More about the book at the CUP site (it’s not available yet — I’m checking the copyedited manuscript right now).
I am now halfway through teaching a graduate course on “Topological methods in algebraic geometry”. The idea is to give a quick, informal introduction to various topics which are important, but a little too advanced to appear in most first-year graduate courses. I hope to communicate a homotopy-theoretic point of view, which can be applied in many different ways.
The topics we have covered so far are:
- Classifying spaces in topology, principal G-bundles
- Spectral sequences
- Rational homotopy theory
- Topology of Lie groups
- Faithfully flat descent
To summarize some of the early parts of the course: every geometer will learn about the Chern classes of a complex vector bundle. My idea for the course was to put this construction in a broader context, by considering principal G-bundles for any group G instead of just the general linear group (or the unitary group). This brings up some of the main ideas of homotopy theory, since the classifying space for principal G-bundles is related to G by taking the loop space. Discussing spectral sequences and rational homotopy theory shows that we can compute a lot in homotopy theory. Using these general methods, we can get a rather strong understanding of the topology of all Lie groups. At this point, the connection with algebraic geometry becomes serious, since many properties of a compact Lie group are best understood by considering its complexification, which is a complex reductive group.
Grothendieck’s theory of faithfully flat descent is a beautiful topic of commutative algebra, which appears in surprisingly few books. It generalizes the idea of constructing sheaves or vector bundles by gluing together sheaves or vector bundles on open subsets. The theory should be useful for understanding principal bundles in algebraic geometry, equivariant Chow groups, and algebraic stacks in the rest of the course. And oh yes, I also want to discuss derived categories. We’ll see how it goes.
Here are the books I have suggested for the course so far.
- R. Bott and L. Tu. Differential forms in algebraic topology.
- J. Milnor and J. Stasheff. Characteristic classes.
- J. McCleary. A user’s guide to spectral sequences.
- Y. Félix, S. Halperin, and J.-C. Thomas. Rational homotopy theory.
- Y. Félix, J. Oprea, D. Tanré. Algebraic models in geometry.
- W. Waterhouse. Introduction to affine group schemes.
And anyone who’s been paying attention knows that important ingredients in the secret sauce can be found here:
- J.-P. Serre. Oeuvres: collected papers, 4 vols.
My paper, On the integral Hodge and Tate conjectures over a number field, has now — after minor revision — been accepted by Forum of Mathematics Sigma, and should be appearing shortly. On the web, lots of people seem to conflate the open access model with editorial slapdashery. As you’d expect from the editorial board, there was no sign of that in my experience with FOM. I received two serious and helpful referee reports. Among other helpful recommendations, one referee pointed me to a very relevant reference that I didn’t even know existed, and the other referee pointed out that I didn’t understand a formula that I’d thought I understood pretty well.
Original from 19 December 2012
The title of this post is, of course, an exaggeration: I already have some version of nearly all my papers on my department webpage and now diligently post new papers on the arXiv. What I mean is that I’ve submitted a paper to one of the new Open Access journals launched by Cambridge University Press, in my case Forum of Mathematics Sigma, where the algebraic geometry strand is edited by Sebastien Boucksom, Ravi Vakil, and Claire Voisin. Sigma has other strands — the nearby algebra strand is edited by Dennis Gaitsgory, Raphaël Rouquier, and Catharina Stroppel. (There is also a second journal, Forum of Mathematics Pi, for papers of broad interest.)
The journals are meant to investigate whether mainstream Open Access can be a large-scale solution in mathematics to the problem of the increasingly expensive subscription model, and there is meant to be no corner-cutting on editorial integrity or publishing standards. Since, to really put this to the test, there needs to be serious volume I thought I’d do my bit to send some their way.
Those interested are strongly encouraged to apply. Applications should be sent directly to the Institute for Advanced Study, using either the Institute’s web site or MathJobs. The deadline for applications is December 1, 2013. Decisions about membership are made by the Institute; please contact them directly if you have questions about the deadline or an application.
The Institute offers many positions at the postdoctoral level, but also hosts mathematicians at all stages of their careers. Some mathematicians receive funding from their home institutions, foundations or governments, while others are supported by the Institute. The Institute mostly supports people who stay either for the full year or for one semester (fall or spring), but others may be interested in attending one of two week-long workshops:
October 13–17, 2014: Fundamental groups and periods.
March 9–13, 2015: Chow groups, motives, and derived categories.
The title of the program is meant to be interpreted broadly. One main theme is the study of the topology of a complex algebraic variety, with Hodge theory as the most powerful method. Another is the study of algebraic varieties over an arbitrary field using etale cohomology and other cohomology theories. A major goal in both theories is to understand algebraic cycles on a given variety. This includes some enormous problems where, nonetheless, progress is being made: the Hodge conjecture, the Tate conjecture, the Bloch–Beilinson conjecture, and so on. The program intends to bring a mix of people interested in various aspects of the subject: motives, K-theory, Chow groups, periods, fundamental groups, derived categories, and so on.
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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