The algebra of coherent algebraic sheaves

359 Pages·2014·1.92 MB·English

by  McLennan A

The Algebra of
Coherent Algebraic Sheaves
with an Explicated Translation of Serre’s
Faisceaux Alg´ebriques Coh´erents
by
Andrew McLennan1
1 School of Economics, Level 6, Colin Clark Building, University of Queensland, St Lucia,
QLD4072Australia,[email protected]. Thisversion: December17,2014.
For Shino, whose advice for those starting out in research is to understand
one paper better than the author.
Contents
Contents iii
Preface vii
Introduction 1
A Elements of Commutative Algebra 5
A1 Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . . 5
A2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
A3 The Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . . . 11
A4 Noetherian and Artinian Rings and Modules . . . . . . . . . . 13
A5 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
A6 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . 22
A7 Principal Ideals, Factorization, and Normality . . . . . . . . . . 27
A8 Noether Normalization and Hilbert’s Nullstellensatz . . . . . . 34
A9 Geometric Motivation . . . . . . . . . . . . . . . . . . . . . . . 35
A10 Associated Primes . . . . . . . . . . . . . . . . . . . . . . . . . 46
A11 Primes Associated to Principal Ideals. . . . . . . . . . . . . . . 50
A12 Primary Decomposition . . . . . . . . . . . . . . . . . . . . . . 52
A13 Chains of Submodules . . . . . . . . . . . . . . . . . . . . . . . 55
A14 Artinian Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
B Elements of Homological Algebra 61
B1 The Five and Snake Lemmas . . . . . . . . . . . . . . . . . . . 62
B2 Complexes, Homology, and Cohomology . . . . . . . . . . . . . 64
B3 Direct and Inverse Limits . . . . . . . . . . . . . . . . . . . . . 68
B4 The Long Exact Sequence . . . . . . . . . . . . . . . . . . . . . 70
B5 Left and Right Exact Functors . . . . . . . . . . . . . . . . . . 74
B6 The Two Main Bifunctors . . . . . . . . . . . . . . . . . . . . . 77
B7 Projective Modules . . . . . . . . . . . . . . . . . . . . . . . . . 80
B8 Injective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 83
B9 Flat Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
C Univariate Derived Functors 91
C1 Projective Resolutions . . . . . . . . . . . . . . . . . . . . . . . 91
iii
iv CONTENTS
C2 Injective Resolutions . . . . . . . . . . . . . . . . . . . . . . . . 92
C3 Univariate Left Derived Functors . . . . . . . . . . . . . . . . . 95
C4 Univariate Right Derived Functors . . . . . . . . . . . . . . . . 101
C5 Short Exact Sequences of Resolutions . . . . . . . . . . . . . . 103
C6 The Long Exact Sequences of Derived Functors . . . . . . . . . 114
D Derived Bifunctors 121
D1 Left Derived Bifunctors . . . . . . . . . . . . . . . . . . . . . . 121
D2 Right Derived Bifunctors . . . . . . . . . . . . . . . . . . . . . 128
D3 Axiomatic Characterizations of Tor and Ext . . . . . . . . . . . 133
D4 The Iterated Connecting Homomorphism . . . . . . . . . . . . 136
D5 Projective and Weak Dimension . . . . . . . . . . . . . . . . . . 139
D6 Ext and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 144
E Derived Rings and Modules, and Completions 149
E1 Completing a Topological Group . . . . . . . . . . . . . . . . . 150
E2 The Completion of a Filtered Group . . . . . . . . . . . . . . . 154
E3 The Associated Graded Group . . . . . . . . . . . . . . . . . . 157
E4 Derived Graded Rings . . . . . . . . . . . . . . . . . . . . . . . 158
E5 Filtered Modules and the Artin-Rees Lemma . . . . . . . . . . 159
E6 Completions of Rings and Modules . . . . . . . . . . . . . . . . 161
E7 Noetherian Completions . . . . . . . . . . . . . . . . . . . . . . 163
F Initial Perspectives on Dimension 165
F1 Two Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . 165
F2 The Hilbert-Poincar´e Series . . . . . . . . . . . . . . . . . . . . 167
F3 The Hilbert Polynomial . . . . . . . . . . . . . . . . . . . . . . 171
F4 The Dimension of a Local Ring . . . . . . . . . . . . . . . . . . 173
F5 Regular Local Rings . . . . . . . . . . . . . . . . . . . . . . . . 176
F6 The Principal Ideal Theorem . . . . . . . . . . . . . . . . . . . 178
G The Koszul Complex 181
G1 Tensor Products of Cochain Complexes . . . . . . . . . . . . . 181
G2 The Koszul Complex in General . . . . . . . . . . . . . . . . . 183
G3 The Koszul Complex . . . . . . . . . . . . . . . . . . . . . . . . 185
G4 Regular Sequences and de Rham’s Theorem . . . . . . . . . . . 187
G5 Regular Sequences in Local Rings . . . . . . . . . . . . . . . . . 190
G6 A Variant of the Koszul Complex . . . . . . . . . . . . . . . . . 192
H Depth and Cohen-Macaulay Rings 197
H1 Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
H2 Cohen-Macaulay Rings . . . . . . . . . . . . . . . . . . . . . . . 200
I Global Dimension 207
I1 Auslander’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 207
CONTENTS v
I2 Minimal Free Resolutions . . . . . . . . . . . . . . . . . . . . . 208
I3 The Hilbert Syzygy Theorem . . . . . . . . . . . . . . . . . . . 212
I4 The Auslander-Buchsbaum Formula . . . . . . . . . . . . . . . 213
I5 A Characterization of Projectivity . . . . . . . . . . . . . . . . 215
I6 Factoriality of Regular Local Rings . . . . . . . . . . . . . . . . 217
Bibliography 221
What It’s All About 223
Coherent Algebraic Sheaves 235
Index 343

Preface
The preface of a book is traditionally devoted to remarks that have some per-
sonalcharacter, and for mostbooksthese are mundaneandreassuring. “Even
in these turbulent times” the author’s feelings gravitate, like a pendulum un-
der the slow influence of friction, to appreciation of his or her parents, the
delightful domestic environment he or she currently enjoys, the support and
encouragement of colleagues, that nice person from the publisher who took
care of all those pesky details, and so forth.
