearly, (R, d) is a DG R0-algebra. We can view any A-algebra B as a DG A-algebra concentrated in degree 0. A homomorphism f : (R, dR) → (S, dS) of DG A-algebras is an A- algebra homomorphism that preserves degrees (f(Rn) ⊂ Sn) such that dSf = fdR. If (R, dR), (S, dS) are DG A-algebras, we define their tensor product R ⊗A S to be the DG A-algebra having a) underlying A-module the usual tensor product R⊗AS of modules, with grading given by R ⊗A S = n≥0 p+q=n Rp ⊗A Sq
- 19. 6 Definition and first properties of (co-)homology modules b) product induced by (x⊗y)(x ⊗y ) = (−1)pq (xx ⊗yy ) for y ∈ Sp, x ∈ Rq c) differential induced by d(x ⊗ y) = dR(x) ⊗ y + (−1)q x ⊗ dS(y) for x ∈ Rq, y ∈ S. Let {(Ri, di)}i∈I be a family of DG A-algebras. For each finite subset J ⊂ I, we extend the above definition to i∈J ARi; for finite subsets J ⊂ J of I, we have a canonical homomorphism i∈J ARi → i∈J ARi. We thus have a direct system of homomorphisms of DG A-algebras. We say that the direct limit is the tensor product of the family of DG A- algebras {(Ri, di)}i∈I. It is a DG A-algebra, that we denote by i∈I ARi (and is not to be confused with the tensor product of the underlying family of A-algebras Ri). A DG ideal I of a DG A-algebra (R, d) is a homogeneous ideal of the graded A-algebra R that is stable under the differential, i.e., d(I) ⊂ I. Then R/I is canonically a DG A-algebra and the canonical map R → R/I is a homomorphism of DG A-algebras. An augmented DG A-algebra is a DG A-algebra together with a sur- jective (augmentation) homomorphism of DG A-algebras p: R → R , where R is a DG A-algebra concentrated in degree 0; its augmentation ideal is the DG ideal ker p of R. A DG subalgebra S of a DG A-algebra (R, d) is a graded A-subalgebra S of R such that d(S) ⊂ S. Let (R, d) be a DG A-algebra. Then Z(R) := ker d is a graded A-subalgebra of R with grading Z(R) = n≥0 Z(R)∩Rn , and B(R) := im(d) is a homogeneous ideal of Z(R). Therefore the homology of R H(R) = Z(R)/B(R) is a graded A-algebra. Example 1.2.2 Let R0 be an A-algebra and X a variable of degree n 0. Let R = R0 X be the following graded A-algebra: a) If n is odd, R0 X is the exterior R0-algebra on the variable X, i.e., R0 X = R01 ⊕ R0X, concentrated in degrees 0 and n. b) If n is even, R0 X is the quotient of the polynomial R0-algebra on variables X(1) , X(2) , . . . , by the ideal generated by the ele- ments X(i) X(j) − (i + j)! i!j! X(i+j) for i, j ≥ 1.
