Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Posts Tagged ‘Weil cohomology’

Toen on homotopy types of algebraic varieties

Posted by Andreas Holmstrom on October 4, 2009

Two recent conversations both reminded me of a short note of Toen, with the title Homotopy types of algebraic varieties. This note explains in only eight pages several exciting ideas, which I find interesting especially because they point towards some possible future interactions between homotopy theory and arithmetic geometry.

He starts out by a conceptual discussion of classical Weil cohomology theories, which were discussed in this earlier post. The idea is that the cohomological invariants should be refined into some notion of “homotopy type”, the relation being somewhat analogous to the relation in algebraic topology, between the cohomology and the homotopy type of, say, a CW complex. He then goes on to sketch how this can be made precise, using the language of stacks and schematic homotopy types.

Towards the end of the paper, he speculates about a possible connection between the homotopy types of a variety and rational points on the variety. The study of rational points is one of the main themes of arithmetic geometry, as they correspond to integer or rational solutions of (systems of) polynomial equations. The famous section conjecture of Grothendieck, explained in these notes of Kim, is supposed to give a conceptual proof of Faltings’ theorem, aka the Mordell conjecture. Faltings’ theorem says that a curve of genus at least 2, defined over \mathbb{Q}, only has a finite number of rational points. Toen suggests a generalization of the section conjecture to higher-dimensional varieties, using his notion of homotopy types.

Another main theme of arithmetic geometry is L-functions of various kinds. To any variety over \mathbb{Q}, one can attach an L-function, which encodes lots of information about the arithmetic properties of the variety. Many outstanding conjectures in number theory are formulated in terms of these functions, for example the Riemann hypothesis and the Birch-Swinnerton-Dyer conjecture, as well as many other more accessible conjectures. The building blocks of an L-function are precisely the various Weil cohomology groups, and one could speculate about the significance of Toen’s conceptual approach to Weil cohomologies. Could it give us some new tools for approaching questions about L-functions? Or could it be that L-functions are not the right thing to consider, but that the notion of homotopy types could lead us to some better objects of study?

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The Chow ring

Posted by Andreas Holmstrom on March 24, 2009

Matt DeLand recently posted a very nice introduction to the Chow ring and Chern classes on Rigorous Trivialities. We will need this material when talking more about Weil cohomology and pure motives, and I will simply refer to his post, and to this introduction by Gillet, for all things related to Chow groups/rings.

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Weil cohomology

Posted by Andreas Holmstrom on March 18, 2009

The most useful and natural notion of cohomology for smooth projective varieties is given by the set of axioms known as Weil cohomology. When studying more general varieties (see this post), we need different notions of cohomology – will come back to this in the future. For the basic definitions and properties of Weil cohomology, I will simply refer to this excellent short note by de Jong. There are a few slightly different ways to define Weil cohomology, but I don’t want to get into a lot of details of this at the moment. The most significant difference between de Jong’s note and some other references is that some authors omit the notion of Tate twist. However, from the point of number theory this is  a bad thing to do.

Looking at varieties over a field k say, we have a range of Weil cohomology theories, each with its own coefficient field. The term “coefficient field” simply means the field K over which our cohomology groups are vector spaces. We list here the most well-known Weil cohomology theories, for various fields k.

For every prime number \ell different from the characteristic of k, we have the \ell-adic cohomology, sometimes referred to as \ell-adic étale cohomology, or just étale cohomology. The coefficient field for \ell-adic cohomology is the field \mathbf{Q}_{\ell} of \ell-adic numbers, so the cohomology groups are vector spaces over this field.

If char(k) = 0, we have algebraic de Rham cohomology, with coefficient field k itself.

If \sigma: k \to \mathbf{C} is an embedding of k into the field of complex numbers, we have the so called Betti cohomology associated to \sigma, which is just the singular cohomology of the variety viewed as a complex variety by means of the embedding \sigma. The singular cohomology here is taken with rational coefficients, so the coefficient field of Betti cohomology is the field \mathbf{Q} of rational numbers.

If k is the field \mathbf{Q}_{p} of p-adic numbers for some prime number p, we have p-adic étale cohomology. (This can also be defined for more general fields similar to \mathbf{Q}_{p} – more about this in a future post.)

If k is a perfect field of characteristic $p > 0$, we have crystalline cohomology. The coefficient field in this case is the fraction field of the ring of Witt vectors of k.

All of these cohomologies come with rich extra structure (in addition to being K-vector spaces), such as for example Galois action or Hodge structure. In coming posts we will try to look at each of these cohomology theories in some detail, with their extra structure, and also define all of the terms left undefined above. We will also look at Grothendieck’s idea of pure motives as a “universal Weil cohomology theory”.

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Varieties

Posted by Andreas Holmstrom on March 5, 2009

The most basic class of geometric object encountered in algebraic geometry is of course varieties. Before talking about cohomology of varieties, it seems sensible to say a few words about different types of varieties. I will assume that you know what a variety is – if not, look at Hartshorne or any other introductory book on algebraic geometry, or online notes of Dolgachev, MilneDebarre, VakilGathmann and other people.

A cohomology theory for varieties will typically be a functor from some category Var of varieties to the category of abelian groups or vector spaces. When reading about cohomology for some class of varieties, there are three key questions to ask about the category of varieties considered. 

Question 1: Are the varieties required to be complete/proper/projective? Although these words don’t mean exactly the same thing, they are morally and for most practical purposes the same. 

Question 2: Are the varieties required to be smooth/nonsingular?

Question 3: What is the base field?

The possible answers to Q1 and Q2 give us four possible classes of varieties: Smooth proper varieties, smooth varieties not necessarily proper, proper varieties not necessarily smooth, and general varieties. The third of these seems to be less common, so excluding it leaves us with the three most important classes of varieties, in increasing complexity:

  • A: Smooth proper varieties (really nice and well-behaved)
  • B: Smooth varieties (a bit more complicated, but still nice)
  • C: Arbitrary varieties (nasty things, very hard to understand)

When seeing a category of varieties being introduced, it is often useful to make an internal note of which of the three situations we are in. For example, someone talking about “quasiprojective nonsingular varieties” would be in class B, someone talking about “smooth projective varieties” would be in class A, and someone talking about “integral separated schemes of finite type over the base field” is in class C. In most texts, the author states in the very beginning what he means by “variety”, and it is often one of the first two. We will see later that the right notion of cohomology depends on which situation we are in.

Question 3 also has a big impact on the study of cohomology theories for the varieties in question. Different cohomology theories are defined for different base fields. The most common base fields are: Finite fields, global fields and local fields, algebraic closures of these fields, and the fields \mathbf{R} and \mathbf{C}.

We will soon start looking at cohomology theories for smooth projective varieties, i.e. Weil cohomology theories.

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