Holmstrom B10 Introduction

This is a first and very rough draft of an attempt to write a “Rough guide to cohomology theories in algebraic geometry”. The plan is that this guide should consists of three parts:

1. A “dictionary” listing all cohomology theories in the mathematical literature, together with a short introduction and basic facts about each of them. Draft versions for such dictionary entries are currently listed here
2. A sequence of short essays (or think of them as blog posts), each one explaining and clarifying some aspect of cohomology theories in algebraic geometry.
3. A “Glossary” of interesting notions, conjectures, ideas, and mathematicians which are related to cohomology in algebraic geometry but which did not find a natural place in one of the previous two parts. A draft version of such a glossary can be found here, but at the time of writing, this glossary really contains a lot of unreadable notes with no structure whatsoever.

The motivation for writing this guide comes mainly from trying to understand the many deep cohomological mysteries of arithmetic geometry. In the setting of algebraic topology, we understand the concept of cohomology very well. In the setting of algebraic geometry over a field, we understand cohomology reasonably well, and although there are still many unsolved questions, at least we know what the important cohomology theories are and what questions to ask about them. However, in arithmetic geometry, it seems like our understanding of cohomology is very rudimentary. Even very basic questions (such as finite generation of various cohomology groups) are mostly completely out of reach, and when it comes to more serious questions such as those related to L-functions and zeta functions, there is a complete lack of tools, ideas, and conceptual frameworks which would allow us to make significant progress. Of course there are some small exceptions, but this is my general feeling. A very long time ago, as I was in the beginning of my PhD, I wrote this note expressing some of this frustration. Anyway, if we understand the situation in topology and algebraic geometry well, it should make us better equipped to start working on the mysteries of arithmetic geometry.

Needless to say, the exposition here will be quite unbalanced, with emphasis on the areas I know well, such as motivic homotopy theory, and gaps in other areas, such as for example p-adic cohomology. But I have aimed for completeness, and am quite sure that at least all major cohomology theories are mentioned. In this first draft I have been extremely brief and sometimes a bit unprecise.

Throughout the posts there will be lots of references, but here I would like to mention two general survey references which are extremely useful for anyone interested in cohomology in algebraic geometry. The first is the two Motives volumes from a conference in Seattle in the early 90s. A scanned version (40MB) can be found here: Volume I and Volume 2. The other is the K-theory handbook, also in two volumes, which is now freely available here. Together all these surveys give a good picture of what research in this area is about.

Other useful overview references: Dugger’s book project Navigating the motivic world

nLab page on B10 Introduction