Early Bourbaki seminars now freely available

Until quite recently, the first 10 volumes of the Bourbaki seminars were under subscriber-access only at NUMDAM, but now they are freely available! These 10 volumes cover the years 1948-1968 and hence contain many gems from the early days of scheme theory and the development of cohomological tools in algebraic geometry.

Motivic homotopy theory in Bonn, and much more

The conference list is now updated with lots of exciting events for the summer. A small sample:

• Higher Structures in Topology and Geometry IV, Göttingen, June 2-4
• Young Topologists Meeting, Copenhagen, June 16-20
• Homotopy theory and derived algebraic geometry, Toronto, Aug 30 – Sep 3
• Geometric aspects of motivic homotopy theory, Bonn, Sep 6-10

I already mentioned the Paris conference on Cohomology of algebraic varieties, Hodge theory, algebraic cycles, and motives in last week of April, but there is still time to register.

Motives in Paris and Regulators in Barcelona

The Events page has been updated again. Some highlights:

• Copenhagen Topology Conference (Jan 8-10)
• Cohomology of algebraic varieties, Hodge theory, algebraic cycles, motives, in Paris (April 26-30)
• Regulators in Barcelona (July 12-22)

Homotopical categories and simplicial sheaves

(This is an expanded version of the 2nd part of a talk I gave last month. For the first part, see this post.)

Homotopical categories

The topic for this post is “homotopical categories”, and their role in algebraic geometry. I want to emphasize that I am very much in the process of learning about all these things, so this post is based more on interest and enthusiasm than actual knowledge. I hope to convey some of the main ideas and why they could be interesting, and come back to the details in many future posts, after having learned more. I apologize for not defining everything carefully, and for brushing the “stable” aspects of the theory, i.e. spectra and sheaves of spectra, under the carpet.

There are many different ways to speak of “homotopical categories”, and I only use this expression because I don’t know of a better thing to call them. The most well-known approach is the language of model categories, invented by Quillen and developed by many others. There are many excellent online introductions, for example Dwyer-Spalinski, Goerss-Schemmerhorn, and appendix A2 of Jacob Lurie’s book on higher topos theory, available on his webpage. Other languages are given by the many different approaches to higher categories; see the nLab page and the survey of Bergner. Still other languages include Segal categories, A-infinity categories, infinity-stacks, and homotopical categories in the precise sense of Dwyer-Hirschhorn-Kan-Smith.

Although I don’t want to go into the details of all these different homotopical/higher-categorical subtleties, I will try to list some of the basic features that “homotopical” categories typically have.

• A homotopical category should behave like a nice category of topological spaces.
• In particular, there should be a class of morphisms called weak equivalences, and:
• To any homotopical category , one should be able to associate a “homotopy category” and a functor which is universal among functors sending weak equivalences to isomorphisms. Morally, is obtained from by “formally inverting the weak equivalences”.
• A homotopical category should admit all limits and colimits, and also homotopy limits and homotopy colimits.
• A homotopical category should be enriched over some kind of spaces, i.e. for any two objects , the set should be a “space” in some sense, for example a simplicial set, a topological space, or a chain complex of abelian groups.

Simplicial objects

Before talking about algebraic geometry, we need to recall some “simplicial language”. The category is defined as follows. Objects are the finite ordered sets of the form . Morphisms are order-preserving functions , i.e. functions such that . If is any category, we define the category of simplicial -objects to be the category in which the objects are the contravariant functors from to , and the morphisms are the natural transformations of functors. There is a functor from to given by sending an object of to the corresponding constant functor, i.e. the functor sending all objects to and all morphisms to the identity morphisms of .

Some examples:

• Take ,  the category of sets. The above construction gives us the category  of simplicial sets. This category is “sort of the same as the category of topological spaces”. The precise statement is that there is a pair of adjoint functors which make  and into Quillen equivalent model categories; in particular, their homotopy categories are equivalent (as categories). For the purposes of algebraic topology, we can work with any of these categories. For example, we can define homotopy groups and various generalized homology and cohomology groups of a simplicial set. The inclusion of into corresponds to viewing a set as a discrete topological space. A weak equivalence between two simplicial sets is a morphism inducing isomorphisms on all homotopy groups.
• Take , the category of abelian groups. There is a forgetful functor from to the category , induced by the forgetful functor from to. The Dold-Kan correspondence tells us that there is an equivalence between and the category of (non-negatively graded) chain complexes of abelian groups. Under this equivalence, homotopy groups of a simplicial abelian group correspond to homology groups of a chain complex.
• Take, the category of-algebras for a commutative ring. Then there is some kind of Dold-Kan correspondence between simplicial algebras and DG-algebras. See Schwede-Shipley for precise statements.
• Take , the category of sheaves of sets on some topological space or site. Then is the category of simplicial sheaves. This category can also be viewed as the category of sheaves of simplicial sets on the site. Any category of simplicial sheaves is a “homotopical category” (I am not making this precise here). For example, one way of defining weak equivalences is to say that a morphism of simplicial sheaves is a weak equivalence iff it induces weak equivalences of simplicial sets on all stalks. 