- 20. 1.2 Differential graded algebras 7 The grading is defined by deg X(m) = nm for m 0. We set X(0) = 1, X = X(1) and say that X(i) is the ith divided power of X. Observe that i!X(i) = Xi . Now let R be a DG A-algebra, x a homogeneous cycle of R of degree n − 1 ≥ 0, i.e., x ∈ Zn−1(R). Let X be a variable of degree n, and R X = R ⊗R0 R0 X . We define a differential in R X as the unique differential d for which R → R X is a DG A-algebra homomorphism with d(X) = x for n odd, respectively d(X(m) ) = xX(m−1) for n even. We denote this DG A-algebra by R X; dX = x . Note that an augmentation p: R → R satisfying p(x) = 0 extends in a unique way to an augmentation p : R X; dX = x → R by setting p(X) = 0. Lemma 1.2.3 Let R be a DG A-algebra and c ∈ Hn−1(R) for some n ≥ 1. Let x ∈ Zn−1(R) be a cycle whose homology class is c. Set S = R X; dX = x and let f : R → S be the canonical homomorphism. Then: a) f induces isomorphisms Hq(R) = Hq(S) for all q n − 1; b) f induces an isomorphism Hn−1(R)/ c R0 = Hn−1(S). Proof a) is clear, since Rq = Sq for q n, b) Zn−1(R) = Zn−1(S) and Bn−1(R) + xR0 = Bn−1(S). Definition 1.2.4 If {Xi}i∈I is a family of variables of degree 0, we define R0 {Xi}i∈I := i∈I R0 R0 Xi as the tensor product of the DG R0-algebras R0 Xi for i ∈ I (as in Definition 1.2.1). If R is a DG A-algebra, we say that a DG A-algebra S is free over R if the underlying graded A-algebra is of the form S = R ⊗R0 S0 {Xi}i∈I where S0 is a polynomial R0-algebra and {Xi}i∈I a family of variables of degree 0, and the differential of S extends that of R. (Caution: it is not necessarily a free object in the category of DG A-algebras.) If R is a DG A-algebra and {xi}i∈I a set of homogeneous cycles of R, we define R {Xi}i∈I; dXi = xi to be the DG A-algebra R ⊗R0 ( i∈I R0 R0 Xi; dXi = xi ), which is free over R. Lemma 1.2.5 Let R be a DG A-algebra, n − 1 ≥ 0, {ci}i∈I a set
- 21. 8 Definition and first properties of (co-)homology modules of elements of Hn−1(R) and {xi}i∈I a set of homogeneous cycles with classes {ci}i∈I. Set S = R {Xi}i∈I; dXi = xi , and let f : R → S be the canonical homomorphism. Then: a) f induces isomorphisms Hq(R) = Hq(S) for all q n − 1; b) f induces an isomorphism Hn−1(R)/ {ci}i∈I R0 = Hn−1(S). Proof Similar to the proof of Lemma 1.2.3, bearing in mind that direct limits are exact. Theorem 1.2.6 Let p: R → R be an augmented DG A-algebra. Then there exists an augmented DG A-algebra pS : S → R , free over R with S0 = R0, such that the augmentation pS extends p and gives an iso- morphism in homology H(S) = H(R ) = R if n = 0, 0 if n 0. If R0 is a noetherian ring and Ri an R0-module of finite type for all i, then we can choose S such that Si is an S0-module of finite type for all i. Proof Let S0 = R. Assume that we have constructed an augmented DG A-algebra Sn−1 that is free over R, such that Sn−1 0 = R0 and the augmentation Sn−1 → R induces isomorphisms Hq(Sn−1 ) = Hq(R ) for q n − 1. Let {ci}i∈I be a set of generators of the R0-module ker Hn−1(Sn−1 ) → Hn−1(R ) (equal to Hn−1(Sn−1 ) for n 1), and {xi}i∈I a set of homogeneous cycles with classes {ci}i∈I. Let Sn = Sn−1 {Xi}i∈I; dXi = xi . Then Sn is a DG A-algebra free over R with Sn 0 = R0 and such that the augmentation pSn : Sn → R extending pSn−1 defined by pSn (Xi) = 0 induces isomorphisms Hq(Sn ) = Hq(R ) for q n (Lemma 1.2.5). We define S := lim − → Sn . If R0 is a noetherian ring and Ri an R0-module of finite type for all i, then by induction we can choose Sn with Sn i an Sn 0 = R0-module of finite type for all i, since if Sn−1 i is an Sn−1 0 -module of finite type for all i, then Hi(Sn−1 ) is an Sn−1 0 -module of finite type for all i. Definition 1.2.7 Let A → B be a ring homomorphism. Let R be a DG A-algebra that is free over A with a surjective homomorphism of DG
