More about the book at the CUP site (it’s not available yet — I’m checking the copyedited manuscript right now).
Category Archives: book
I’m about to deliver the final manuscript of my book Group Cohomology and Algebraic Cycles to Cambridge University Press. As a sort of early advertisement, here’s the draft preface. I thank them at the end of the preface, but I’d like to say here too: I thank Ben Antieau and Peter Symonds for many useful suggestions.
Group cohomology reveals a deep relation between algebra and topology. A group determines a topological space in a natural way, its classifying space. The cohomology ring of a group is defined to be the cohomology ring of its classifying space. The challenges are to understand how the algebraic properties of a group are related to its cohomology ring, and to compute the cohomology rings of particular groups.
A fundamental fact is that the cohomology ring of any finite group is finitely generated. So there is some finite description of the whole cohomology ring of a finite group, but it is not clear how to find it. A central problem in group cohomology is to find an upper bound for the degrees of generators and relations for the cohomology ring. If we can do that, then there are algorithms to compute the cohomology in low degrees and therefore compute the whole cohomology ring.
Peter Symonds made a spectacular advance in 2010: for any finite group G with a faithful complex representation of dimension n at least 2 and any prime number p, the mod p cohomology ring of G is generated by elements of degree at most n2 (Symonds 2010). Not only is this the first known bound for generators of the cohomology ring; it is nearly an optimal bound among arbitrary finite groups, as we will see.
This book proves Symonds’s theorem and several new variants and improvements of it. Some involve algebro-geometric analogs of the cohomology ring. Namely, Morel–Voevodsky and I independently showed how to view the classifying space of an algebraic group G (for example, a finite group) as a limit of algebraic varieties in a natural way. That allows the definition of the Chow ring of algebraic cycles on the classifying space BG (Morel and Voevodsky 1999, prop. 2.6; Totaro 1999).
A major goal of algebraic geometry is to compute the Chow ring for varieties of interest, since that says something meaningful about all subvarieties of the variety.
The fact that not all the cohomology of BG is represented by algebraic cycles (even for abelian groups G) is the source of Atiyah-Hirzebruch’s counterexamples to the integral Hodge conjecture (Atiyah and Hirzebruch 1962; Totaro 1997, 1999). It is a natural problem of “motivic homotopy theory” to understand the Chow ring and more generally the motivic cohomology of classifying spaces BG. Concretely, computing the Chow ring of BG essentially amounts to computing the Chow groups of the quotients by G of all representations of G. Such quotients are extremely special among all varieties, but they have been fundamental examples in algebraic geometry for more than 150 years. Computing their Chow groups is a fascinating problem. (Rationally, the calculations are easy; the interest is in integral or mod p calculations.)
Bloch generalized Chow groups to a bigraded family of groups, now called motivic cohomology. A great achievement of motivic homotopy theory is the proof by Voevodsky and Rost of the Bloch–Kato conjecture (Voevodsky 2011, theorem 6.16). A corollary, the Beilinson–Lichtenbaum conjecture, says that for any smooth variety over a field, a large range of motivic cohomology groups with finite coefficients map isomorphically to etale cohomology. Etale cohomology is a more computable theory, which coincides with ordinary cohomology in the case of complex varieties. Thus the Beilinson–Lichtenbaum conjecture is a powerful link between algebraic geometry and topology.
Chow groups are the motivic cohomology groups of most geometric interest, but they are also farthest from the motivic cohomology groups that are computed by the Beilinson–Lichtenbaum conjecture. A fundamental difficulty in computing Chow groups is “etale descent”: for a finite Galois etale morphism X → Y of schemes, how are the Chow groups of X and Y related? This is easy after tensoring with the rationals; the hard case of etale descent is to compute Chow groups integrally, or with finite coefficients. Etale descent is well understood for etale cohomology, and hence for many motivic cohomology groups with finite coefficients.
The problem of etale descent provides some motivation for trying to compute the Chow ring of classifying spaces of finite groups G. Computing the Chow ring of BG means computing the Chow ring of certain varieties Y which have a covering map X → Y with Galois group G (an approximation to EG → BG) such that X has trivial Chow groups. Thus the Chow ring of BG is a model case in seeking to understand etale descent for Chow groups.
Chow rings can be generalized in various ways, for example to algebraic cobordism and motivic cohomology. Another direction of generalization leads to unramified cohomology, cohomological invariants of algebraic groups (Garibaldi, Merkurjev, and Serre 2003), and obstructions to rationality for quotient varieties (Bogomolov 1987; Kahn and Ngan 2012). All of these invariants are worth computing for classifying spaces, but we largely focus on the most classical case of Chow rings. Some of our methods will certainly be useful for these more general invariants. For example, finding generators for the Chow ring (of any algebraic variety) automatically gives generators of its algebraic cobordism, by Levine and Morel (2007, theorem 1.2.19).
