Motivic Cohomology I: correspondences

Where we’re at, and where we’re going

Over the course of the next 3 months I’m going to learn the basics of motivic cohomology (and the construction of Voevodsky’s geometric motives). During this time, I’ll make it a habit to make a blog post covering what I’ve learned each week. The goal will be, at the end of this semester, to have a concise overview of the theory in the form of ~12 blog posts. There will be four parts: I. Background, II. tbd, III. tbd, IV. Comparison results and computations.

Warning: Regarding motivation and historical accuracies be very wary – as I probably have not checked any of the sources to repeat this. Be also wary of the maths but, to a lesser extent as I will double check this. By convention, all my schemes are separated and of finite type over a field.

Motivation

There are a lot of individual sources for the motivation of motives in algebraic geometry. A great source is James Milne’s article Motives – Grothendieck’s Dream, where he surveys the original development for the concept of a motive. While the things we discuss are related to the ideas of the above-mentioned article, they are strictly different as well.

In particular, Grothendieck did define a category of pure motives which is constructed from the category of smooth projective schemes over a field k, which I’ll write as \mathsf{SmPr}_k. We will start with a different category, namely smooth schemes over k (i.e. we don’t require the projective hypothesis). Grothendieck wanted to extend his ideas to the category of all schemes over k, and use this new category of “mixed motives” to define a universal Weil cohomology theory from which all other Weil cohomology theories would be realizations of the universal one.

We’re going to do something a bit different. We’ll try to construct a (co)homology theory which is an algebraic version of singular (co)homology (parentheses because I think it’s actually homology). In the process we’ll also construct a category DM_{gm}^{eff}(k,R) of geometric motives, which will contain Grothendieck’s pure motives as a full subcategory in special cases (probably will discuss this in more detail in part IV.). If, in addition, k admits a resolution of singularities, then DM_{gm}^{eff}(k,R) will contain the motive M(Y) for any scheme Y over k. There still isn’t a defined category of mixed motives, but it could be possible we will describe the derived category of such a category (that which is currently lacking is a defined way of picking out such a category given its derived category).

Alright, I’ll stop talking about things I don’t know about and start describing what I can just copy from various sources, all the while introducing errors due to total ignorance.

Finite Correspondences

We’re going to describe a category \mathsf{Cor}_k which in some sense should linearize the category, \mathsf{Sm}_k, of smooth schemes over k. We will keep the same objects as \mathsf{Sm}_k but we are going to enlarge the set of morphisms between any two smooth schemes X,Y so that \text{Hom}_{\mathsf{Cor}}(X,Y) becomes an abelian group containing \text{Hom}_{\mathsf{Sm}}(X,Y).

But how do we go about getting more morphisms between schemes? One could observe any morphism f:X\rightarrow Y determines a subscheme of the product X\times Y by the embedding X\xrightarrow{\Delta} X\times X\xrightarrow{\text{id}\times f} X\times Y called the graph of f; we will denote by \Gamma_f the graph as a subscheme.

Remark: we can naively describe the graph on points by x\mapsto (x,x)\mapsto (x,f(x)). Of course points don’t always make sense in algebraic geometry but, if we adopt the functor point-of-view then this is exactly what happens.

If we have another morphism g:Y\rightarrow Z then we can “compose” two graphs by taking the intersection of the products \Gamma_f\times Z and X\times \Gamma_g and then pushing forward the intersection to X\times Z. Naively using points we could write

(\{(x,f(x))\}\times Z)\cap (X\times\{(y,g(y))\})=\{(x,f(x),g\circ f(x))\}\mapsto \{(x,g\circ f (x))\}

This gives us a way to describe morphisms f:X\rightarrow Y as closed subschemes of the product X\times Y which are isomorphic to X under the first projection \text{pr}_1.