- 22. 1.2 Differential graded algebras 9 A-algebras R → B inducing an isomorphism in homology. Then we say that R is a free DG resolution of the A-algebra B. Corollary 1.2.8 Let A → B be a ring homomorphism. Then a free DG resolution R of the A-algebra B exists. If A is noetherian and B an A- algebra of finite type, then we can choose R such that R0 is a polynomial A-algebra of finite type and Ri an R0-module of finite type for all i. Proof Let R0 be a polynomial A-algebra such that there exists a sur- jective homomorphism of A-algebras R0 → B. (If A is noetherian and B an A-algebra of finite type, then we can choose R0 a polynomial A- algebra of finite type.) Now apply Theorem 1.2.6 to R0 → B. Definition 1.2.9 Let R be a DG A-algebra that is free over R0, i.e., R = R0 {Xi}i∈I . For n ≥ 0, we define the n-skeleton of R to be the DG R0-subalgebra generated by the variables Xi of degree ≤ n and their divided powers (for variables of even degree 0). We denote it by R(n). Thus R(0) = R0, and if A → B is a surjective ring homomorphism with kernel I and R a free DG resolution of the A-algebra B with R0 = A, then R(1) is the Koszul complex associated to a set of generators of I. Lemma 1.2.10 Let A be a ring and B an A-algebra. Let A → S R → B be a commutative diagram of DG A-algebra homomorphisms, where S is a free DG resolution of the S0-algebra B and R is a DG A-algebra that is free over A. Then there exists a DG A-algebra homomorphism R → S that makes the whole diagram commute. Proof Let R(n) be the n-skeleton of R. Assume by induction that we have defined a homomorphism of DG A-algebras R(n − 1) → S so that the associated diagram commutes. We extend it to a DG A-algebra homomorphism R(n) → S keeping the commutativity of the diagram. a) If n = 0, R(0) = R0 and R0 → S0 exists because R0 is a polyno- mial A-algebra.
- 23. 10 Definition and first properties of (co-)homology modules b) If n is odd, let R(n) = R(n−1) {Ti}i∈I . We have a commutative diagram R(n − 1)n ⊕ i∈I R0Ti R(n − 1)n−1 R(n − 1)n−2 R(n)n − − − −→ R(n)n−1 − − − −→ R(n)n−2 R(n − 1)n @ @ R Sn − − − −→ Sn−1 − − − −→ Sn−2 and therefore a homomorphism R(n)n → ker(Sn−1 → Sn−2) = im(Sn → Sn−1), and so there exist an R0-module homomorphism R(n)n → Sn extending R(n − 1)n → Sn. By multiplicativity using the map R(n)n → Sn, we extend R(n − 1) → S to a homo- morphism of DG A-algebras R(n) → S. c) For even n ≥ 2, suppose that R(n) = R(n − 1) {Xi}i∈I . As above, we define R(n)n → Sn and then extend it to R(n) → S by multiplicativity using divided power rules based on the binomial and multinomial theorems. In more detail, suppose the map R(n)n → Sn is defined by Xi → v t=1 atY (rt,1) 1 · · · Y (rt,m) m ∈ Sn, where the at are coefficients in S0, the Yi are variables with deg Yi 0, and the divided powers Y (rt,j ) j have integer exponents rt,j ≥ 0. (Of course, for deg Yj odd and r 1, we understand Y (r) j = 0.) Then for l 0, the image of X (l) i is determined by the familiar divided power rules† (a) (Y1 + · · · + Yv)(l) = α1+···+αv=l α1,...,αv≥0 Y (α1) 1 · · · Y (αv) v ; and (b) (Y1Y2)(l) = Y l 1 Y (l) 2 (if deg Y1 and deg Y2 ≥ 2 are even). Thus R(n) → Sn is given by X (l) i → α1+···+αv=l α1,...,αv≥0 v t=1 aαt t (Y (rt,1) 1 )αt · · · (Y (rt,m) m )αt αt! , † Both are justified by observing that the two sides agree on multiplying by l!.
