Motive-ating the Weil Conjecture Proof

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 Image may be NSFW.
Clik here to view. be a smooth, proper algebraic variety. In previous posts, we have seen the power of studying Image may be NSFW.
Clik here to view. by introducing the ring Image may be NSFW.
Clik here to view. whose elements are classes of subvarieties in Image may be NSFW.
Clik here to view. $H^*(X \times X)$ and where multiplication is given by Image may be NSFW.
Clik here to view. $\alpha \cdot \beta = (\pi_{13})_* \left( \pi_{12}^* \alpha \cup \pi_{23}^* \beta \right)$ , with Image may be NSFW.
Clik here to view. $\pi_{ij}$ the three projections from Image may be NSFW.
Clik here to view. $X \times X \times X$ to Image may be NSFW.
Clik here to view. $X \times X$ . We are now going to create a category where Image may be NSFW.
Clik here to view. will be the endomorphism ring of Image may be NSFW.
Clik here to view..

The first thing we have to think about is what we mean by the cohomology of Image may be NSFW.
Clik here to view. $X \times X$ . 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 Image may be NSFW.
Clik here to view. $X \times X$ , 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 Image may be NSFW.
Clik here to view., let Image may be NSFW.
Clik here to view. C(Z) be the Image may be NSFW.
Clik here to view. $\mathbb{Q}$ -vector space spanned by algebraic cycles in Image may be NSFW.
Clik here to view.. Let Image may be NSFW.
Clik here to view. A(Z) be a quotient of Image may be NSFW.
Clik here to view. C(Z) which is small enough that the operations of cup-product, pull-back and push-forward are well defined. There are four standard choices of Image may be NSFW.
Clik here to view. A(Z) , presented in order of increasingly coarse equivalence relations:

Rational Equivalence: two cycles are equivalent if they can be linked by a family parametrized by Image may be NSFW.
Clik here to view. $\mathbb{P}^1$ .
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 Image may be NSFW.
Clik here to view., two cycles are the same if they have the same image in Image may be NSFW.
Clik here to view..
Numerical Equivalence: Two cycles Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. are equivalent if, for all algebraic cycles Image may be NSFW.
Clik here to view. of complementary dimension, Image may be NSFW.
Clik here to view. $\int C_1 D = \int C_2 D$ .

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.

Image may be NSFW.
Clik here to view. A(Z) is graded by dimension of cycle. We’ll write Image may be NSFW.
Clik here to view. $A^{i}(Z)$ for the subspace spanned by algebraic cycles of codimension Image may be NSFW.
Clik here to view.. (Here and throughout the post, dimensions are algebraic. So, if Image may be NSFW.
Clik here to view. is a variety over Image may be NSFW.
Clik here to view. $\mathbb{C}$ , we are talking about cycles whose real codimension is Image may be NSFW.
Clik here to view..)

Now, let Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. be smooth proper varieties, of dimensions Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view.. The category of motives will have objects Image may be NSFW.
Clik here to view. [X] and Image may be NSFW.
Clik here to view. [Y] (there will be other objects, which we will introduce later). We define Image may be NSFW.
Clik here to view. $\mathrm{Hom}([X],[Y])$ to be Image may be NSFW.
Clik here to view. $A^{d}(X \times Y)$ . We define the composition map Image may be NSFW.
Clik here to view. $\mathrm{Hom}([X],[Y]) \times \mathrm{Hom}([Y], [Z]) \to \mathrm{Hom}([X], [Z])$ by Image may be NSFW.
Clik here to view. $\alpha \circ \beta = (\pi_{13})_* \left( \pi_{12}^* \alpha \cup \pi_{23}^* \beta \right)$ , where Image may be NSFW.
Clik here to view. $\pi_{ij}$ is the projection from Image may be NSFW.
Clik here to view. $X \times Y \times Z$ onto the Image may be NSFW.
Clik here to view.-th and Image may be NSFW.
Clik here to view.-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 Image may be NSFW.
Clik here to view. $\mathrm{Hom}$ as above. We’ll fill in the details below. For any cohomology theory Image may be NSFW.
Clik here to view. H^* and any index Image may be NSFW.
Clik here to view., the map Image may be NSFW.
Clik here to view. $X \mapsto H^i(X)$ is a functor from motives to abelian groups. There is a contravariant functor from smooth proper varieties to motives; a map Image may be NSFW.
Clik here to view. $f: Y \to X$ is sent to its graph, considered as a cycle in Image may be NSFW.
Clik here to view. $X \times Y$ .

One thing that I wondered about when I saw this definition: Why not take Image may be NSFW.
Clik here to view. $\mathrm{Hom}(X,Y)$ to be all of Image may be NSFW.
Clik here to view. $A(X \times Y)$ , rather than just the codimension Image may be NSFW.
Clik here to view. part? If we did this, then Image may be NSFW.
Clik here to view. $X \mapsto H^*(X)$ would be a functor, but the individual components Image may be NSFW.
Clik here to view. $X \mapsto H^i(X)$ 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.

