This post concludes a series of posts I’ve been writing on the attempt to prove the Weil Conjectures through the Standard Conjectures. (Parts 1, 2, 3, 4, 5.) In this post, I want to explain the idea of the category of motives. In the modern formulation of algebraic topology, cohomology theories are functors from some category of spaces to the category of abelian groups. The category of motives is meant to be a universal category through which any such functor should factor, when the source space is the category of algebraic varieties. At least in the early days of the subject, the gold test of this theory was the question of whether the Weil Conjectures could be proved entirely in this universal setting. Nowadays, this question is still open, but the use of motives has grown. To my limited understanding, this growth has two reasons: among number theorists, it has become clear that motivic language is an excellent way to formulate results on Galois representation theory; among birational geometers and string theorists, many applications have been found for motivic integration. There will be a bunch of category theory in this post, which I hope will make it more attractive to the tensor category crowd.
I am much less comfortable with this topic than the other posts in this series; my understanding doesn’t go much further than Milne’s survey article. So I’m going to make this post a pretty short introduction to the main ideas. That will be the end of my expository posts; I also want to write one more post raising some questions about motives that seem natural to me.
The Ring of Correspondences
Let be a smooth, proper algebraic variety. In previous posts, we have seen the power of studying by introducing the ring whose elements are classes of subvarieties in and where multiplication is given by , with the three projections from to . We are now going to create a category where will be the endomorphism ring of .
The first thing we have to think about is what we mean by the cohomology of . Since motives are supposed to be a universal cohomology theory, we don’t want to bias definitions by building in a particular cohomology theory at the beginning.
The most naive solution would be just to take the vector space of all cycles on , and replace cup product with actual intersection. The reason we can’t do this is that actual intersection will give the wrong answers. For example, consider a line in the projective plane. We want its self intersection to be a point, not
itself. So we need to pass to some sort of theory where intersection products work correctly, which means something like cohomology.
Fortunately, Fulton, MacPherson and others did a magnificent job creating a purely algebraic theory of intersection of algebraic cycles, as summarized in Fulton’s book Intersection Theory. Following the ideas of that book, for a smooth proper algebraic variety , let be the -vector space spanned by algebraic cycles in . Let be a quotient of which is small enough that the operations of cup-product, pull-back and push-forward are well defined. There are four standard choices of , presented in order of increasingly coarse equivalence relations:
- Rational Equivalence: two cycles are equivalent if they can be linked by a family parametrized by .
- Algebraic Equivalence: two cycles are equivalent if they can be linked by a family parameterized by some connected algebraic variety (equivalently, by some connected algebraic curve).
- Cohomological Equivalence: For your favorite cohomology theory , two cycles are the same if they have the same image in .
- Numerical Equivalence: Two cycles and are equivalent if, for all algebraic cycles of complementary dimension, .
The theory of motives can be built using any of these, and they all have strengths and weaknesses. If rational or algebraic equivalence is used, then the endomorphism rings of our objects will contain negligible morphisms. (See Noah and my posts on this in a different context.) Cohomological equivalence builds into the theory a particular choice of cohomology theory. Numerical equivalence is extremely hard to test, as it is hard to deal with the quantification over all cycles. Grothendieck’s Conjecture D states that numerical and cohomological equivalence are, in fact, the same. The others are all known to be different.
is graded by dimension of cycle. We’ll write for the subspace spanned by algebraic cycles of codimension . (Here and throughout the post, dimensions are algebraic. So, if is a variety over , we are talking about cycles whose real codimension is .)
Now, let and be smooth proper varieties, of dimensions and . The category of motives will have objects and (there will be other objects, which we will introduce later). We define to be . We define the composition map by , where is the projection from onto the -th and -th factors.
The category of motives is defined to be, roughly, the category whose objects are smooth, proper algebraic varieties and whose morphisms are given by as above. We’ll fill in the details below. For any cohomology theory and any index , the map is a functor from motives to abelian groups. There is a contravariant functor from smooth proper varieties to motives; a map is sent to its graph, considered as a cycle in .