We now summarize the contents. Continue reading
I’ll be spending the academic year 2012-13 at UCLA, and so leaving one well-stocked university library for another. Nevertheless, like every mathematician, I have some favorite books — for research, teaching, and finishing my book — that I can’t be without. After a lot of fussing, in an effort to pack light, I’ve chosen a core traveling library to take from Cambridge to Los Angeles:
Adem/Milgram, Cohomology of finite groups
Atiyah/Macdonald, Introduction to commutative algebra
Benson, Representations and cohomology, I and II
Benson, Polynomial invariants of finite groups
Bloch, Lectures on algebraic cycles
Brown, Cohomology of groups
Eisenbud, Commutative algebra with a view toward algebraic geometry
Fulton, Intersection theory
Fulton, Introduction to toric varieties
Fulton/Harris, Representation theory
Garibaldi/Merkurjev/Serre, Cohomological invariants of algebraic groups
Griffiths/Harris, Principles of algebraic geometry
Hartshorne, Algebraic geometry
Kobayashi, Hyperbolic manifolds and holomorphic mappings
Kollár, Lectures on resolution of singularities
Kollár, Shafarevich maps and automorphic forms
Kollár/Mori, Birational geometry of algebraic varieties
Lazarsfeld, Positivity in algebraic geometry, I and II
Milne, Etale cohomology
Mumford/Fogarty, Geometric invariant theory
Mukai, An introduction to invariants and moduli
Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture
Serre, Cohomologie galoisienne
Serre, Linear representations of finite groups
Seshadri, Fibrés vectoriels sur les courbes algebriques
Voisin, Hodge theory and complex algebraic geometry, I and II
I started this blog about a year ago briefly recommending Rob Lazarsfeld’s book Positivity in Algebraic Geometry, which gives bite-size treatments of many topics hard to find elsewhere.
I’d like to make a stronger case now because it’s an important book. People often give me credit for knowing a lot just because I know what’s in it. It’s rarely on my shelves because it’s almost always in a stack near where I’m working. When I lost my copies in transit between MSRI and Cambridge, I replaced them immediately.*
The title might sound, on the face of it, like something specialized or technical. In fact, positivity is arguably the fundamental difference between algebraic geometry and topology. For example, the intersection multiplicity of two distinct complex curves which meet at a point in a complex algebraic surface S is always positive. As a result, if you know the homology classes of the two curves, then you know the total intersection number N from the cohomology ring of S, and that implies that the “physical” number of intersection points is at most N. This is completely false in topology: you can push around one submanifold to meet another submanifold in as many points as you like. The result is that just knowing the homology class of an algebraic curve controls its geometric properties (it can’t wiggle too much). Much of algebraic geometry builds on this kind of rigidity.
*Not at all painful because Lazarsfeld insisted Springer publish in paperback and keep the price down. Losing Kollár’s Rational Curves on Algebraic Varieties, on the other hand…
Some people like schemes too much. Others don’t like them at all. Here is my view on how to like schemes for the right reasons.
There are several good things about schemes. I usually study varieties over a field, but there are various ways one can encounter more general schemes in the course of a given problem.
One good thing is the way schemes automatically include information about “multiplicities”. For example, given a polynomial f in one variable over a field k, we would like to say that the number of zeros of f is equal to the degree of f. But there are obvious obstacles to that statement. For example, the polynomial x2+1 over the real numbers actually has no roots in R, so we are really talking about roots of f in some algebraic closure of k. But even then, you have to keep track of “multiplicities” for the statement about deg(f) to be true; e.g., the polynomial (x-1)2 has a root at x=1 “with multiplicity 2″.
Both these complications are avoided by the language of schemes. In that language, the zero set of a polynomial f(x) of degree d over a field k is always a 0-dimensional closed subscheme of the affine line A1 over k, and that subscheme always has degree d. This shows two different good aspects of the notion of a scheme. First, a subscheme Y of a variety X determines not only a subset Y(k) of X(k) (where X(k) means the set of solutions in k of the equations defining a variety X), it also determines a subset Y(E) of X(E) for every extension field E of k. (For example, the subscheme Y of the affine line over the real numbers defined by the equation x2+1 = 0 is not the empty scheme, even though Y(R) is the empty set, because we can see that Y(C) is not empty.) Second, a subscheme Y of a variety X contains more subtle “nilpotent” information than just the subsets Y(E) of X(E) for field extensions E of k. For example, the subscheme Y of the affine line over the real numbers defined by the equation (x-1)2=0 is different from the point Z defined by x-1=0, even though Y(E)=Z(E) for all field extensions E of the real numbers. We picture Y as a “fat point”, a point together with an “infinitesimal neighborhood of length 1″.
In this simple situation of polynomials in one variable, one could live without the geometric language of schemes. But for the same phenomena of “multiplicities” and so on in higher dimensions, the language of schemes is unavoidable (and useful). For example, consider the intersection of two smooth curves at a point in a smooth surface. If they intersect transversely, then the intersection (as a scheme) is “reduced”, i.e., it’s just the point. But if the intersection is not transverse, then the intersection is a 0-dimensional subscheme of degree greater than 1, i.e. it’s a “fattened” or “non-reduced” version of the point. (This is relevant even in the familiar situation of varieties over the complex numbers.)