Idempotent Completion

The first technical point is that we want the category of motives to idempotently complete. That means that, if Image may be NSFW.
Clik here to view. is a variety, and Image may be NSFW.
Clik here to view. $\pi$ is an idempotent in the ring Image may be NSFW.
Clik here to view. $\mathrm{Hom}([X],[X])$ , then we adjoin a formal image of the map Image may be NSFW.
Clik here to view. $\pi$ . I wrote more about this process here.

Top and bottom dimensional cohomology, tensor product structure

Let Image may be NSFW.
Clik here to view. be smooth and projective of dimension Image may be NSFW.
Clik here to view.. There are two important classes in Image may be NSFW.
Clik here to view. $\mathrm{Hom}([X],[X])$ : we define Image may be NSFW.
Clik here to view. e_0 to be Image may be NSFW.
Clik here to view. $\{ \mathrm{pt} \} \times X$ and Image may be NSFW.
Clik here to view. e_d to be Image may be NSFW.
Clik here to view. $\{ X \} \times \{ \mathrm{pt} \}$ . The maps Image may be NSFW.
Clik here to view. e_0 and Image may be NSFW.
Clik here to view. e_d are easily seen to be idempotent. We temporarily define Image may be NSFW.
Clik here to view. M^0(X) and Image may be NSFW.
Clik here to view. M^d(X) to be their images.

For any Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view., the cycle Image may be NSFW.
Clik here to view. $\{ \mathrm{pt} \} \times Y$ gives a map from Image may be NSFW.
Clik here to view. M^0(X) to Image may be NSFW.
Clik here to view. M^0(Y) . Moreover, this map is an isomorphism with inverse Image may be NSFW.
Clik here to view. $\{ \mathrm{pt} \} \times X \subset Y \times X$ . In short, Image may be NSFW.
Clik here to view. M^0(X) and Image may be NSFW.
Clik here to view. M^0(Y) are canonically isomorphic for all Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view..
For this reason, we will engage in the standard abuse of notation and talk about THE object Image may be NSFW.
Clik here to view. $\boldsymbol{1}$ , which is Image may be NSFW.
Clik here to view. M^0(X) for every Image may be NSFW.
Clik here to view..

If Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. have the same dimension, then Image may be NSFW.
Clik here to view. $X \times \{ \mathrm{pt} \}$ gives a similar canonical isomorphism between Image may be NSFW.
Clik here to view. M^d(X) and Image may be NSFW.
Clik here to view. M^d(Y) . Again, we will abuse notation and talk about THE object Image may be NSFW.
Clik here to view. L^d , which is Image may be NSFW.
Clik here to view. M^d(X) for every Image may be NSFW.
Clik here to view.-dimensional Image may be NSFW.
Clik here to view..

We can describe these structures more concisely by putting a tensor structure on the category of motives. We define Image may be NSFW.
Clik here to view. $[X_1] \otimes [X_2]$ to be the object Image may be NSFW.
Clik here to view. $[X_1\times X_2]$ . Given Image may be NSFW.
Clik here to view. $f_i \in \mathrm{Hom}(X_i, Y_i)$ , the map Image may be NSFW.
Clik here to view. $f_1 \otimes f_2$ is the obvious product cycle. (One must also extend this definition to the case where Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. are defined as the images of some idempotents; we leave this to the reader.) Then one can check that Image may be NSFW.
Clik here to view. $\boldsymbol{1}$ is the tensor identity, and Image may be NSFW.
Clik here to view. L^d is Image may be NSFW.
Clik here to view. $L^{\otimes d}$ , where we abbreviate Image may be NSFW.
Clik here to view. L^1 to Image may be NSFW.
Clik here to view..

If we are working over Image may be NSFW.
Clik here to view. $\mathbb{F}_p$ , then Frobenius acts on Image may be NSFW.
Clik here to view. $L^{d}$ by Image may be NSFW.
Clik here to view. p^d . Many of the classical spaces of algebraic geometry decompose, in the category of motives, as a direct sum of copies of Image may be NSFW.
Clik here to view. $L^{\otimes k}$ for various Image may be NSFW.
Clik here to view.‘s; this is why the number of points on them is given by polynomials in Image may be NSFW.
Clik here to view.. 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.

Dualization

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 Image may be NSFW.
Clik here to view. $\Delta$ be the class of the diagonal in Image may be NSFW.
Clik here to view. $(X \times X) \times X$ ; this gives a map Image may be NSFW.
Clik here to view. $[X] \otimes [X] \to [X]$ in the category of motives. In particular, Image may be NSFW.
Clik here to view. [X] 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 Image may be NSFW.
Clik here to view. $\Delta \circ e_d$ , where Image may be NSFW.
Clik here to view. $d=\dim X$ . This is a map Image may be NSFW.
Clik here to view. $[X] \otimes [X] \to L^{d}$ . In classical algebraic topology, one would choose an isomorphism between the top cohomology of Image may be NSFW.
Clik here to view. and the bottom (an orientation) and this map would be the Poincare duality pairing.

