![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/88749-250411023245-f32ed805/85/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/88749-250411023245-f32ed805/85/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/88749-250411023245-f32ed805/85/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/88749-250411023245-f32ed805/85/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. meet with a single specimen of the tall candelabra-formed cactuses (Cerei), so common on those high grass-plains, that is not loaded with their weight. In spite of their working in the dark, in spite of their subterranean tunnels, their strongholds, and the fecundity of their queens, the termites, even when their swarms do not expose themselves to the dangers already mentioned, are subject to the attacks of innumerable foes—ant-eaters, birds, and a whole host of insects—that do man no little service by keeping them within bounds. One of their most ferocious enemies is a species of black ant, which, on the principle of setting one thief to catch another, is used by the negroes of Mauritius for their destruction. When they perceive that the covered ways of the termites are approaching a building, they drop a train of syrup as far as the nearest encampment of the hostile army. Some of the black ants, attracted by the smell and taste of their favourite food, follow its traces and soon find out the termite habitations. Immediately part of them return to announce the welcome intelligence, and after a few hours a black army, in endless columns, is seen to advance against the white-ant stronghold. With irresistible fury (for the poor termites are no match for their poisonous sting and mighty mandibles) they rush into the galleries, and only retreat after the extirpation of the colony. Mr. Baxter (‘Eight Years’ Wanderings in Ceylon’) once saw an army of black ants returning from one of these expeditions. Each little warrior bore a slaughtered termite in his mandibles, rejoicing no doubt in the prospect of a quiet dinner-party at home. Even man is a great consumer of termites, and they are esteemed a delicacy by the natives, both in the old and in the new world. In some parts of the East Indies the people have an ingenious way of emptying a termite-hill, by making two holes in it, one to the windward and the other to the leeward, placing at the latter opening a pot rubbed with an aromatic herb to receive the insects, when driven out of their nest by the smoke of a fire made at the former
- 28. breach. In South Africa the general way of catching them is to dig into the ant-hill, and when the builders come forth to repair the damage, to brush them off quickly into the vessel, as the ant-eater does into his mouth. They are then parched in iron pots over a gentle fire, stirring them about as is done in roasting coffee, and eaten by handfuls, without sauce or any other addition, as we do comfits. According to Smeathman, they resemble in taste sugared cream, or sweet almond paste, and are, at the same time, so nutritious that the Hindoos use them as a restorative for debilitated patients. While most termites live and work entirely under covered galleries, the marching white ant (T. viarum) exposes itself to the day. Smeathman, on one occasion, while passing through a dense forest, suddenly heard a loud hiss like that of a serpent; another followed, and struck him with alarm; but a moment’s reflection led him to conclude that these sounds proceeded from white ants, although he could not see any of their huts around. On following this noise, however, he was struck with surprise and pleasure at perceiving an army of these creatures emerging from a hole in the ground, and marching with the utmost swiftness. Having proceeded about a yard, this immense host divided into two columns, chiefly composed of labourers, about fifteen abreast, following each other in close order, and going straight forward. Here and there was seen a soldier, carrying his vast head with apparent difficulty, at a distance of a foot or two from the columns; many other soldiers were to be seen, standing still or passing about, as if upon the look-out lest some enemy should suddenly surprise their unwarlike comrades. But the most extraordinary and amusing part of the scene was exhibited by some other soldiers, who having mounted some plants, ten or fifteen inches from the ground, hung over the army marching below, and by striking their jaws upon the leaves at certain intervals, produced the noise above mentioned; to this signal the whole army immediately returned a hiss and increased their pace. The soldiers at these signal-stations sat quite still during these intervals of silence, except now and then making a slight turn of the head, and seemed