The standard textbook on schemes is Hartshorne’s Algebraic Geometry. Unfortunately the actual definition of schemes is done in a very dry way there. Still, there is a huge amount of illuminating stuff in that book: the informal essay introducing schemes on pp. 55-59 is good, many exercises are interesting, many good examples and so on. By the end of chapter II on schemes, he gets to crucial geometric ideas like divisors (section 6) and differentials (i.e., the tangent bundle, or more precisely its dual). Chapters IV and V on curves and surfaces are appealing and (I think) can be read before going through all the earlier parts of the book. Appendices A, B, C introducing more advanced topics in an informal way are even better: don’t miss these.
There are by now lots of other books introducing schemes, although none has displaced Hartshorne. I have heard good things about Mumford’s Red Book of Varieties and Schemes, but I haven’t read it myself.
I don’t remember exactly when or why I wrote the list below. I suspect it was a couple of years ago for Part III algebraic geometry students. If you want to be a mathematician, there’s no substitute for knowing some math. You might as well learn it from great writers.
“To learn to write well, one should read Serre, Bott, Milnor,…”
(I think I’m quoting Steve Hurder here, but I believe this too.)
The following list includes both short, readable books that everyone should read and longer reference books.
The standard topics in pure mathematics are: real analysis including Lebesgue integration (I recommend Royden, Real Analysis); complex analysis; topology (I recommend Armstrong, Basic Topology); and algebra including Galois theory. Fourier series are also fundamental; I recommend Dym and McKean, Fourier Series and Integrals, with a variety of applications in a short space.
Part III-level books
Representation theory: Serre, Linear Representations of Finite Groups. Fulton-Harris, Representation Theory (of semisimple Lie algebras, or equivalently of compact Lie groups). By concentrating on examples, Fulton-Harris’s book is wonderfully readable although somewhat long.
Commutative algebra: Atiyah and Macdonald, Introduction to Commutative Algebra. Very clear in a short space.
Number theory: Serre, A Course in Arithmetic. Cassels, Local Fields.
Topology: Bott and Tu’s Differential Forms in Algebraic Topology is a very readable introduction to smooth manifolds and goes far; everyone should read it. Hatcher, Algebraic Topology.
Riemannian geometry: Gallot-Hulin-Lafontaine’s Riemannian Geometry is one of several gentle introductions. Warner’s Foundations of Differentiable Manifolds and Lie Groups is heavier, but is indispensable for giving the only understandable proof of the Hodge theorem for a Riemannian manifold.
Analysis: Royden, Real Analysis. Lieb and Loss, Analysis.
Algebraic geometry: Hartshorne, Algebraic Geometry. Griffiths and Harris, Principles of Algebraic Geometry. These are long references, indispensable for the working algebraic geometer (emphasizing algebraic and analytic approaches, respectively). Huybrechts’s Complex Geometry is a good simplification of Griffiths-Harris.
On more specific topics in algebraic geometry, some outstanding books are Mukai, An Introduction to Invariants and Moduli, and Mumford, Abelian Varieties. There are several other great books (both easier and harder) by Mumford. Borel, Linear Algebraic Groups.
Topology: Milnor’s Characteristic Classes and Morse Theory are magnificent books: short, readable, with a tremendous range of applications. Everyone should read them. There are several other great books by Milnor.
McCleary’s A User’s Guide to Spectral Sequences covers a lot of algebraic topology beyond the basics. Thurston, Three-Dimensional Geometry and Topology.
Symplectic geometry: Arnold, Mathematical Methods in Classical Mechanics.
Homological algebra: Brown, Cohomology of Groups, is an excellent book applying topological ideas to algebra. Weibel, An Introduction to Homological Algebra. Benson, Representations and Cohomology (2 vols.) S. MacLane, Categories for the Working Mathematician.
Number theory: Serre, Local Fields, among several other great books. Lang, Algebraic Number Theory. Miyake, Modular Forms. Silverman, The Arithmetic of Elliptic Curves.
Geometric group theory: Serre, Trees. De la Harpe, Topics in Geometric Group Theory, gives quick treatments of a rich variety of topics.
Analysis: Zimmer, Basic Results in Functional Analysis, treats the fundamental topics and applications in a very short space. Krylov, Lectures on Elliptic and Parabolic Equations in H¨older Spaces, is one of the few graduate-level introductions to serious PDE theory. The big reference books are Gilbarg-Trudinger, Elliptic Partial Differential Equations of Second Order, and Evans, Partial Differential Equations.
Dynamical systems: Walters, An Introduction to Ergodic Theory, is a standard short introduction. Hasselblatt and Katok, An Introduction to the Modern Theory of Dynamical Systems, is the standard big reference book.
Everyone interested in algebraic geometry, number theory, and many aspects of topology and group theory should look at Serre’s Oeuvres: Collected Papers. Atiyah’s Collected Papers are fundamental for topology, with links to analysis and differential geometry.
Why not think about reading Rob Lazarsfeld’s Positivity in Algebraic Geometry I. Classical Setting: Line Bundles and Linear Series? Two or three people in the past couple of weeks have asked me questions that this book sheds light upon, and I’ve decided that more people need to know about this excellent book.