- 24. 1.3 Second definition 11 where the monomial (Y (rt,1) 1 )αt · · · (Y (rt,m) m )αt αt! equals • 1 if αt = 0; • 0 if αt ≥ 2 and rt,j = 0 for every j with deg Yj even positive; • (rt,j αt)! αt!(rt,j !)αt × (Y (rt,1) 1 )αt · · · Y (rt,j αt) j · · · (Y (rt,m) m )αt if αt = 1, or if for some j deg Yj is even and positive and rt,jαt ≥ 1; note that the coefficient (rt,j αt)! αt!(rt,j !)αt is an integer. Using the formula Y (p) i Y (q) i = (p+q!) p!q! Y (p+q) i , we see that (Y (rt,i) i )αt = (rt,iαt)! (rt,i!)αt Y (rt,iαt) i , and so this definition does not depend on the chosen j. A straightforward computation (easier if we multiply “formally” by p!q!), shows that under this map, X (p) i X (q) i and (p+q)! p!q! X (p+q) i have the same image. Remarks i) The assumption that S is free over S0 is only used to avoid defin- ing divided powers structure. ii) For the definition of Hn(A, B, M), for n = 0, 1, 2, we use free DG resolutions only up to degree 3, and so we could have used sym- metric powers resolutions instead of divided powers resolutions (since they agree in degrees ≤3). However, in Chapter 4 we use minimal resolutions and there we need divided powers. 1.3 Second definition Definition 1.3.1 Let A → B be a ring homomorphism. Let e: R → B be a free DG resolution of the A-algebra B. Let J = ker(R ⊗A B → B, x ⊗ b → e(x)b). Let J(2) be the graded R0⊗A B-submodule of R ⊗A B generated by the products of the elements of J and the divided powers X(m) , m 1 of variables of J of even degree ≥ 2. Note that J(2) is a subcomplex of R0 ⊗A B-modules of J. We define the complex ΩR|A ⊗R B := J/J(2) , which is in fact a complex of B-modules. In degree 0 it is isomorphic to ΩR0|A ⊗R0 B, where ΩR0|A is the usual
- 25. 12 Definition and first properties of (co-)homology modules R0-module of differentials of the A-algebra R0. For, we have an exact sequence of R0-modules defined by the multiplication of R0 (considering R0 ⊗A R0 as an R0-module multiplying in the right factor) 0 → I → R0 ⊗A R0 → R0 → 0, which splits, and so applying − ⊗R0 B we obtain an exact sequence 0 → I ⊗R0 B → R0 ⊗A B → B → 0, showing that I ⊗R0 B = J0. On the other hand, the exact sequence of R0-modules 0 → I2 → I → ΩR0|A → 0 gives an exact sequence I2 ⊗R0 B = (I ⊗R0 B)2 = J2 0 → I ⊗R0 B = J0 → ΩR0|A ⊗R0 B → 0, and therefore J0/J (2) 0 = J0/J2 0 = ΩR0|A ⊗R0 B. In degree 1, (J/J(2) )1 = J1/J0J1 = (R1 ⊗A B)/J0(R1 ⊗A B) = (R1 ⊗A B)⊗R0⊗AB B = R1 ⊗R0 B is the free B-module obtained by base extension of the free R0-module R1. Similarly, in degree 2, (J/J(2) )2 = J2/(J0J2 + J2 1 ) = (R2/R2 1) ⊗R0 B. In general, for n 0, (ΩR|A ⊗R B)n = (R(n)/R(n − 1))n ⊗R0 B. Definition 1.3.2 We say that an A-algebra P has property (L) if for any A-algebra Q, any Q-module M, any Q-module homomorphism u: M → Q such that u(x)y = u(y)x for all x, y ∈ M, and for any pair of A-algebra homomorphisms f, g: P → Q such that im(f − g) ⊂ im(u), there exists a biderivation λ: P → M such that uλ = f − g P λ f g M u − − → Q. Here we say that λ is a biderivation to mean that λ is A-linear and λ(xy) = f(x)λ(y) + g(y)λ(x). Lemma 1.3.3 Let A be a ring, P an A-algebra. a) If P is a polynomial A-algebra, then P has property (L). b) If P has property (L) and S is a multiplicative subset of P, then S−1 P has property (L).
