Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

What is cohomology?

Posted by Andreas Holmstrom on December 13, 2008

Cohomology (or homology) means different things to different people. The common theme of all notions of cohomology, is the idea of using algebraic invariants to study geometric objects. More precisely, a cohomology theory is a functor from a geometric category (for example CW complexes or schemes) to an algebraic category (for example abelian groups, vector spaces, or modules). This is an extremely powerful idea, as the algebraic objects are often easier to work with, so a problem in geometry can be solved by transferring it to algebra.

We will be a bit sloppy in that we won’t really distinguish between cohomology and homology in the discussion below. Homology usually refers to functors which are covariant, while cohomology refers to functors which are contravariant.

Most commonly, the word cohomology is used to refer to singular cohomology, one of the fundamental notions of algebraic topology. More generally, algebraic topology studies and makes use of generalized cohomology and homology theories, such as K-theory, complex cobordism, and stable homotopy groups. Good online references for these things include this book of May (pdf) and the books of Hatcher.

In mathematics as a whole, there are over 400 different notions of cohomology. The reason for this multitude of cohomologies seems to be that almost any interesting functor from geometry to algebra is referred to as a cohomology theory, regardless of its properties. One of the very few things that all cohomology theories seem to have in common, is the appearance of long exact sequences, which is one of the most important tools for doing actual computations. More generally, the power of cohomology comes from the use of homological algebra, see for example these lecture notes (pdf) of Schapira.

Most of the cohomology theories in mathematics seem to appear in algebraic and arithmetic geometry. Many of these have helped solve some of the major mathematical problems of the past century. I will come back with more posts discussing these in more detail.

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5 Responses to “What is cohomology?”

  1. Just curious- why is the word “group cohomology” applied to a covariant functor? Is it just because group homology already exists? What about sheaf cohomology then?

    • Andreas Holmstrom said

      I think the behaviour of group cohomology is contravariant in the first variable and covariant in the second. You have to be a bit careful though in order to state precisely on which category you have your functor, see for example the brief and clear discussion in Brown: Cohomology of groups, starting at the bottom of p 78. Here is a direct link to the page at Google Books, hope it works.

      A different answer is that the group cohomology of a group is related to the singular cohomology of the corresponding Eilenberg-MacLane space, and similarly for group homology and singular homology. This is probably the historical reason for the terminology. You might enjoy Mac Lane’s survey on the origin of group cohomology, which should be freely available here.

    • Andreas Holmstrom said

      Aha, I now saw that you just wrote a nice post on group cohomology. And I forgot to say that for sheaf cohomology it’s similar, it behaves contravariantly in the first variable and covariantly in the second.

      • Ok, so it’s basically restriction in the first variable (or corestriction, I forget which). Thanks!

        By the way, for sheaf cohomology are you referring to the Ext functors of two sheaf variables, or is the underlying topological space one of the variables?

      • Andreas Holmstrom said

        I come from algebraic geometry, so when talking about sheaf cohomology I think of the scheme (or the manifold) as the first variable and the sheaf as the second variable. But any Hom or Ext bifunctor (when taking two sheaves or other objects in an abelian category as input) would be contravariant in the first variable and covariant in the second.

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