- 29. as solicitous to keep their posts as regular sentinels. After marching separately for twelve or fifteen paces, the two columns of this army again united, and then descended into the earth by two or three holes. Mr. Smeathman watched them for more than an hour, without perceiving their numbers to increase or diminish. Both the labourers and soldiers of this species are furnished with eyes. One of the many unsolved mysteries of termite life is whence they derive the large supplies of moisture with which they not only temper the clay for the construction of their long covered ways above ground, but keep their passages uniformly damp and cool below the surface. Yet their habits in this particular are unvarying, in the seasons of drought as well as after rain; in the most arid positions; in situations inaccessible to drainage from above, and cut off by rocks and impervious strata from springs from below. Struck with this wonderful phenomenon, Dr. Livingstone raises the question whether the termites may not possess the power of combining the oxygen or hydrogen of their vegetable food by vital force, so as to form water; and indeed it is highly probable that they are endowed with some such faculty, which, however wonderful, would still be far less astonishing than the miracles of their architectural instinct. After having described the miseries which the tropical insects inflict upon man—how they suck his blood, destroy his rest, exterminate his cattle, devour the fruits of his fields and orchards, ransack his chests and wardrobes, feast on his provisions, and plague and worry him wherever they can—it is but justice to mention their services. Among the insects which are of direct use to us, the silk-worm (Bombyx mori) is by far the most important. Originally a native of tropical or sub-tropical China, where the art of making use of its filaments seems to have been discovered at a very early period, it is now reared in countless numbers far and wide over the western world, so as to form a most important feature in the industrial resources of Europe. Thousands of skilful workmen are employed in spinning and weaving its lustrous threads, and thousands upon
- 30. thousands, enjoying the fruits of their labours, now clothe themselves, at a moderate price, in silken tissues which but a few centuries back were the exclusive luxury of the richest and noblest of the land. Besides the silk-worm, we find many other moths in the tropical zone whose cocoons might advantageously be spun, and only require to be better known to become considerable articles of commerce. The tusseh-worm (Bombyx mylitta) of Hindostan, which lives upon the leaves of the Rhamnus jujuba furnishes a dark- coloured, coarse, but durable silk; while the Arandi (B. cynthia), which feeds upon the foliage of the castor-oil plant (Ricinus communis), spins remarkably soft threads, which serve the Hindoos to weave tissues of uncommon strength. In America, there are also many indigenous moths whose filaments might be rendered serviceable to man, and which seem destined to great future importance, when trade, quitting her usual routine, shall have learnt to pry more closely into the resources of Nature. While the Cocci, or plant bugs, are in our country deservedly detested as a nuisance, destroying the beauty of many of our garden plants by their blighting presence, two tropical members of the family, as if to make up for the misdeeds of their relations, furnish us—the one with the most splendid of all scarlet dyes, and the other with gumlac, a substance of hardly inferior value. The English gardener spares no trouble to protect his hot- and greenhouse plants from the invasion of the Coccus hesperidum; but the Mexican haciendero purposely lays out his Nopal plantations that they may be preyed upon by the Coccus cacti, and rejoices when he sees the leaves of his opuntias thickly strewn with this valuable parasite. The female, who from her form and habits might not unaptly be called the tortoise of the insect world, is much larger than the winged male, and of a dark-brown colour, with two light spots on the back, covered with a white powder. She uses her little legs only during her first youth, but soon she sucks herself fast, and