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- 27. repressive laws revived, 108 ; when subject to inquisitorial jurisdiction, 141 ; shatter a crucifix, 267 ; popular feeling against, 356 ; finance war of Granada, 356 ; their expulsion urged by Torquemada, 357 ; they plead with the Sovereigns, 358 ; Dominicans preach against them, 359; letterof, 361 ; calumniated, 363 ; appeals of, 365 ; banished, ^67 et seq. ; exploited, 368 ; attempts to convert them, 369 ; encouraged by their rabbis, 370 ; exodus from Spain, 371 ; their sufferings, 372 ; apostates, 373 Juan, Prince — illness of, 359 Judaizers — 93 ; discovered. 10 1 : in Seville, 109, in edict of grace to, 120; trapped, 121 ; signs by which known, 121 et seq. ; seek absolution in Rome, 132 ; number convicted in Toledo, 256 ; Auto of in Rome, 391 Lachaves, Juan Gutierrez de — appointed assessor, 136 ; councillor of the Suprema, 137 La Gardia, The Holy Child of — crucn d, 269 ; legend of, 271 et seq. ; Testimonio quoted, 276 ; paternity of, 329 ; why crucified, 337 ; evidence considered, 346 et seq. ; discrepancies in evidence, 3 50 et seq. ; an operation in magic, 353 ,* worship of, 354 La Gardia, Sacristan of — arrested, 346 Lea, H. C. — on solicitation, 172 Lecky, W. E. H. — on persecution, 9 Llorente, J. A. — sketch of career, 6 et seq. ; on ritual murder, 78 ; on blood-lust of inquisitors, 117 ;
- 28. Index 401 on Quemadero, 127 ; on Torquemada, 136; on solicitation, 171 ; on trials in Zaragoza, 225 ; on case of Aranda, 381 ; on false witnesses, 388 Loeb, Isidore — his theory on the affair of La Gardia, 319, 348 Maldonado, Alonso — conspires against Cabrera, 61 Manrique, Gomez — arrests Toledo conspirators, 241 Manrique, Inigo — appointed to assist Torquemada, 383 Marin^eus, Lucius — on Isabella's reforms, 69 Martin, Alonso, reputed father of Santo Nino, 329 Martinez, Hernando, Canon of Ecija, denounces Jews, 82 ; defies authority, 83 ; causes massacre in Seville, 84 Medina, Juan Ruiz de — 109 Medina Sidonia, Duke of — NewChristians shelter in his dominions, 112 Medina, Tristan de — appointed assessor, 1 36 ; councillor of the Suprema, 137 Mendoza, Pedro Gonzalez de — Primate of Spain, 97 ; entrusted with conversion of Jews, 99 ; introduction of Inquisition ascribed to, 100 ; delegated to appoint inquisitors in Castile, 109 ; . instrumental in the proclamation of the edict of grace, 120 Mendoza, Salazar de — on foundation of Kingdom of Spain, 72 ; ascribes introduction of Inquisition to Cardinal Mendoza, 100 Merlo, Diego de — charged with conversion of Jews, 107 Mili tia Chris ti — 2 2 7 Monterubio, Fr. Pedro de — sent to Zaragoza, 221 Montfort, Simon de — 33 Moors — see Moslem Moreno, Martinez — his Historia del Santo Niilo, 269 ; on miracles of Nifto, 355 Morillo, Fr. Miguel — inquisitor in Seville, 109 ; vindictive procedure of, 116; his hatred of the Jews, 126; Pope protests against his rigour, 128; confirmed in office by Torquemada, 136 Moriscoes — immunity enjoyed by, 376 26 Moslem — in Peninsula, 89 ; banished, 375 ; in Granada, 376 Negat/vos — 194 ; deemed impenitent, 197 Nero — persecutes Christians, 19 New-Christians — 87 ; objects of malevolence, 93 ; in offices of eminence, 94 ; fly from Seville, 112 ; terrorized, 114 ; their peril, 125 ; seek refuge in Rome, 128 ; complain to Pope, 129 ; in Aragon, 215 ; appeal against tribunal of Zaragoza, 216 ; their despair, 217 ; their panic in Zaragoza, 223 ; seek secret absolutions, 257 ; swindled, 258 NicjEA — Council of, 23 Ocana, Juan de — incriminated by Benito Garcia, 284 ; arrested, 286 ; incriminated by Yuce Franco, 318; tortured, 324 ; confrontation of, 327 ; further