What we’re about to do isn’t really a new idea. For a long time it had been noticed that some objects which we would normally think of as functions are not functions. For example, z=f(w)=w^{1/n} is a “function” which gives generically n (at 0 and \infty it is single valued – but everywhere else it gives back exactly n values) complex values z for any input  w, so can not be an honest function.  If we wanted to still consider this a function we have to make a choice of wedge of the complex plane on which there is only one value returned as z for any input w. It was Riemann (I’m pretty sure) who had the idea to, instead, consider the graph of the transpose of such f! Using our example, the function z=f(w)=w^{1/n}, a generically n-valued function, should instead give an object S_n=\{ (z,w)\in \mathbb{P}^1(\mathbb{C})\times \mathbb{P}^1(\mathbb{C}):z^n=w\} which projects to the Riemann sphere \mathbb{P}^1(\mathbb{C}) in the w coordinate with generically n preimages.

With this in mind we’ll define the group \text{Cor}(X,Y), which will be our morphisms between X,Y in the category \mathsf{Cor}_k, to be the what we would get if we allowed multivalued functions between schemes. That is, if a function defines a graph which is an isomorphism with the first factor under the projection, a correspondence will define a subscheme of the product which is finite and surjective under the projection to the first factor.

To give a formal definition let us assume the case X is connected. Otherwise, X=\amalg_i X_i is the disjoint union of smooth schemes X_i, and we will define \text{Cor}(X,Y)=\oplus_i \text{Cor}(X_i,Y).

(1.1) Definition: an elementary correspondence from X to Y is a closed integral (i.e. reduced and irreducible) subscheme W\subset X\times Y which is finite and surjective under the first projection to X. We let \text{Cor}(X,Y) be the free abelian group generated by elementary correspondences. When considered as an element of \text{Cor}(X,Y) we denote the elementary correspondence W by [W].

Remark: The definition of an elementary correspondence makes sense when X is smooth and Y is any scheme. Having Y arbitrary in \text{Cor}(X,Y) will allow us, eventually, to define the notion of a motive for any scheme Y.

We call the elements of \text{Cor}(X,Y) finite correspondences. The finite correspondences should be thought of as keeping track of subschemes of X\times Y which are finite and surjective over X along with their geometric multiplicity. We can even show there is a uniquely determined assignment Z\mapsto \sum_i n_i[W_i] where [W_i] are the irreducible components of Z and n_i=\ell (\mathcal{O}_{Z,W_i}) are their geometric multiplicities.

The above means we can associate to any morphism f:X\rightarrow Y an element [\Gamma_f], which is the image of the graph X\xrightarrow{\Delta}X\times X\xrightarrow{\text{id}\times f} X\times Y. If X is connected, then since X is smooth it is reduced, the graph will be integral and hence [\Gamma_f] is an elementary correspondence in this case.

Now our aim will be to define an associative and bilinear composition of correspondences which generalizes the aforementioned one described for the graphs of morphisms (this will show we can realize \mathsf{Sm}_k as a subcategory of \mathsf{Cor}_k – once we fully define that as well). First, we must recall some notions of intersection theory.

By a cycle in a scheme X I mean an integral closed subscheme of X. I write Z(X) for the free abelian group generated by the cycles of X. Note that we have \text{Cor}(X,Y)\subset Z(X\times Y). For a proper (for example a finite) map f:X\rightarrow Y we have a \mathbb{Z}-linear homomorphism f_\ast:\mathbb{Z}(X)\rightarrow \mathbb{Z}(Y) defined by f_\ast([W])=d[V] where V=f(W) and d=[k(V):k(W)] the degree as a field extension if \text{dim}(W)=\dim(V) and d=0 otherwise. There is also an intersection product, \cap defined for cycles which when taken between two cycles of proper intersection, say Q,P, we find [P\cap Q]=\sum_i n_i [W_i] where W_i are the irreducible components of the intersection and n_i are their geometric multiplicity.

We need one more fact before defining the composition of correspondences.