I could easily profess to such sentiments, but it would be evasive, because
what everyone really wants to know is: Why in the world is an economist
writing a book of algebra and algebraic geometry? What sort of hubris might
inspire him to think he has any competence for such a task? What could he
possibly hope to gain? And in the face of these questions, how do I conceive
of my efforts, and what sort of “public face” am I trying to present to the
world?
The actual answers are quite a bit less dramatic than this sounds. I am
a mathematical economist, which means that if I am perhaps not exactly a
mathematician, I am certainly not exactly not a mathematician. From the
point of view of pure mathematics, mathematical economics is a fringy thing,
perhaps mildly interesting, but suspiciously justified by appeals to values be-
yond mathematics, and inessential to the central thrust and foundations of
the discipline. Be that as it may, it does presenta rich menu of technical chal-
lenges, and is perhaps not more distant from the main currents of research
than various other subfieldswithin mathematics proper. Over the years it has
attracted the interest of many mathematicians, including the Fields medalists
Stephen Smale and Pierre-Louis Lions. Relative to other specializations, the
technical foundations of mathematical economics are quite broad. Economic
phenomena can be modelled in many ways, so if there’s a tool out there that
can be put to use, probably somebody will do so eventually. Real analysis,
topology, functional analysis, and mathematical statistics underly fundamen-
tal economic models.
In the mid 1980’s it occurred to me that algebraic geometry might have
some relevance to game theory, because the notion of Nash equilibrium is
a matter of polynomial equations and inequalities. (It turns out that the
seemingly nearby but actually quite distant field of semi-algebraic geometry
does indeed provide quite useful results and insights.) So, I walked across
vii
viii PREFACE
campus and started attending a course based on Hartshorne (1977). The first
chapter was notimpossible(if oneaccepted thecited resultsfromalgebra)but
after that I quickly became hopelessly confused. At first I could understand
the logic of the definitions related to scheme theory, if not their motivations,
but before too long I was just lost in the jungle at night.
I didlearn that, along withlogic, topology, measuretheory, andfunctional
analysis, the reformulation of algebraic geometry in the 1960’s was one of the
profoundtransformationsofmathematicsduringthe20th century,andthatifI
didn’tfindsomewaytolearnmore,largeswathsofcontemporarymathematics
would be far beyond my comprehension and appreciation. But I think that
mypersistencereallyhadmoretodowithnotwantingtoaccept suchadefeat.
Now Hartshorne makes it pretty clear that he expects a strong background
in algebra, so I obtained various books and read each one up to some point.
It’s a gorgeous subject, but it has its own motivations and internal agenda,
as did each of the authors. Certainly I learned a lot, but each time I tried to
return to Hartshorne I was rebuffed, and this was also the case after I read
lower level books on algebraic geometry.
About a decade ago it occurred to me that I might try reading Serre’s
“Faisceaux Alg´ebrique Coh´erents” (henceforth FAC) which was obviously an
important milestone in the history of the subject, presumably much closer to
the original motivations and ways of thinking, and universally praised in the
highest terms. My high school French is barely adequate for mathematics,
and it is a journal article, not a textbook, so this also proved quite difficult.
However, I had the thought that instead of reading it, it might work better to
prepare a translation. This had the advantage of slowing me down, so that I
couldpatiently workthrougheachlogical detail. Bothformyownbenefit,and
becauseIcouldimagineitbecomingaccessibletoreadersatamuchlowerlevel
than would otherwise be the case, I interpolated explanatory remarks when
Serre elided some details, appealed to some not entirely elementary result,
wrote in a way that later became obsolete, and so forth. This had the effect of
creating a sense of dialogue, making it at least a quite original mathematical
document. Everything seemed to be going nicely, and “working” on it was a
delightfully relaxing activity.