We’d rather not choose an isomorphism between Image may be NSFW.
Clik here to view. $L^{\otimes d}$ and Image may be NSFW.
Clik here to view. $\boldsymbol{1}$ . (Among reasons, Frobenius acts on them differently.) Instead, what we do is to formally invert Image may be NSFW.
Clik here to view. (with respect to tensor product). More specifically, we define Image may be NSFW.
Clik here to view. $\mathrm{Hom}(X \otimes L^{a}, Y \otimes L^b)$ to be whichever of Image may be NSFW.
Clik here to view. $\mathrm{Hom}(X, Y \otimes L^{b-a})$ and Image may be NSFW.
Clik here to view. $\mathrm{Hom}(X \otimes L^{a-b}, Y)$ involves a positive exponent. One can show that this is also Image may be NSFW.
Clik here to view. $A^{\dim X + b-a}(X \times Y)$ . So this provides another answer to the question “where did the other graded pieces of Image may be NSFW.
Clik here to view. $A^*(X \times Y)$ go?”

We then define Poincare duality to be true, declaring that the dual of Image may be NSFW.
Clik here to view. $[X] \otimes L^{a}$ is Image may be NSFW.
Clik here to view. $[X] \otimes L^{-d-a}$ . Extending to classes which are defined as images of idempotents is, again, left to the reader.

Summary

An object of the category of motives is a triple Image may be NSFW.
Clik here to view. $(X, \pi, k)$ , where Image may be NSFW.
Clik here to view. is a smooth proper variety, Image may be NSFW.
Clik here to view. $\pi$ an idempotent in Image may be NSFW.
Clik here to view. $\mathrm{Hom}([X],[X])$ and Image may be NSFW.
Clik here to view. an integer. One should think of this as “the image of Image may be NSFW.
Clik here to view. $\pi:[X] \to [X]$ , tensored with Image may be NSFW.
Clik here to view. L^k .” See Milne’s article for precise definitions of morphisms and tensor structure.

A major issue: The missing grading

We have talked about the idempotents Image may be NSFW.
Clik here to view. e_0 and Image may be NSFW.
Clik here to view. e_d , 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 Image may be NSFW.
Clik here to view. $\mathbb{F}_p$ , using homological equivalence motives, and using the Weil Conjectures. The point is that the characteristic polynomials of Frobenius on the different Image may be NSFW.
Clik here to view. H^i(X) are relatively prime, since their eigenvalues have different norms. So one can find some polynomial Image may be NSFW.
Clik here to view. e_i(F) which acts by Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. H^i(X) and by Image may be NSFW.
Clik here to view. on the other Image may be NSFW.
Clik here to view. H^j(X) . This polynomial, interpreted as an endomorphism of the motive Image may be NSFW.
Clik here to view. [X] , is the desired projector. Of course, this is not helpful to those who want to find a new proof of the Weil conjectures.

Review Question

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!

Motive-ating the Weil Conjecture Proof

The Ring of Correspondences

Idempotent Completion

Top and bottom dimensional cohomology, tensor product structure

Dualization

Summary

A major issue: The missing grading

Review Question

Trending Articles

Practice Sheet of Right form of verbs for HSC Students

Download: FK ft Shenky – Nakuyewa ”Prod by: Shenky”

How to win at Markstrat (Markstrat Tips and Tricks) – Vodites

Ominde Commission Report and Recommendations – Ominde Report of 1964

Bureau of Internal Revenue: Regional Offices (Directory)

GO 53 on Enhancement of Ex-gratia upto 5 Lakhs Toddy Tappers in Telangana

Cakewalk CA-2A Leveling Amplifier v2.0.1.97 WiN, v2.0.1.96 OSX Incl Keygen

Mp3 Download: Mdu - Kunjenjenjena

How the kill the job , when DTP request running for long hours.

Microsoft Intune から展開しているアプリのアップデートについて

18-year-old girl was beaten for half an hour by two Northampton men in 'an...

Car crash in Dunton Bassett leaves driver in critical condition

Macky 2, Two Others In Road Accident

Application log 00000000000000089514: Could not convert queue DLVST90CLNT

Detroit mafia: D’Anna Brothers agree to plea deal

Delivery block field greyed out using VA02

Muloraki Au

【個人撮影】スマホのプライベート映像♪「中に出さないで///」カラオケ屋での生ハメ撮りが流出ｗ【リベンジポルノ】＠PornHub

BREAKING NEWS: Diamond Platnumz Is Reported Dead After Ghastly Car Accident

FIAT 500 B0111 B0112