- 31. COCHINEAL. henceforward remains immovably attached to the spot she has chosen, while her mate continues to lead a wandering life. While thus fixed like an oyster, she swells or grows to such a size that she looks more like a seed or berry than an insect; and her legs, antennæ, and proboscis, concealed by the expanding body, can hardly be distinguished by the naked eye. Great care is taken to kill the insects before the young escape from the eggs, as they have then the greatest weight, and are most impregnated with colouring matter. They are detached by a blunt knife dipped in boiling water to kill them, and then dried in the sun, when they have the appearance of small, dry, shrivelled berries, of a deep-brown purple or mulberry colour, with a white matter between the wrinkles. The collecting takes place three times a year in the plantations, where the insect, improved by human care, is nearly twice as large as the wild coccus, which in Mexico is gathered six times in the same period. Although the collecting of the cochineal is exceedingly tedious—about 70,000 insects going to a single pound—yet, considering the high price of the article, its rearing would be very lucrative, if both the insect and the plant it feeds upon were not liable to the ravages of many diseases, and the attacks of numerous enemies. The conquest of Mexico by Cortez first made the Spaniards acquainted with cochineal. They soon learnt to value it as one of the most important products of their new empire, and in order to secure its monopoly, prohibited, under pain of death, the exportation of the insect, and of the equally indigenous Nopal, or Cactus cochinellifer, supposing it not to be able to live upon any other plant. In the year 1677, however, Thierry de Meronville, a Frenchman, made an effort to deprive them of the exclusive possession of the treasure they guarded with such jealous care. Under a thousand dangers, and by means of lavish bribery, he succeeded in transporting some
- 32. of the plants, along with their costly parasite, to the French colony of San Domingo; but, unfortunately, his perseverance did not lead to any favourable results, and more than a century elapsed after this first ineffectual attempt before the rearing of cochineal extended beyond its original limits. In the year 1827, M. Berthelot, director of the botanical garden at Orotava, was more fortunate in introducing it into the Canary Islands, where it thrives so well upon the Opuntia Ficus indica, that Teneriffe rivals Mexico in its production. At present Cochineal is not only raised in many other parts of the tropical world, but even in Spain, near Valencia and Malaga. The Coccus which produces lac, or gumlac, is a native of India, and thrives and multiplies best on several species of the fig-tree. A cheap method having been discovered within the last few years of separating the colouring matter which it contains from the resinous part, it has greatly increased in commercial importance. In the tropical zone we find that not only many birds and several four-footed animals live chiefly, or even exclusively, on insects, but that they are even consumed in large quantities, or eaten as delicacies, by man himself. The nomade of the Sahara and the South African bushman hail the appearance of locust swarms as a season of plenty and good living, and ants’ eggs eke out the meagre bill of fare of the wild Indians on the banks of the Orinoco. Several of the large African caterpillars are edible, and considered as a great delicacy by the natives. On the leaves of the Mopané tree, in the Bushman country, the small larvæ of a winged insect, a species of Psylla, appear covered over with a sweet gummy substance, which is collected by the people in great quantities, and used as food. Another species in New Holland, found on the leaves of the Eucalyptus, emits a similar secretion, which, along with its insect originator, is scraped off the leaves and eaten by the aborigines as a saccharine dainty.
- 33. DIAMOND BEETLE. BUPRESTIS GIGAS. The chirping Cicadæ, or frog-hoppers, which Aristotle mentions as delicious food, are still in high repute among the American Indians; and the Chinese, who allow nothing edible to go to waste, after unravelling the cocoon of the silkworm, make a dish of the pupæ, which the Europeans reject with scorn. The Goliath beetles of the coast of Guinea are roasted and eaten by the natives, who doubtless, like many other savages, not knowing the value of that which they are eating, often make a bonne bouche of what an entomologist would most eagerly desire to preserve. Several of the more brilliant tropical beetles are made use of as ornaments, not only by the savage tribes, but among nations which are able to command the costliest gems of the East. The golden elytra of the Sternocera chrysis and Sternocera sternicornis serve to enrich the embroidery of the Indian zenana, while the joints of the legs are strung on silken threads, and form bracelets of singular brilliancy. The ladies in Brazil wear necklaces composed of the azure green and golden wings of lustrous Chrysomelidæ and Curculionidæ, particularly of the Diamond beetle (Entimus nobilis); and in Jamaica, the elytra of the Buprestis gigas are set in ear-rings, whose gold- green brilliancy rivals the rare and costly Chrysopras in beauty.
- 35. MALAY PIRATES.