One thing that I wondered about when I saw this definition: Why not take to be all of , rather than just the codimension part? If we did this, then would be a functor, but the individual components would not. Since (in this series of posts) our motivating goal is to understand the action of Frobenius on various cohomology groups, it would be unfortunate if we could not study one cohomology group at a time.
The first technical point is that we want the category of motives to idempotently complete. That means that, if is a variety, and is an idempotent in the ring , then we adjoin a formal image of the map . I wrote more about this process here.
Top and bottom dimensional cohomology, tensor product structure
Let be smooth and projective of dimension . There are two important classes in : we define to be and to be . The maps and are easily seen to be idempotent. We temporarily define and to be their images.
For any and , the cycle gives a map from to . Moreover, this map is an isomorphism with inverse . In short, and are canonically isomorphic for all and .
For this reason, we will engage in the standard abuse of notation and talk about THE object , which is for every .
If and have the same dimension, then gives a similar canonical isomorphism between and . Again, we will abuse notation and talk about THE object , which is for every -dimensional .
We can describe these structures more concisely by putting a tensor structure on the category of motives. We define to be the object . Given , the map is the obvious product cycle. (One must also extend this definition to the case where and are defined as the images of some idempotents; we leave this to the reader.) Then one can check that is the tensor identity, and is , where we abbreviate to .
If we are working over , then Frobenius acts on by . Many of the classical spaces of algebraic geometry decompose, in the category of motives, as a direct sum of copies of for various ‘s; this is why the number of points on them is given by polynomials in . I love spaces like this, and have spent a lot of time thinking about them but, for number theorists, the interesting parts of the theory only come when we get beyond these examples.
We have put a tensor structure on the category of motives, which means we know how to formally take the tensor product of two motives. We would like to also take the dual of a motive, so that we can formulate statements like Poincare duality in the theory of motives.
Guided by Poincare duality, we can write down the intersection product. Let be the class of the diagonal in ; this gives a map in the category of motives. In particular, acquires a ring structure, as a cohomology theory should have, but this is not what I want to focus on right now. Rather, I want to look at , where . This is a map . In classical algebraic topology, one would choose an isomorphism between the top cohomology of and the bottom (an orientation) and this map would be the Poincare duality pairing.
We’d rather not choose an isomorphism between and . (Among reasons, Frobenius acts on them differently.) Instead, what we do is to formally invert (with respect to tensor product). More specifically, we define to be whichever of and involves a positive exponent. One can show that this is also . So this provides another answer to the question “where did the other graded pieces of go?”
We then define Poincare duality to be true, declaring that the dual of is . Extending to classes which are defined as images of idempotents is, again, left to the reader.
An object of the category of motives is a triple , where is a smooth proper variety, an idempotent in and an integer. One should think of this as “the image of , tensored with .” See Milne’s article for precise definitions of morphisms and tensor structure.
A major issue: The missing grading
We have talked about the idempotents and , which project onto top and bottom cohomology. One of the great missing results of the category of motives is that no one can show that projectors onto the other cohomology groups exist. Specifically, this is Grothendieck’s Conjecture C. Until then, we have the possibility that the universal cohomology theory is not graded, which would seem peculiar. I think this is one of the things that made people hesitate to embrace motives.
I think (please correct me if I am wrong) that such projectors are known to exist for varieties over , using homological equivalence motives, and using the Weil Conjectures. The point is that the characteristic polynomials of Frobenius on the different are relatively prime, since their eigenvalues have different norms. So one can find some polynomial which acts by on and by on the other . This polynomial, interpreted as an endomorphism of the motive , is the desired projector. Of course, this is not helpful to those who want to find a new proof of the Weil conjectures.
If you want to test your understanding of this series of posts, you should attempt to rephrase all of the Standard Conjectures as statements about the category of motives.
Thanks for reading to the end!