Threading homology through algebra: selected patterns

267 Pages·2006·1.222 MB·English

by  Giandomenico Boffi, David Buchsbaum

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Threading Homology Through
Algebra: Selected Patterns
GIANDOMENICO BOFFI
Universita` G. d’Annunzio
DAVID A. BUCHSBAUM
Department of Mathematics, Brandeis University
·
CLARENDON PRESS OXFORD
2006
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A coloro che amo
To Betty, wife and lifelong friend.
Though she can’t identify each tree, she shares with me the delight
of walking through the forest.
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PREFACE
From a little before the middle of the twentieth century, homological methods
have been applied to various parts of algebra (e.g. Lie Algebras, Associative
Algebras, Groups [finite and infinite]). In 1956, the book by H. Cartan and
S. Eilenberg, Homological Algebra [33], achieved a number of very important
results: it gave rise to the new discipline, Homological Algebra, it unified the
existing applications, and it indicated several directions for future study. Since
then, the number of developments and applications has grown beyond counting,
and there has, in some instances, even been enough time to see various methods
threadingtheirwaythroughapparentlydisparate,unrelatedbranchesofalgebra.
What we aim for in this book is to take a few homological themes (Koszul
complexesandtheirvariations,resolutionsingeneral)andshowhowtheseaffect
the perception of certain problems in selected parts of algebra, as well as their
successinsolvinganumberofthem.Theexpectationisthataneducatedreader
will see connections between areas that he had not seen before, and will learn
techniques that will help in further research in these areas.
What we include will be discussed shortly in some detail; what we leave out
deserves some mention here. This is not a compendium of homological algebra,
nor is it a text on commutative algebra, combinatorics, or representation the-
ory; although, it makes significant contact with all of these fields. We are not
attemptingtoprovideanencyclopedicwork.Asaresult,weleaveoutvastareas
of these subjects and only select those parts that offer a coherence from the
point of view we are presenting. Even on that score we can make no claim to
completeness.
OurChapterI,called“RecollectionsandPerspectives,”reviewspartsofPoly-
nomial Ring and Power Series Ring Theory, Linear Algebra, and Multilinear
Algebra, and ties these with ideas that the reader should be very familiar with.
As the title of the chapter suggests, this is not a compendium of “assumed
known” items, but a presentation from a certain perspective—mainly homolo-
gical.Forexample,almosteveryoneknowsaboutdivisibilityandfactoriality;we
give a criterion for factoriality that ties it immediately to a homological inter-
pretation (and one which found significant application in solving a long-open
question in regular local ring theory).
The next three chapters of this book pull together a group of classical results,
allcomingfromandgeneralizingthetechniquesassociatedwiththeKoszulcom-
plex. Perhaps the major result in Chapter II, on local rings, is the homological
characterizationofaregularlocalringbymeansofitsglobaldimension.Section
II.6includesaproofofthefactorialityofregularlocalringswhichismuchcloser
to the original one, rather than the Kaplansky proof that is frequently quoted.
viii Preface
We have also included a section on multiplicity theory, mainly to carry through
thethemeoftheKoszulcomplex,andasectionontheHomologicalConjectures,
as they provide a good roadmap for still open problems as well as a historical
guidethroughmuchofwhathasbeengoingonintheareathisbookissketching.
Chapter III deals with a class of complexes developed with the following aim in
view:toassociateacomplextoanarbitraryfinitepresentationmatrixofamod-
ule(theKoszulcomplexdoesthisforacyclicmodule),andtohavethatcomplex
playthesameroleintheproofofthegeneralizedCohen–MacaulayTheoremthat
theKoszulcomplexplaysintheclassicalcase.Wehavemadeanexplicitconnec-
tion,intermsofachainhomotopy,betweenanolder,“fatter”classofcomplexes,
andaslimmer,more“svelte”class.Wehavealsoincludedalastsectioninwhich
we define a generalized multiplicity which has found interesting applications, of
late. Chapter IV applies some of the properties of these complexes to a system-
atic study of finite free resolutions, ending in a “syzygy-theoretic” proof of the
unique factorization theorem (or “factoriality”) in regular local rings.
The last three chapters and the Appendix not only focus on determinantal
ideals and characteristic-free representation theory, but also involve a good deal
of combinatorics. Chapter V employs the homological techniques developed in
the previous part in the study of a number of types of determinantal ideals,
