Weil cohomology

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

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

If , we have algebraic de Rham cohomology, with coefficient field itself.

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

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

If 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 .

All of these cohomologies come with rich extra structure (in addition to being -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”.