Serre’s style is very gentle throughout, and up to a certain point FAC is
effectively self-contained, but then there are a flurry of citations to Cartan
and Eilenberg’s Homological Algebra (henceforth CE) which would appear in
print the following year. I acquired this (still very useful) book, and set about
figuringoutwhattheseresultswere. Itquicklyemergedthattheywerecentral
to Serre’s project, and that my translation couldn’t succeed unless the reader
could access them easily. At the same time CE was not an acceptable source,
sincewhatthereaderofFACneedsismixedinwithagreatmanyotherthings,
and some of the cited results are exercises. No other source seemed suitable,
so I set out to write a minimal treatment, working backward in CE in order
to extract only what was required. The result was a “Supplement” consisting
PREFACE ix
of several dozen pages.
Within FAC the work went forward again, until I reached the last few
pages, where I learned, somewhat to my horror, that Serre appeals to various
results of commutative algebra that are not in Atiyah and McDonald’s An
Introduction to Commutative Algebra. Again, I set out to extract a minimal
treatment from various sources, and the supplemental material expanded ac-
cordingly. Eventually it became clear that I would need to take control of the
subject from the beginning, so I wrote what is now Chapter A, even though
this material is basic and very well treated elsewhere. Somehow the “Supple-
ment” ballooned to over two hundred pages, dwarfing (at least in bulk) the
original intent and spirit of the project.
By this point you have probably figured out that I enjoy writing mathe-
matics. Early in my career I had great difficulties with writing (had it not
been for the advent of personal computers my career might have been lost) so
I tried to take that aspect of the work seriously, tracking down written advice
about exposition and (more usefully) thinking about what it was that made
the writings of John Milnor, Michael Spivak, J.S. Milne, Allen Hatcher, and
others, work so well. I also put a lot of effort into writing, learning much from
various mistakes and other experiences. Perhaps most important, I developed
a taste for mathematical exposition as a medium of aesthetic expression. If
you asked why I spent so much time on this project, I would say that no one
would think it odd if I spent the occasional Saturday afternoon dabbling in
watercolors, and this really isn’t any different.
As you might expect, I have strong and well developed views concerning
mathematical exposition, but for the most part I hope that they are better
expressed implicitly in the text than I could state them here. I should say
two things about the book. First, it is the sort of book I would like for
myself, insofar as it is meant to be read, not “studied.” (Readers who like to
work exercises should have no difficulty finding them elsewhere.) I am a busy
professional who doesn’t like being told that he can’t learn a subject without
going back to the course work ghetto, or doing endless problem sets. As much
as possible, I have tried to craft a book that can be absorbed and appreciated
with minimal effort. The reader should be aware of two particular aspects
of this. I have kept the coverage almost as minimal as possible, subject to
the nature of the project. (At a certain point I thought that factoriality of
regular local rings would be required. This turned out to be wrong, but it
would be a shame to stop within spitting distance of this glorious theorem.)
Sometimes I have added inessential results that illustrate or apply the ideas
under discussion, but I have deliberately avoided trying to make the coverage
of any topic “complete.” Second, I have allowed the organization complete
freedom to fall in line with the logic of the material. Possibly this book might
serve as the main text of quite a nice course, but relative to any established
curriculum or concept of what every young algebraist needs to know, and
when she needs to know it, there are large and obvious gaps.
x PREFACE
The second point is that this is the work of an amateur, in several senses.
It was pursued for its own sake, outside of any strategy for “career develop-
ment.” (At best it might push my reputation in economics further sideways.)
Consequently work on it proceeded in “slow cooking mode,” and I could in-
dulge a kind of perfectionism that the pressure to publish can easily quash.
Also, I am no expert in commutative algebra, and perhaps was better able to
appreciate the logic of the material as something fresh, and to convey some
sense of that to the reader. In retrospect the surprising degree of coherence it
attained is, I think, ultimately a reflection of Serre’s long range vision.
Expository projects in this spirit will almost certainly continue to be not
that well rewarded, because accomplishment in research will continue to be
the only acceptable qualification for membership in the academy. (As an
economist I could advance various points of view concerning whether that is a
good or bad thing, but I see no likelihood that it will change soon.) Perhaps
the example of this book may inspire others to think that such work can, in
and of itself, be more than ample reward.
Ordinarily at this point in a preface there would be a long list of names
of all the people who provided feedback, encouragement, and various forms
of assistance. However, economists tend to be quite dubious when they learn
that one of their colleagues is indulging a taste for pure math, so I have been
completely secretive about this project while it was underway. Hopefully
things won’t be that bad ex post, when people see that during the decade or
so that I have been noodling around with FAC, I have also done roughly the
usual amount of the usual sort of research. But just to be on the safe side, if
you happen to meet an economist, please don’t tell them about this, OK?