Homotopical categories in algebraic geometry

Now to algebraic geometry. Through a few examples I want to argue that homotopical categories (in particular categories of simplicial sheaves) provide a useful and natural setting for certain aspects of algebraic geometry.

Firstly, let’s consider the general problem of viewing a cohomology theory as a representable functor. In algebraic topology, the Brown representability theorem says that any generalized cohomology group is representable, when viewed as a functor on the homotopy category of topological spaces. In other words, there is a space such that the cohomology of a space is given by , where the is taken in the homotopy category. Examples include the Eilenberg-MacLane spaces , which represent the singular cohomology groups , and the space , which represents K-theory. The existence of a long exact sequence relating the cohomology groups for various corresponds to the fact that the different Eilenberg-MacLane spaces fit together to form a so called spectrum. The Brown representability theorem is best expressed using the language of spectra, i.e. stable homotopy theory, but I want to postpone a discussion of this to a future post. An interesting aspect of Brown representability for singular cohomology is that by identifying the coefficient group with the corresponding Eilenberg-MacLane space, the two arguments of a singular cohomology group , namely the space and the coefficient group , suddenly are on equal footing. By this I mean that they both live in the same category of topological spaces, rather than in the two separate worlds of topological spaces and abelian groups, respectively.  

In classical algebraic geometry, there is no analogue of Brown representability. Most cohomology theories are of the form , where is some kind of variety, and is a sheaf of abelian groups. One may ask if there is a way to express such a cohomology group as a representable functor. In order to obtain a picture parallell to the topological picture above, a necessary requirement is to have a homotopical category in which the variety and the sheaf both live as objects, “on equal footing”. One possibility for such a category is some category of simplicial sheaves. In order to explain how this works, let us fix some category of varieties, for example the category smooth varieties over some base field . Let us also fix some Grothendieck topology on this category, for example the Zariski topology, the Nisnevich topology, the etale topology, or some flat topology. This defines a site, and we can speak of sheaves on this site, i.e. contravariant functors on , satisfying a “glueing” or “descent” condition with respect to the given topology.

Since Grothendieck, we are familiar with the idea of identifying a variety with the sheaf of sets that it represents, by the Yoneda embedding. We mentioned earlier that for any category , there is a functor . Taking to be the category of sheaves of sets, we get a functor from sheaves of sets to simplicial sheaves. In particular, any variety can be viewed as a simplicial sheaf, by composing the Yoneda embedding with the canonical functor from sheaves of sets to simplicial sheaves. 

We also want to show that a sheaf of abelian groups can be viewed as a simplicial sheaf. We can regard any abelian group as a chain complex, by placing it in degree zero, and placing the zero group in all other degrees. This gives an embedding of the category of abelian groups into the category of chain complexes, and by composing with the Dold-Kan equivalence we get a functor from abelian groups to simplicial sets. This induces a functor from sheaves of abelian groups to simplicial sheaves. More generally, any complex of sheaves of abelian groups can be viewed as a simplicial sheaf.

Now one could hope for an analogue of Brown representability, namely that the sheaf cohomology group could be expressed as , where the Hom is taken in the homotopy category of simplicial sheaves. It seems to be the case that something along these lines should be true. For example, this nLab page on cohomology seems to imply that all forms of cohomology should be of this form, at least sheaf cohomology groups of the type just described. Also, Hornbostel has proved a Brown representability theorem in the setting of motivic homotopy theory.

There are many other phenomena in algebraic geometry which also seem to indicate that categories of simplicial sheaves might be more natural to study than the smaller categories of schemes and varieties we typically consider. Some examples (longer explanations of these will have to wait until future posts):