- 36. U CHAPTER XX. THE MALAYAN RACE. Physical Conformation of the Malays—Betel Chewing— Their Moral Character—Limited Intelligence of the Malays—Their Maritime Tastes—Piracy—Gambling— Cock-fighting—Running A-muck!—Fishing—Malayan Superstitions—The Battas—Their Cannibalism— Eating a Man alive—The Begus—Aërial Huts— Funeral Ceremonies—The Dyaks—Head-hunting— The Sumpitan—Large Houses. nlike the apathetic Indian hunter, whose wishes are bounded by the forest or the savannah, where the chase provides him with a scanty subsistence, or the good-humoured Negro who, fond of agriculture, and attached to the soil on which he was born, never thinks of wandering of his own free will to distant countries, the roving race of the Malays has scattered its colonies far and wide over the Indian Archipelago. The colour of the various tribes of this remarkable people is a yellowish-brown, and varies but little throughout the numerous islands extending from Sumatra and the peninsula of Malacca to the Moluccas. The hair is black, coarse and straight, the beard scanty. The stature is below the average European size, the breast well developed, the limbs meagre. The face is broad and somewhat flat, with high cheek-bones, a small nose, a large mouth with broad lips,
- 37. and black eyes with angular orbits. The children and young people of both sexes are often really handsome in face and graceful in figure, but as they advance in age their features become hard, and frequently present a repulsive appearance. Like most nations in a rude state of society, they are in the habit of permanently disfiguring parts of the body under the idea of ornament. Considering blackness a becoming colour for the teeth— for dogs, they say, have them white—they file the enamel so that the bone may be tinged by the juice of the pungent betel, which, wrapped round the nut of the areca palm, and mixed with lime, they are in the habit of chewing from morning till night. This combination, besides discolouring the teeth, has the disgusting property of dyeing the saliva of so deep a red that the lips and gums appear as if coloured with blood; yet it is in universal use throughout the whole Indian Archipelago, and, as excuses are never failing to justify bad habits, is said to have tonic effects and to promote digestion. The Malays are not a demonstrative people; their behaviour towards strangers is marked by a reserve, a distrust, or even a timidity which inclines the observer to tax with exaggeration the wild and bloodthirsty character which is generally ascribed to their race. The feelings of astonishment, admiration, and fear are never openly expressed, and their slow and considerate speech shows how careful they are not to give offence. To indulge in a joke is quite contrary to their natural disposition, and they deeply feel, and are ever ready to resent, a breach of etiquette or a personal affront. The higher classes are extremely polite, and have all the quiet manners and dignity of the best educated Europeans. But this external polish is united with a reckless cruelty and contempt for human life which forms the dark side of their character. Hence it is not to be wondered at that different authors give us such totally contradictory accounts of them. An old traveller, Nicolo Conti, who wrote in 1430, says that ‘the inhabitants of Java and Sumatra surpass all other people in cruelty,’ while Drake praises their love of truth and justice. Mr. Crawfurd
- 38. describes the Javanese as a peaceable and industrious people, but Barbosa, who visited Malacca about the year 1660, informs us that they are extremely cunning and great cheats; that they seldom speak the truth, and are ever ready for a villanous deed. Their intelligence seems to be incapable of any higher flight. They comprehend nothing which goes beyond the simplest combination of ideas, and have little taste and energy to obtain an increase of knowledge. The civilisation they possess shows no traces of original growth, but is entirely confined to those nations or states which have adopted the Mahometan religion, or in still earlier times received their culture from India. It must, however, be remarked in their favour that the curse both of domestic tyranny and of a foreign yoke weighs heavily upon them, and that the extension of European domination in the Indian Ocean has been as fatal to their race as it has been in America and Africa to the Red-skin and the Negro. ‘The first voyagers from the west,’ says Rajah Brooke, ‘found the natives rich and powerful, with strong established governments, a flourishing literature, and a thriving trade with all parts of the eastern world. The rapacious European has reduced them to their present abject condition. Their governments have been broken up; the old states decomposed by treachery, by bribery, and intrigue; their possessions wrested from them under flimsy pretences, their trade restricted, their vices encouraged, and their virtues repressed.’ ‘Among the Malays of the present day,’ says Newbold, ‘we look in vain for that desire of knowledge which excited their ancestors to transplant the flowers of Arabian literature among their own forests. Works of science are now no longer translated from the Arabic, and creations of the imagination have almost ceased to appear. The few children educated among them learn nothing but to mumble in an unknown tongue a few passages from the Koran, entirely neglecting arithmetic and the acquirement of any useful manual art or employment. Painting, sculpture, architecture, mechanics, geography, are totally unknown to the Malays. Their literature