(1.2) Lemma: suppose Y is a normal scheme. Let W\subset X\times Y be finite and surjective over X, and similarly for V\subset Y\times Z. Then W\times Z and X\times V intersect properly, and each component of their pushforward to X\times Z is finite and surjective over X.

References: [MVW], Lemmas 1.4-1.7.

(1.3) Definition: let [W]\in \text{Cor}(X,Y) and [V]\in \text{Cor}(Y,Z) be elementary correspondences. We define their composition to be the class [(\text{pr}_{13})_\ast(W\times Z \cap X\times V)] in \text{Cor}(X,Z).

(1.4) Proposition: composition of correspondences is associative and bilinear.

Proof.  Let \alpha\in \text{Cor}(X,Y), \beta\in \text{Cor}(Y,Z), and \gamma\in \text{Cor}(Z,W). Then, with abuse of notation letting p stand generically for the projection maps,

\gamma \circ (\beta \circ \alpha) = p_\ast( X\times \gamma \cap p_\ast (X\times \beta \cap \alpha \times Z)\times W)

= p_\ast(X\times \gamma \cap p_\ast ( X\times \beta\cap \alpha\times Z)\times W))

= p_\ast (p_\ast(X\times Y \times \gamma) \cap p_\ast ( X\times \beta\cap \alpha\times Z)\times W))

= p_\ast(p_\ast(X\times Y\times \gamma)\cap p_\ast (X\times \beta \times W\cap \alpha \times Z\times W))

= p_\ast(p_\ast(X\times Y \times \gamma \cap X\times \beta \times W\cap \alpha\times Z\times W))

= p_\ast( p_\ast(X\times Y\times \gamma \cap X\times \beta \times W)\cap \alpha \times W)

= p_\ast (X\times p_\ast(Y\times \gamma \cap \beta\times W) \cap \alpha \times W) = (\gamma\circ \beta)\circ \alpha

For the second part of the claim, a similar computation (and symmetry for composition on the other side) yields

\gamma\circ (f+g)= p_\ast(X\times \gamma \cap (f+g)\times Z)

= p_\ast (X\times\gamma \cap (f\times Z+g\times Z))

=p_\ast(X\times Y \cap f\times Z + X\times \gamma \cap g\times Z)

=p_\ast(X\times \gamma \cap f\times Z) + p_\ast(X\times \gamma \cap g\times Z)=\gamma\circ f+ \gamma \circ g.\square

All together we have proved the following:

there is a category \mathsf{Cor}_k whose objects are the smooth schemes over k and whose morphisms are the abelian groups \text{Hom}_{\mathsf{Cor}}(X,Y):=\text{Cor}(X,Y). Together with the last proposition we’ve defined an additive category; the 0 object of this category is the empty scheme, and the direct sum is given by the disjoint union of schemes. Further, our computation of the composition of graphs from earlier essentially goes through as well, so that we have a faithful embedding \mathsf{Sm}_k\hookrightarrow \mathsf{Cor}_k.

We’ll end with two observations about \mathsf{Cor}_k. First, it is a symmetric monoidal category. This will be useful later but for now all it means is that X\otimes Y: X\times_k Y is a symmetric, i.e. X\otimes Y\cong Y\otimes X, product-like operation between the objects of this category. The second observation is that we could have, in the definition, used the free R-module generated by elementary correspondences, i.e. define a new module \text{Cor}_R(X,Y)=\text{Cor}(X,Y)\otimes R. There would be almost no change to the rest of the development so we will keep this in mind and, whenever we wish to consider the R-linear category with objects smooth schemes and morphisms \text{Cor}_R(X,Y) we will write \mathsf{Cor}_k(R).


References:

[MVW] Motivic Cohomology – Mazza, Voevodsky, Weibel. Link. (Try Google if the link expires).

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One comment on “Motivic Cohomology I: correspondences

  1. […] last left off with, Motivic Cohomology I: correspondences. In this post it was described how we could construct a new category so that we have a faithful […]

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