• It seems to be the case that almost any geometric object generalizing the concept of a variety can be thought of as a simplicial sheaf. Examples: Simplicial varieties, stacks, algebraic spaces.
• Deligne’s groundbreaking work on Hodge theory in the 70s (see Hodge II and Hodge III) uses in a crucial way that the singular cohomology of a complex variety can be defined on the larger category of simplicial varieties. Simplicial varieties are special cases of simplicial sheaves, and I believe it should be true that functors on simplicial varieties can be extended to simplicial sheaves.
• Simplicial varieties/schemes also pop up naturally in other settings. For example, Huber and Kings need K-theory of simplicial schemes for their work on the motivic polylogarithm.
• As already indicated, simplicial sheaves appears to be the most natural domain of definition for many different kinds of cohomology theories.
• Morel and Voevodsky’s  A1-homotopy theory (also known as motivic homotopy theory) is based on categories of simplicial sheaves for the Nisnevich topology.
• Brown showed that Quillen’s algebraic K-theory can be thought of as “generalized sheaf cohomology”, where the coefficients is no longer a sheaf of abelian groups, but a simplicial sheaf.
• The work of Thomason relating algebraic K-theory and etale cohomology uses the language of simplicial sheaves.
• Simplicial sheaves provide a natural language for “resolutions”. For example, it gives a unified picture of the two methods for computing sheaf cohomology: Cech cohomology and injective resolutions.
• Simplicial sheaves seems to be the most natural language for descent theory.
• Toen‘s work on higher stacks can be formulated in terms of simplicial sheaves. 
• Homotopy categories of simplicial sheaves can be thought of a generalization of the more classical derived categories of sheaves. The homotopical point of view seems to clarify some unpleasant aspects of the classical theory of triangulated categories. 

See also the nLab entry on motivation for sheaves, cohomology, and higher stacks.

Questions

I hope to come back to many of these examples in detail. For now, I  just want to list a few questions which I find intriguing. 

• To define a category of simplicial sheaves, we must choose a Grothendieck topology. How does this choice affect the properties of the category we obtain? Morel and Voevodsky work with the Nisnevich topology, Huber and Kings work with the Zariski topology, and Toen (at least sometimes) works with some flat topology. For some purposes, it seems to be the case that we don’t need a topology at all, instead we can just work with simplicial presheaves. What is the role of the Grothendieck topology? 
• Most of the above examples are developed for varieties over a base field of characteristic zero. Based on the above, it seems reasonable to believe that simplicial sheaves are useful in this case, but what if the base scheme is field of characteristic p, a local ring, a Dedekind domain, or something even more general? Is it the case that simplicial sheaves is the most natural language for understanding cohomology theories for arithmetic schemes, such as schemes which are flat and of finite type over ? Are simplicial sheaves important in number theory/Arakelov theory/geometry over the field with one element? What are the obstacles to “doing homotopy theory over an arithmetic base”?

Obviously I hope that there will be interesting answers to these questions, but I am still completely in the dark as to what these answers might be.

What is geometry?

(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 , 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, -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 (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  and hence to . 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, -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.

Varieties

The most basic class of geometric object encountered in algebraic geometry is of course varieties. Before talking about cohomology of varieties, it seems sensible to say a few words about different types of varieties. I will assume that you know what a variety is – if not, look at Hartshorne or any other introductory book on algebraic geometry, or online notes of Dolgachev, Milne, Debarre, Vakil, Gathmann and other people.

A cohomology theory for varieties will typically be a functor from some category of varieties to the category of abelian groups or vector spaces. When reading about cohomology for some class of varieties, there are three key questions to ask about the category of varieties considered. 

Question 1: Are the varieties required to be complete/proper/projective? Although these words don’t mean exactly the same thing, they are morally and for most practical purposes the same. 

Question 2: Are the varieties required to be smooth/nonsingular?

Question 3: What is the base field?

The possible answers to Q1 and Q2 give us four possible classes of varieties: Smooth proper varieties, smooth varieties not necessarily proper, proper varieties not necessarily smooth, and general varieties. The third of these seems to be less common, so excluding it leaves us with the three most important classes of varieties, in increasing complexity:

• A: Smooth proper varieties (really nice and well-behaved)
• B: Smooth varieties (a bit more complicated, but still nice)
• C: Arbitrary varieties (nasty things, very hard to understand)

When seeing a category of varieties being introduced, it is often useful to make an internal note of which of the three situations we are in. For example, someone talking about “quasiprojective nonsingular varieties” would be in class B, someone talking about “smooth projective varieties” would be in class A, and someone talking about “integral separated schemes of finite type over the base field” is in class C. In most texts, the author states in the very beginning what he means by “variety”, and it is often one of the first two. We will see later that the right notion of cohomology depends on which situation we are in.

Question 3 also has a big impact on the study of cohomology theories for the varieties in question. Different cohomology theories are defined for different base fields. The most common base fields are: Finite fields, global fields and local fields, algebraic closures of these fields, and the fields and .

We will soon start looking at cohomology theories for smooth projective varieties, i.e. Weil cohomology theories.

Cohomology breadcrumb trail, the beginning

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…

What is cohomology?

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.