Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Posts Tagged ‘geometry’

What is geometry?

Posted by Andreas Holmstrom on April 20, 2009

(This is an extended set of notes for Part 1 of a talk I gave a few days ago at the Young Researchers in Mathematics conference in Cambridge. Part 2 will be posted soon.)

What is geometry? What is a “space”? When is an object “geometric”? Everyone would agree that a manifold is a geometric object, and similarly for a CW complex, and probably also for a scheme. But what about a group – is it a geometric object? What about a noncommutative ring? These and other mathematical objects form categories – but when should a category be regarded as a geometric category?

The question “What is geometry” is of course very naive, but I believe it is still of some interest. First of all, it is interesting from a historical point of view to look at the answers given at various points in history, and how our idea of geometry has developed over time. Secondly, when developing new forms of geometry, where sometimes even the fundamental definitions and constructions are not completely in place, it could possibly be helpful to have spent some time reflecting on what geometry really is. The new forms of geometry I have in mind include Arakelov geometry, geometry over \mathbf{F}_1, derived algebraic geometry, and various forms of noncommutative geometry. Thirdly, one would like to understand for what kinds of objects one can define “cohomology”. For example, we can define various forms of cohomology for manifolds, schemes, Lie algebras, associative algebras, groups, rigid analytic spaces, \mathbf{C}^{*}-algebras, ring spectra, categories, stacks, operads, and many other things. What exactly do these objects have in common?

What follows is a list of suggestions for conceptual answers to the question “What is geometry?”. Of course the answers are complementary, each of them capturing some particular aspect of what geometry is. 

1. For a very long time, geometry was the same thing as Euclidean geometry, and to say or think something else was almost unheard of. Only in the 19th century did Western mathematicians begin to realize that there could actually be other forms of geometry.

2. The famous Erlangen program, formulated by Klein in 1872, gave a unification of the various types of geometry existing at the time, focusing on the notion of symmetry, and on properties invariant under symmetry groups. These ideas had a huge impact on the development of Lie theory and various other subjects in geometry and physics. 

3. There is something called “Cartan geometries” (developed by Élie Cartan), which appears to be a further generalization of the Erlangen program, including Riemannian geometry in the picture. I have not found a good online source, but there is a book by Sharpe

4. One important way of approaching geometry is to shift focus from the geometric object to some set of functions on the object. For example, one could replace a topological space by the ring of continuous complex-valued functions on the space, or replace an algebraic variety with the ring of polynomial functions on the variety. In many cases, this process gives an equivalence of categories. This approach is the standard way of introducing Grothendieck‘s schemes, and is also the basic idea of noncommutative geometry.

5. Closely related to the previous item is the idea of defining geometry as the study of locally ringed spaces. In their really nice introduction to algebraic geometry, Demazure and Gabriel define a geometric space to be a locally ringed space

6. One could “define” a geometric category as a category admitting an interesting functor to \mathbf{Hot} (the homotopy category of topological spaces). Some examples to motivate this approach: For categories of (well-behaved) topological spaces with some extra structures, e.g. smooth manifolds, there is a forgetful functor to \mathbf{Top} and hence to \mathbf{Hot}. For the category of groups, and more generally the category of small categories, we have the classifying space functor. For the category of (non-negatively graded) chain complexes of abelian groups, we have the Dold-Kan correspondence, which gives a functor to simplicial abelian groups and hence to Hot (more about this example in Part 2 of the talk).

7. Another way of “defining” a geometric category could be: A category admitting some notion of cohomology. The problem with this definition is of course that it is hard to define what exactly we mean by cohomology, but it should be a functor to some abelian category, producing long exact sequences and spectral sequences in ways similar to what we observe in topology and algebraic geometry.

8. There are various approaches to “homotopical categories”, and we could define geometry as the study of these categories. The most well-known approach is probably Quillen’s notion of model categories. There are many other approaches and languages as well, for example various notions of infinity-categories, homotopical categories in the sense of Dwyer-Hirschhorn-Kan-Smith, higher stacks, Segal categories, simplicial sheaves, simplicial categories, A^{\infty}-categories, and more. I will say more about this in Part 2 of the talk.

9. In the 60s, Lawvere developed the concept of a “theory”. As a special case, there is something called a “geometric theory”, which could maybe serve as a way to define what geometry is. For more about this, see the online book by Barr and Wells, in particular sections 4.5 and 8.3.

10. Some people would argue that everything is geometry.

Some remarks: There are probably many other approaches to answering the question we started with. It seems to me that a good definition of geometry should (1) allow for noncommutative structures, and (2) agree with the principle that everything algebraic is also geometric. 

In Part 2 of the talk (to be posted soon!) I will say something about point no 8, homotopical categories, and try to show that they can be useful in algebraic geometry.

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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…

Posted in Cohomology breadcrumb trail | Tagged: , , | 3 Comments »