- 39. declined with the fall of their empire in the Archipelago, nor could it well be expected to flourish under the Upas trees of Portuguese intolerance, Dutch oppression, and British apathy.’ Essentially maritime in their tastes, the Malays have been named the Phœnicians of the East; but not satisfied with the peaceful pursuits of the fisherman or the merchant, many of them infest the Indian Ocean as merciless pirates. Encouraged by the weakness and distraction of the old- established Malay governments, the facilities offered by natural situation, and the total absence of all restraint from European nations, except now and then the destruction of some mud fort or bamboo-village, which is soon rebuilt, the Illanuns, the Balagnini, and other sea-robbing tribes, issue forth like beasts of prey, enslave or murder the inhabitants on the coasts or at the entrance of rivers, and attack ill-armed or stranded European vessels. The Illanuns of Mindanao are particularly noted for their daring and long-protracted piratical excursions, which they undertake in large junks with sails, netting, and heavy guns. On one occasion the ‘Rajah Brooke’ met eighteen Illanun boats on neutral ground, and learned from their two chiefs that they had been two years absent from home; and from the Papuan slaves on board it was evident that their cruise had extended from the most eastern islands of the Archipelago to the north-western coast of Borneo. The Balagnini inhabit a cluster of small islands in the vicinity of Sooloo, where they probably find encouragement and a slave market. They cruise in large prahus, and to each of these a fleet boat or ‘sampan’ is attached, which on occasion can carry from ten to fifteen men. They seldom have large guns like the Illanuns, but, in addition to their other arms, brass pieces, carrying from a one- to a three-pound ball. They use long poles with barbed iron points, with which, during an engagement or flight, they hook their prey. By means of their sampans they are able to capture all small boats; and it is a favourite device with them to disguise one or two men, whilst the rest lie concealed in the bottom of the boat, and thus to surprise
- 40. prahus at sea, and fishermen or others at the mouths of rivers. Their cruising grounds are very extensive; they frequently make the circuit of Borneo; Gillolo and the Moluccas lie within their range, and it is probable that Papua is occasionally visited by them. The Borneans, from being so harassed by these freebooters, who yearly take a considerable number of this unwarlike people into slavery, call the easterly monsoon ‘the pirate wind.’ Their own native governments are probably without exception participators in or victims to piracy, and in many cases both—purchasing from one set of pirates and enslaved and plundered by another; and whilst their dependencies are abandoned, the unprotected trade goes to ruin. Thus piracy rests like a blighting curse upon lands pre-eminently blessed by Nature, and proves as ruinous to the welfare of the Eastern Archipelago as the black stain of the African slave trade to that of the Negroes. The Malays are inveterate gamblers, and, perhaps for want of some nobler object on which to expend their mental energies, carry the mania of betting at cock-fights to a ruinous excess. Passionately addicted to this favourite amusement, they will lose all their property on a favourite bird, and having lost that, stake their families, and after the loss of wife and children, their own personal liberty, being prepared to serve as slaves in case of losing. Whole poems are devoted to enthusiastic descriptions of cock-fighting, which is regulated by universally acknowledged laws as minute as those of the Hoyleian Code. The birds are not trimmed as in England, but fight in full feather, armed with straight or curved artificial spurs, sharp as razors and about two and a half inches long. Large gashes are inflicted by these murderous instruments, and it rarely happens that both cocks survive the battle. One spur only is used, and is generally fastened near the natural spur on the inside of the left leg. Cocks of the same colour are seldom matched. The weight is adjusted by the setters-to, passing them to and from each other’s hands as they sit facing each other in the cock-pit. Should there be any difference, it is brought down to an equality by the spur being fixed so many scales higher
- 41. on the leg of the heavier cock, or as deemed fair by both parties. In adjusting these preliminaries the professional skill of the setters-to is called into action, and much time is taken up in grave deliberation, which often terminates in wrangling. The birds, after various methods of irritating them have been practised, are then set to. During the continuance of the battle the excitement and interest taken by the gambling spectators in the barbarous exhibition is vividly depicted in their animated looks and gestures. The Malays who are not slaves go always armed; they would think themselves disgraced if they went abroad without their crees or poniards, which, to render them more formidable, are often steeped in poison. These weapons, which thus afford them the ready means for avenging an affront, are probably the chief cause which renders their outward deportment to each other remarkably punctilious and courteous, but they sometimes become highly dangerous in the hands of a people whose