namely Pfaffians and powers of Pfaffians. In Chapter VI we develop the basics
of a characteristic-free representation theory of the general linear group (which
has already made its appearance in earlier chapters). Because of the generality
aspired to, heavy use is made of letter-place methods, an idea used more by
combinatorialiststhanbycommutativealgebraists.Assomeoftheproofsrequire
more detail than is probably helpful for those encountering this material for the
firsttime,wedecidedtoplacethesedetailsinaseparateAppendix:AppendixA.
Much of the development of this chapter rests heavily on the notion of straight
tableaux introduced by B. Taylor. In Chapter VII we first present a number of
results that immediately follow from this more general theory. Then examples
are given to indicate what further use has been made of it, and in most cases
referencesaregiventodetailedproofs.Itisinthispartofthechapterthatwesee
theimportantinfluenceoftheworkofA.Lascouxincharacteristiczero.Wegive
someofthebackgroundtotheHashimotoexampleofthedependenceoftheBetti
numbers of determinantal ideals on characteristic. We deal with resolutions of
Weylmodulesingeneral,andskew-hooksinparticular,andwemakeconnections
with intertwining numbers, Z-forms, and several other open problems.
Theintendedreadershipofthisbookrangesfromthird-yearandabovegradu-
ate students in mathematics, to the accomplished mathematician who may or
maynotbeinanyofthefieldstouchedon,butwhowouldliketoseewhatdevel-
opments have taken place in these areas and perhaps launch himself into some
of the open problems suggested. Because of this assumption, we are allowing
ourselves to depend heavily on material that can be found in what we regard as
comprehensive and accessible texts, such as the textbook by D. Eisenbud. We
may at times, though, include a proof of a result here even if it does appear in
suchatext,ifwethinkthatthemethodofproofistypicalofmanyofthatkind.
CONTENTS
I Recollections and Perspectives 1
I.1 Factorization 1
I.1.1 Factorization domains 1
I.1.2 Polynomial and power series rings 6
I.2 Linear algebra 8
I.2.1 Free modules 8
I.2.2 Projective modules 13
I.2.3 Projective resolutions 17
I.3 Multilinear algebra 21
I.3.1 R[X ,...,X ] as a symmetric algebra 22
1 t
I.3.2 The divided power algebra 28
I.3.3 The exterior algebra 30
II Local Ring Theory 37
II.1 Koszul complexes 38
II.2 Local rings 43
II.3 Hilbert–Samuel polynomials 46
II.4 Codimension and finitistic global dimension 50
II.5 Regular local rings 54
II.6 Unique factorization 56
II.7 Multiplicity 59
II.8 Intersection multiplicity and the homological conjectures 64
III Generalized Koszul Complexes 69
III.1 A few standard complexes 69
III.1.1 The graded Koszul complex and its “derivatives” 70
III.1.2 Definitions of the hooks and their explicit bases 72
III.2 General setup 80
III.2.1 The fat complexes 82
III.2.2 Slimming down 83
III.3 Families of complexes 85
III.3.1 The “homothety homotopy” 88
III.3.2 Comparison of the fat and slim complexes 91
III.4 Depth-sensitivity of T(q;f) 94
III.5 Another kind of multiplicity 99
IV Structure Theorems for Finite Free Resolutions 103
IV.1 Some criteria for exactness 104
IV.2 The first structure theorem 110
x Contents
IV.3 Proof of the first structure theorem 115
IV.3.1 Part (a) 115
IV.3.2 Part (b) 118
IV.4 The second structure theorem 119
V Exactness Criteria at Work 127
V.1 Pfaffian ideals 128
V.1.1 Pfaffians 128
V.1.2 Resolution of a certain pfaffian ideal 131
V.1.3 Algebra structures on resolutions 132
V.1.4 Proof of Part 2 of Theorem V.1.8 134
V.2 Powers of pfaffian ideals 136
V.2.1 Intrinsic description of the matrix X 137
V.2.2 Hooks again 138
V.2.3 Some representation theory 139
V.2.4 A counting argument 140
V.2.5 Description of the resolutions 143
V.2.6 Proof of Theorem V.2.4 145
VI Weyl and Schur Modules 149
VI.1 Shape matrices and tableaux 149
VI.1.1 Shape matrices 149
VI.1.2 Tableaux 153
VI.2 Weyl and Schur modules associated to shape matrices 154
VI.3 Letter-place algebra 156
VI.3.1 Positive places and the divided power algebra 156
VI.3.2 Negative places and the exterior algebra 159
VI.3.3 The symmetric algebra (or negative letters and places) 164
VI.3.4 Putting it all together 164
VI.4 Place polarization maps and Capelli identities 165
VI.5 Weyl and Schur maps revisited 167
VI.6 Some kernel elements of Weyl and Schur maps 169
VI.7 Tableaux, straightening, and the straight basis theorem 174
VI.7.1 Tableaux for Weyl and Schur modules 174
VI.7.2 Straightening tableaux 176
VI.7.3 Taylor-made tableaux, or a straight-filling algorithm 181
VI.7.4 Proof of linear independence of straight tableaux 183
VI.7.5 Modifications for Schur modules 186
VI.7.6 Duality 187
VI.8 Weyl–Schur complexes 187
VII Some Applications of Weyl and Schur Modules 193
VII.1 The fundamental exact sequence 193
VII.2 Direct sums and filtrations for skew-shapes 197
VII.3 Resolution of determinantal ideals 199