- 29. incriminates Yuce and Ca Franco, 329 33° ' further admissions of, 341 ; burnt, 344 Ojeda, Fr. Alonso de — urges establishment of Inquisition, 93 ; resisted by Isabella, 97 ; renews efforts, 98 ; supplied with fresh argument, 10 1 ; charged with conversion of Jews, 107 ; at burning of Susan, 117; dies of plague, 1 1 8 Optatus — urges massacre of the Donatists, 25 Orozco, Sebastian de — 239 ; on plot in Toledo, 241 ; on first Auto de Fe in Toledo, 244 Ortega, Juan — organizes Hermandad, 56 Pantigoso, Juan de — Yuc6 Franco's advocate, 297 Paramo, Ludovicus X — on source of Inquisition, 17 ; ascribes to Mendoza introduction of Inquisition to Castile, 100 Pecuniary Penances, 150 Pegna, Francesco, the scholiast, 143 ; on canonical purgation, 160 ; on children of heretics, 164 ; on examination of accused, 173 ; enjoins guile, 174 et seq. ; his honesty, 180; on torture, 185; on execution of innocent men, 197 ; on formal intercession, 204 ; on Auto de Fe, 205 Pelagius — heresy of, 24
- 30. 402 Index Penitentiaries — ordered by Torquemada, 237 Perejon, David — in affair of La Gardia, 318, 325 Pico della Mirandola, Giovanni — eludes Inquisition, 264 Pius IX, Pope — canonizes Arbue9, 230 Priscillian — burnt, 27 Pulgar, Hernando del — on state of Castile, 53 ; on Isabella's reforms, 69 ; on judaizing, 71 ; a New-Christian, 94 ; on Mendoza's catechism, 100 Quemadero — built, 127 ; demolished by Bonaparte's soldiers, 128 Quintanilla, Alonso de — Isabella's chancellor, 56 Raymond of Toulouse — 33 Relapsos — 149, 194 ; denned, 198 Riario, Raffaele, 67 Ribera, Hernando de — in affair of La Gardia, 291, 326; convicted, 347 Rios, Amador de los — on first appearance of Jews in Spain, 73 ; on Jewish community in thirteenth century, 75 ; on ritual murder, 80 ; on Susan's daughter 115; on banishment of Jews, 369 Ritual Murder — charges of, 78 et seq. Rodrigo, F. J. Garcia — 8 ; on Susan's conspiracy, 116; on Quemadero, 128 ; on torture, 187; on prisons, 263 ; on fanaticism, 393 Rule, Dr. W. H.— -8, 31 ; on Quemadero, 128 St. Hilaire, Rosseeuw— on Torquemada, 6 ; on Isabella's reforms, 69 St. Peter the Martyr — Confraternity of, 117, 227 Sanbenito — revived by Torque mada, 149 ; its origin and history, 206 et seq. ; considered salutary by Torquemada, 209 ; its various forms, 209 ; preserved after Autos de Fe, 255 Sanc — Yuc6 Franco's attorney, 297 ; abandons case, 341 Sanchez de la Fuente, Francisco — appointed assistant to Torquemada, 383 Sanchez, Guillerme — procures his brother's release, 226 ; arrested, 227 Sanchez, Juan Pedro — conspires against Inquisition, 217 ; burnt in effigy, 222 ; arrested in Toulouse, 226 ; released, 226 ; his befrienders arrested, 227 San Martino, Fr. Juan