Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Cohomology breadcrumb trail, the beginning

Posted by Andreas Holmstrom on February 25, 2009

On of the main reasons for setting up this blog is that I would like to write a reasonably coherent set of notes, giving an overview of cohomology theories in algebraic geometry. Actually “overview” might not be the right word, I am rather thinking of a “fil d’Ariane”, or “breadcrumb trail”, which would allow a serious student to obtain some kind of overview if she so wished. The notes I would like to write is the kind of thing I wish someone had given me when I started my graduate studies. At that time, I tried to think about various problems in number theory, but found that I always ran into trouble with various kinds of cohomology, and I could not make any sense or see much pattern in them. I asked five different mathematicians what a cohomology theory actually is, in algebraic geometry, and I got five different answers. When I a few months into my graduate studies listened to a talk by Guido Kings, and he used “rigid syntomic cohomology” as if nothing could have been more basic or natural, I decided I would start writing down notes and collecting facts and references with the aim of one day in the distant future becoming fluent in cohomological language. That day is still rather far away, but at least I hope I have come to the point where writing down a set of rough notes would help my own thought processes. So this is what I will try to do, and if anyone else gets any benefit from this, that would be an extra bonus. I will use the tag “Cohomology breadcrumb trail” for posts which belong to these notes.

One of the things that makes algebraic geometry difficult and interesting, is that there are lots of different kinds of geometric objects. Examples include various classes of varieties, various kinds of more general schemes (for example over arithmetic rings), different kinds of stacks, algebraic spaces, motives, and simplicial sheaves. There are also notions such as log geometry, rigid geometry, derived algebraic geometry, and various forms of noncommutative geometry. For each of these types of geometry, there is a a number of different cohomology theories which can be used to define invariants of the geometric objects.  

The multitude of cohomology theories is frequently a source of confusion. To mention just one single example, people often talk about the “universal cohomology theory”. However, “universal” can mean different things, and depending on what you mean, the universal cohomology theory can be Grothendieck’s Chow motives, Voevodsky’s motivic cohomology, or the algebraic cobordism theory of Levine and Morel. I hope to be able to clarify this and many other similar things, and to give a short introduction to all kinds of cohomology in algebraic geometry. This might of course be too ambitious a goal, but there is no harm in trying…

3 Responses to “Cohomology breadcrumb trail, the beginning”

  1. Thanks for your blog, I really appreciate any efforts on explaining the different cohomology theories to the beginner in algebraic geometry.

    You have talked about the different geometries. Maybe you could start by explaining these? Of course, varieties should be known to the reader of this blog, schemes maybe, too. The notion of stacks has no geometric meaning for me yet and I don’t know algebraic spaces or motives.

    I would also be very happy if you pointed me to the sources from which you learnt cohomology theories.

    I just know sheaf cohomology (thus standard singular & deRham & simplicial) and now start learning etale cohomology – mainly because I am interested in Grothendiecks etale fundamental group. So I hope you will explain the homotopy theory in between, too.

  2. Thomas said

    A try to answer:

    On etale cohomology I’d recomend as easy to read entry Tamme’s nice “Etale Cohomology” and the first ca. two chapters of Milne’s book resp. it’s shorter online version for etale, henselian etc. algebraic issues, then the beautifull SGA 4 1/2 .

    Conc the etale fundamental group, Pop’s “Anabelian Phenomena in Geometry and Arithmetic”, Makoto Matsumoto “Galois representations in fundamental
    groups and their Lie algebras”. Short surveys on the fundamental group and many interesting other things like semi-stable reduction, the anabelian program, e.g. Mézard “Fundamental group”, are in “Courbes semi-stable et groupe fundamental en géometrié algébraique”.

  3. Thomas said

    Here is a related thread in an other blog.

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