nervous temperament is liable to sudden explosions of frantic rage. Like the old Berserks of the heroic ages of Scandinavia, a Malay is capable of so far working himself into fury, of so far yielding to some spontaneous impulse, or of so far exciting himself by stimulants, as to become totally regardless of what danger he exposes himself to. In this state, which is called ‘running a-muck,’ he rushes forth as an infuriated animal and attacks all who fall in his way, until he is either struck down like a wild beast, or having expended his morbid rage he falls down exhausted. The Malays are bad agriculturists and artisans but excellent sportsmen. From the small birds which they entangle in their snares to the large animals of the forest, which they shoot or entrap in pit- falls, or destroy by spring-guns, nothing worth catching escapes their attention. Such is their delight in fishing, that even women and children may be seen in numbers during the rains angling in the swampy rice grounds. Spearing excursions against the swordfish are undertaken during the dark of the moon by the light of torches. A good eye, a steady hand are necessary, and a perfect knowledge of the places where the fish are to be found. Each canoe carries a
- 42. steersman, a man with a long pole to propel the vessel, and a spearsman, who, armed with a long slender javelin having a head composed of the sharpened spikes of the Nibong palm, and holding in his left hand a large blazing torch, takes his station at the stern of the canoe. They thus glide slowly and noiselessly over the still surface of the clear water, till the rays of the flambeau either attract the prey to the surface or discover it lying seemingly asleep at a little depth below. The sudden splash of the swiftly descending spear is heard, and the fish is the next moment seen glittering in the air, either transfixed by the spikes or caught in the interstices as the weapon is withdrawn. As a natural consequence of their extreme ignorance, the Malays, even the best educated, are inordinately superstitious, and people the invisible world with a host of malignant spirits. The Pamburk roams the forest, like the wild huntsman of the Haruz, with demon dogs, and the storm fiend Hantu Ribut howls in the blast and revels in the whirlwind. Tigers are considered in many instances to be the receptacles of the souls of departed human beings, and they believe that some men have the faculty of transforming themselves at pleasure into tigers, and that others enjoy the privilege of invulnerability. They rely firmly on the efficacy of charms, spells, amulets, talismans, lucky and unlucky moments, magic, and judicial astrology. To pull down or repair a seriously damaged house is considered unlucky, so that whenever a Malay has occasion to build a new house he leaves the old one standing. While the coasts of Borneo and Sumatra are occupied by the more civilised Mahometan Malays, the interior of these vast islands is inhabited by nations, probably of the same race, who, secluded from the rest of the world, exhibit in their customs a strange and almost incredible mixture of good and evil, of humane tendencies and diabolical barbarism. Thus the Battas, who next to the Malays are the most numerous people of Sumatra, have the same polite and ceremonious manner, they possess an ancient code of law, they write books, and are fond
- 43. of music, they build commodious houses, which they ornament with tasteful carvings, they wear handsome tissues and know the art of smelting and amalgamating metals; they are extremely good- natured, and yet they not only eat human flesh, but eat it under circumstances of unexampled atrocity. According to their own traditions, their ancestors knew nothing of this horrid practice, which was first instigated by the demon of war about the year 1630, and from being originally an act of vengeance or fury, became at length one of their institutions in times of peace, and is now legally sanctioned as a punishment for certain heavy crimes. In some cases the delinquent is first killed and then eaten, in others he is eaten alive, an aggravated punishment which, however, is only reserved for traitors, spies, and enemies seized arms in hand. Before the day appointed for execution, messengers are sent to all friends and allies, and preparations made as for a great festival. The victim, tied to a stake, awaits his horrible fate, while the air resounds with music and the clamour of hundreds of spectators. The rajah of the village steps forward, draws his knife, addresses the assembly, relates the crimes which justify the sentence, and says that now the moment is come for punishing the doomed wretch, whom he describes as a hellish scoundrel, as a Satan in a human form. At these words the actors in the shocking drama about to be performed feel, as they say, an invincible longing to swallow a piece of the villain’s flesh, as they then feel sure that he can do them no further harm, and impatiently brandish their knives. The rajah or the injured person, such is his privilege, now cuts off the first piece of flesh, which he generally selects from the inner side of the forearm (this being esteemed the most delicate morsel), or from the cheek when sufficiently fat, holds it up triumphantly, and tastes some of the flowing blood, his eyes at the same time sparkling with delight. He then hurries to one of the fires that have been kindled close by to broil his piece of meat before swallowing it, while the whole troop falls upon the miserable wretch, who, hacked to pieces, and bleeding from a hundred wounds, in a few moments expires. The avidity with which they devour his quivering flesh,