de — inquisitor in Seville, 109 ; vindictive procedure of, 116 ; hatred of Jews, 126 ; Pope protests against rigour of, 128 ; confirmed in office by Torquemada, 136 Santa Cruz, Gaspar de — escapes to Toulouse, 228 ; amends imposed upon his son, 228 Santangel, Luis de — conspires against Inquisition, 217 ; arrested, 221 Santiago — Knights of, 59 ; GrandMastership of, 60 Santillana, Francisco de — 106 Santo Domingo, Fr. Fernando de — delegated to try affair of La Gardia, 289 ; at Auto de Fe, 343 Santo Nino — see La Gardia, Holy Child of Sauli, Manuel — conspires, 115;
- 31. burnt, 116 Secret Absolutions — 257 ; bulls of, 251 Secular Arm — euphemistic expression, 194 ; abandonment to, 204 Segovia — riots in, 6b Seneor, Abraham — 365 Seville — visited by Isabella, 63 judaizing in, 109, in ; Inquisition established in, 1 14 et seq. ; first burnings in, 118 ; numerous arrests in, 119 ; number burnt in, 127 ; permanent tribunal established in by Torquemada, 1 36 Siliceo, Cardinal Juan Martinez — discovers Jewish letter, 361 Sixtus IV, Pope — opposed by Isabella, 67 ; orders Inquisition, 89 ; grants bull for establishment of Inquisition in Castile, 107 ; protests against rigour of Seville inquisitors, 128 ; revokes right of Sovereigns to appoint inquisitors, 129 ; appoints inquisitors, 131 ; letter of to Isabella, 133 Solares, Alfonso, 380 Solicitation — sin of, 169 Solis, Alonso de — charged with conversion of Jews, 107 Suarez de Fuentelsaz, Alonso — appointed assistant to Torque
- 32. Index 403 mada, 383 ; virtually supersedes Torquemada, 384 Suprema, Council of — 137 Susan, Diego de — conspiracy of, 114 ; betrayed by his daughter, 115 ; burnt, n6etseq. Tablada — meadows of, 118; permanent burning platform erected there, 127 Tazarte, Yuce — procures consecrated wafer, 306 ; enchantment performed by, 308 ; his sorceries examined, 320 Teruel — in revolt, 231 Toledo — tribunal established in, 136, 239; plot against Inquisition in, 240 ; activity of Inquisition in, 243 ; first Auto de F6 in, 244 ; second Auto in 246 ; secular arm, 247 ; burning-place of, 251 ; further Autos in, 252 et seq. ; Judaizers convicted in, 256 Torquemada, Fr. Juan de (Cardinal of San Sisto) — 94, 104 Torquemada, Lope Alonso de — 104 Torquemada, Pero Fernandez de — 105 Torquemada, Fr. TomXs de — advocates Inquisition, 102 ; his name and family, 104 ; Prior of Santa Cruz, 105 ; Isabella's confessor, 105 ; influence with Isabella, 106 ; asceticism of, 106 ; withdraws to Segovia, 107 ; delegated to appoint inquisitors in Castile, 109 ; appointed inquisitor bv Pope, 131 ; created GrandInquisitor of Spain, 135 ; reconstitutes the Holy Office, 136 ; president of the Suprema, 137 ; assembles his subaltern inquisitors, 138 ; formulates his code, 142 ; the articles of his first instructions, 144 et seq. ; revives sanbenito, 149 and 209 ; decrees secrecy, 157; on prosecution of the dead, 161 ; seeks to extend inquisitorial jurisdiction, 168 ; on negativos, 197 ; on relapsos, 200 ; his power, 214 ; stirs Aragonese tribunal into activity, 215 ; convenes council at Tarragona, 216; delegates Arbues and Yuglar, 217 ; his action on murder of Arbues, 221 ; orders proclamation of Autos, 222 ; attempts to withstand papal authority, 225 ; resisted in Aragon, 231 ; his decrees of 1485, 233 ; ordered by Pope to re-edit his code of terror, 235 ; his decrees of 1488, 236 ; orders building of penitentiaries, 237 ; renders delation compulsory, 242; his fanatical hatred of Jews, 243 ; complaints of his rigour, 256 ; resents papal interference, 257 protests to Pope, 260 ; his wealth, 260 ; his character, 261 ; treatment of his sister, 261 ; builds Monastery of St. Thomas, 262 ; fanaticism of, 263 ; arrogance of, 264 ; violates equity, 266 ; urges expulsion of Jews, 268 ; accused of inventing
- 33. affair of La Gardia, 269 ; intends to direct trial of Y. Franco, 288 ; entrusts this to his delegates, 289 ; goes to Andalusia, 292 ; in connection with affair of La Gardia, 353 ; exploits the affair, 354, 356 ; advocates banishment of Jews, 357, 363 ; purity of his aims, 364 ; rebukes Sovereigns, 366 ; desires conversion of Jews, 369 ; irresistible, 374 ; his service to Spain, 376 ; confirmed in office by Alexander VI., 377 ; protests against papal briefs, 378 ; his enemies increasing, ib. ; ascendancy of , 379; prosecutes bishops, 380 ; appeals to Pope against him, 382; his power curtailed, 383 ; virtually superseded, 384 ; crippled by gout, 385 ; last instructions of, 386 et seq. ; his death, 392 ; his epitaph, 394 Torralba, Bartolome — conspires, 115 ; burnt, 116 Torre, De la — conspires, 240 ; arrested, 241 Torrejoncillo, Fr. Francisco de — scurrilous publication of, ^ 36b Torture — by inquisitors, 155 ; when employed, 184 et seq. ; the five degrees of, 188 ; engines employed, 189 et seq. ; ratification of confession, 192 Trasmiera, Diego Garcia de — in praise of secrecy, 157; on Mercy and Justice, 211 ; on murder of Arbues, 221 ; on Autos de Fe, 222 Triana, Castle of — prison of the Inquisition, 119
- 34. 4°4 Index Uranso, Vidal de — conspires against Inquisition, 218 ; murders Arbues, 219 ; put to torture, 221 ; his confession betrays all sympathizers, 222 Val, Domingo de — crucified by Jews, 78 Valencia — resists Inquisition, 231 ; attempted crucifixion in, 360 Valencia, Poncio de — councillor of Suprema, 137 Valencia, Captain- General of — humiliated, 264 Valerian — 21 Vaudois — see Waldenses Vazquez, Martin — Yuce Franco's advocate, 297 Vegas, Damiano de — his Memoria of the Santo Nino, 269 V ERGUENZA 244 Villada, Dr. Pedro de — Provisor of Astorga, 282 ; examines Benito Garcia, 283 ; delegated to try affair of La Gardia, 289 ; visits Yuce Franco in prison, 306 ; enjoins Yuce Franco to make full confession, 316 ; at Auto de Fe, 343 Villa Real — tribunal established in by Torquemada, 1 36 Vincent Ferrer, St. — converts Jews, 85 Voltaire — on Auto de Fe, 201 Waldenses — 32 Wendland, P. — on ritual murder, 80 XlMENES DE ClSNEROS, FRANCISCO —385 Yusuf Ben Techufin — defeats Christians, 52 Zamarra — see Sanbenito Zaragoza — Inquisition established in, 216 ; first Auto held in, 217 ; riot in, 220 ; Autos during i486 in 222 ; reign of terror in, 223 Zosimus, Pope — banishes Pelagius, 24 Primed in Great Britain by Hazell, Watson A Vinet/, Ld^ London and Ayleabury.
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