- 44. untouched by his shrieks and supplications, is the more to be wondered at as in other cases they show themselves susceptible of a tender pity for the sufferings of others. As if scenes like these were not sufficiently horrible, it has even been affirmed that the Battas eat their aged parents alive, but we hardly need the authority of Dr. Junghuhn, who, during a residence of two years among the Battas, only heard of three cases of public cannibalism, that this report has no foundation in truth. So much, however, is certain, that this singular people have a great liking for human flesh, and in all cases where a simple execution takes place seize the opportunity of quietly carrying home some favourite joint. The Battas have no priests, no temples, no idols. 23 They believe in a number of evil spirits, or Begus, who have their seat in the various diseases of the human body, and in a few good spirits, or Sumongot, the immortal souls of great forefathers, who reside on the high mountain tops. The souls only of such persons as die of a violent death ascend into the invisible land of immortality, and this may be some consolation to the poor wretches whom they horribly cut up at their cannibal feasts, while all persons dying of illness are considered as having fallen into the power of the Begus, and as totally annihilated. They have no idea of a Supreme Being, and their only religious ceremony, if such it may be called, is that on festival occasions they scatter rice to the four quarters of the wind, in order to propitiate the Begus. In consequence of the general state of anarchy in which their unfortunate country is plunged, they live in small fortified villages, surrounded by palisades and deep ditches so as to leave but two gates for a passage. As in the feudal times, eminences strong by nature are frequently selected for the sites of these settlements, where the Batta, though removed from the more fruitful plains, cultivates his small field of mountain rice in greater security. In some districts, where hostile invasions are less to be feared, he possesses, besides his village residence, a detached hut in a forest clearance near some
- 45. river navigable by canoes. To be out of the reach of wild animals or inundations, these huts are frequently built on trees whose central branches have been lopped off, while the outer ones have been left standing, so as to afford a grateful shade to the little aërial dwelling. From this eminence, which the proprietor reaches by a ladder from twenty-five to thirty feet high, he looks down complacently upon his paddy field below, and as he is no sportsman, the undisturbed denizens of the forest afford him many a pastime. Monkeys gambol without fear on the trees around him; long-tailed squirrels leap from bough to bough; elephants bathe in the river; lemurs and fox-bats fly about in the evening; stags feed in the thicket beneath; and the only enemy he seeks to destroy is the Leguan lizard, who, intent on plundering his hen-roost, lies concealed among the reeds on the river’s bank. The Battas, having frequently suffered by foreign invasions, suspect all strangers of evil intentions, and desire to be as little as possible disturbed by their visits. For this reason, as well as for additional security against hostile incursions, they have no roads nor bridges, and as the villages are generally many miles apart and separated from each other by jungles or woods, this total want of the means of communication presents an almost insuperable obstacle to the traveller. Their distrust of strangers extends even to the members of their own nation, so that Battas of one province cannot enter another without running the risk of being seized as spies and eaten alive. While two of the great events of human life—birth and marriage —pass almost unnoticed among the Battas, the third and last act of this ‘strange eventful history’ gives rise to ceremonies which one would hardly expect to meet with among a nation of cannibals. When the rajah of a large village dies, his body is kept so long in the house, until the rice which is sown on the day of his death by his son or his brother comes to maturity. When the rice is about to ripen, a buffalo is killed, and its bones sent round to all friends and relations among the rajahs of the neighbourhood as an invitation to
- 46. the burial, which is to take place on the tenth day after the reception of these strange missives. Every rajah who accepts the invitation is obliged to bring with him a buffalo. The coffin is placed on a bier before the house, and on the arrival of the guests their buffaloes are tied to strong poles close by. The wives, sons, and other near relations of the deceased, now walk seven times with loud lamentations round the buffaloes, after which the oldest or first wife breaks a pot of boiled rice grown from the seed sown on the dying day on the forehead of one of the buffaloes. This is the signal for a frantic explosion of grief among the mourning women, whose piercing cries are accompanied by the incessant beating of drums and brass kettles in the house. After this lugubrious scene, which soon terminates with the real or feigned exhaustion of the actors, each of the rajahs now in his turn walks seven times round the buffalo which he brought with him, and kills it with a stroke of his lance. The coffin is then removed to the burial-place, and placed on the side of the open grave, amid the profound silence of the assembly. Its lid is opened, and the eldest son of the deceased, stepping forward, looks at the corpse, the face of which is turned towards the sun, and, raising his hand to the sky, says, ‘Now, father, thou seest for the last time the sun, which thou wilt never see again.’ After this short but affecting allocution the lid is closed and the coffin lowered into the grave, upon which the company returns to the village, where meantime the slaughtered buffaloes have been made ready for the funeral feast. Their horns, skulls, and jaw-bones, fastened to stakes, are placed as ornaments round the grave, which has no other monument or inscription. On each of the two following days some food is carried to it, a welcome treat for the dogs, and then it is consigned to the neglect which is the ultimate fate of all. The mystical sowing of rice, and the touching words spoken at the grave, prove that the Battas, though without any fixed religious worship, have still religious feelings, and may serve to confirm the truth of the remark, that there is no nation, however barbarous, which does not show at least some traces of a belief in the Divinity,
- 47. and reveal, however obscurely, that man has been born for something higher than a mere animal existence. Among the Dyaks, a name indiscriminately applied to all the wild people on the island of Borneo, we find no less revolting customs than among the Battas of Sumatra. They are hunters of their kind, not merely for the sake of an unnatural feast, but simply for the sake of collecting heads. Skulls are the commonest ornaments of a Dyak house, and the possession of them is the best token of manly courage. A Dyak youth is despised by all the maidens of his village as long as he has not cut off the head of an enemy or waylaid a stranger; returning from a successful chase with one of these ghastly trophies, he is welcomed as a hero. The head is stuck upon a pole, and old and young dance around it, singing and beating gongs. Murder of the most revolting atrocity, which anywhere else would make its perpetrator be considered the enemy of his kind, is thus by a horrible perversity one of the elements of courtship. The same atrocious custom is found among the Harafuras of Celebes, the Nias Islanders, and some other Malay nations of the Indian Archipelago. When the Harafuras go to war, they first steal some heads, boil them, and drink the broth to render themselves invulnerable. The Minkokas of Celebes limit the custom of taking heads to funeral or festive occasions, more especially on the death of their rajah or chief. When this occurs they sally forth, with a white band across their forehead to notify their object, and destroy alike their enemies and strangers. From twenty to forty heads, according to the rank of the deceased rajah, being procured, buffaloes are killed, rice boiled, and a solemn funeral feast is held, and, whatever time may elapse, the body is not previously buried. The heads, on being cleaned, are hung up in the houses of the three principal persons of the tribe, and regarded with great veneration and respect. The national weapon of the Dyaks, though not in use among all their tribes, is the Sumpitan, a blow-pipe about five feet long, with an arrow made of wood, thin, light, sharp-pointed, and dipped in the
- 48. poison of the upas tree. As this is fugacious, the points are generally dipped afresh when wanted. For about twenty yards the aim is so true that no two arrows shot at the same mark will be above an inch or two apart. On a calm day the utmost range may be a hundred yards. Though impregnated with a poison less deadly than the Wourali of the American Indians, yet the shafts of the sumpitan are formidable weapons from the frequency with which they can be discharged, and the skill of those who use them. The arrows are contained in a bamboo case, hung at the side, and at the bottom of this quiver is the poison of the upas. When they face an enemy the box at the side is open, and, whether advancing or retreating, they fire the poisoned missiles with great precision. The style of building of the Dyaks is very peculiar; most of their villages consisting of a single house, in which from fifteen to twenty families live together, in separate compartments. The floor of these long buildings, which are thatched with palm leaves, rests on piles about six or ten feet from the ground, and the simple furniture consists of some mats, baskets, and a few knives, pots, a very primitive loom, and some dried heads by way of ornament. Though habitual assassins from ignorance and superstitious motives, the Dyaks are said to be of a mild, good-natured, and by no means bloodthirsty character. They are hospitable when well used, grateful for kindness, industrious and honest, and so truthful that the word of one of them might safely be taken before the oath of half-a-dozen civilised Malays. The celebrated traveller, Mrs. Ida Pfeiffer, who had the courage to wander among the Dyaks, and the good fortune to return with her head on her shoulders, speaks highly of their patriarchal life, the love they have for their children, and the respectful conduct of the children towards their parents. As to their personal appearance, she affirms that, though some authors describe them as fine men, they are only a little less ugly
- 49. than the Malays.
- 50. CORAL ISLAND.
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