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Navigating the Motivic World Daniel Dugger Navigating the motivic world Daniel Dugger Department of Mathematics, University of Oregon, Eugene, OR 97403 E-mail address: [email protected] Contents Introduction 1 Part 1. Motivation 3 Chapter 1. Introduction to the Weil conjectures 5 1. A first look 5 2. Formal statement of the conjectures 10 3. Zeta functions 13 4. A plan to prove the conjectures 16 5. Some history of the proofs of the conjectures 20 A. Computer calculations 22 B. Computations for diagonal hypersurfaces 27 Chapter 2. Topological interlude: the cohomology of algebraic varieties 37 1. Lefschetz theory 37 2. The Hard Lefschetz theorem 39 3. The Hodge index theorem 41 4. Hodge theory 43 5. Correspondencesandthecohomologyofmanifolds 46 Chapter3. AsecondlookattheWeilconjectures 57 1. Weil cohomology theories 58 2. The K¨unneth conjecture 59 3. The Lefschetz standard conjecture 62 4. Algebraic preliminaries 65 5. The Hodge standard conjecture 67 6. Hodge decompositions in characteristic p 72 7. The Tate conjecture 80 8. The Weil conjectures for abelian varieties 81 Part 2. Machinery 83 Chapter 4. Introduction to ´etale cohomology 85 1. Overview of some key points 85 2. Topological perspectives 87 3. Rigid open covers and generalized Cechcomplexesˇ 92 4. Cohomology via ´etale coverings 98 5. Etale´ maps in algebraic geometry 107 6. Systems of approximations 108 7. Hypercoversand´etalehomotopytypes 110 3 4 CONTENTS 8. Etale´ cohomology and ´etale K-theory 112 Chapter 5. Sheaves and homotopy theory 113 Chapter 6. Topological interlude: Lefschetz pencils 115 1. Background 116 2. The topology of the sum-of-squares mapping 123 3. Lefschetz pencils 129 4. The Picard-Lefschetz formulas 132 5. Construction of Lefschetz pencils 137 6. Leftover proofs and geometrical considerations 143 7. Proof of the variation formula 147 Chapter 7. Deligne’s proof of the Riemann hypothesis 153 1. Grothendieck L-functions 153 2. First reductions of the proof 155 3. Preliminaries on the symplectic group 158 4. The fundamental estimate 159 5. Completion of the proof 162 Part 3. Algebraic K-theory 163 Chapter 8. Algebraic K-theory 165 Part 4. Motives and other topics 167 Chapter 9. Motives 169 1. Topological motives 169 2. Motives for algebraic varieties 172 3. Constructing categories of motives 177 4. Motiviccohomologyandspacesofalgebraiccycles 178 Chapter 10. Crystalline cohomology 181 Chapter 11. The Milnor conjectures 183 1. The conjectures 183 2. Proofoftheconjectureonthenormresiduesymbol 190 3. Proofoftheconjectureonquadraticforms 198 4. QuadraticformsandtheAdamsspectralsequence 201 1. Some examples of the Milnor conjectures 206 2. MoreonthemotivicAdamsspectralsequence 209 Appendix. Bibliography 213 Introduction 1 Part 1 Motivation CHAPTER 1 Introduction to the Weil conjectures The story of the Weil conjectures has many layers to it. On the surface it might seem simple enough: the conjectures postulated the existence of certain formulas for the number of solutions to equations over finite fields. The rather amazing thing, however, is that the conjectures also provided a link between these formulas and the world of algebraic topology. Understanding why such a link should exist—and in the process, proving the conjectures—was one of the greatest mathematical achievements of the twentieth century. It is also one which has had lasting implications. Work on the Weil conjectures was one of the first places that deep algebraic-topological ideas were developed for varieties over arbitrary fields. The continuation of that development has taken us through Quillen’s algebraic K-theory and Voevodsky’s motivic cohomology, and is still a very active area of research. In Sections 1 and 2 of this chapter we will introduce the Weil conjectures via several examples. Section 3 discusses ways in which the conjectures are analogs of properties of the classical Riemann zeta function. Then in Sections 4 and 5 we outline the cohomological approach to the problem, first suggested by Weil and later carried out by Grothendieck and his collaborators. The chapter has two appendices, both dealing with further examples. Appendix A introduces the reader to some tools for computer calculations. Appendix B treats the class of examples originally handled by Weil, which involve an intriguing connection with algebraic number theory. 1. A first look Let’s dive right in. Suppose given polynomials f1,...,fk Z[x1,...,xn]. Fix a prime p, and look at solutions to the equations ∈ f1(x1,...,xn)= f2(x1,...,xn)= . = fk(x1,...,xn)=0 where x ,...,x F m and the coefficients of the f ’s have been reduced modulo 1 n ∈ p i p. Let Nm denote the number of such solutions. Our task will be to develop a formula for Nm as a function of m. In the language of algebraic geometry, the mod p reductions of the fi’s define an algebraic variety X = V (f1,...,fk) over the field Fp. The set of points of this variety defined over the extension field Fpm is usually denoted X(Fpm ), and we have Nm = #X(Fpm ). Example 1.1. Consider the single equation y2 = x3 + x and take p = 2. Over F2 there are exactly two solutions for (x, y), namely (0, 0) and (1, 0). Over 2 F4 = F2[ω]/(ω +ω+1) one has four solutions: (0, 0), (1, 0), (ω,ω), and (ω+1,ω+1). So we have N1 = 2 and N2 = 4. 5 6 1. INTRODUCTION TO THE WEIL CONJECTURES We will mostly want to talk about projective varieties rather than affine va- n n rieties. If F is a field, let A (F ) = (x1,...,xn) : xi F and let P (F ) = n+1 n{ ∈ } [A (F ) 0]/F ∗ where F ∗ acts on A (F ) by scalar multiplication. Given homo- − geneous polynomials fi F [x0,...,xn], consider the set of common solutions to n ∈ the fi’s inside of P (F ). These are the F -valued points of the projective algebraic variety X = V (f1,...,fk). 2 3 2 Example 1.2. Consider the single equation y z = x + xz . Over F2 there are 3 three solutions in P , namely [0, 0, 1], [1, 0, 1], and [0, 1, 0]. Over F4 there are five solutions: [0, 0, 1], [1, 0, 1], [ω,ω, 1], [ω +1,ω +1, 1], [0, 1, 0]. Given homogeneous polynomials f ,...,f Z[x ,...,x ], we will be con- 1 k ∈ 1 n+1 cerned with counting the number of points of V (f1,...,fk) defined over Fpm . We will start by looking at two elementary examples which can be completely under- stood by hand. d d d+1 Example 1.3. X = P . As P (F m ) = [A 0]/(F m )∗ we have p − p (pm)d+1 1 N (X)= − =1+ pm + p2m + + pdm. m pm 1 · · · − r r Example 1.4. X = Gr2(A ), the variety of 2-planes in A . The points of X are linearly independent pairs of vectors modulo the equivalence relation given by the action of GL2. To specify a linearly independent pair, one chooses a nonzero vector v1 and then any vector v2 which is not in the span of v1. The number of ways to make these choices is [(pm)r 1] [(pm)r pm]. Similarly, the number of m 2 − m· 2 m − elements of GL (F m ) is [(p ) 1] [(p ) p ]. Hence one obtains 2 p − · − [(pm)r 1] [(pm)r pm] N (X)= − · − =1+ pm +2p2m +2p3m +3p4m +3p5m + m [(pm)2 1] [(pm)2 pm] · · · − · − 6 To take a more specific example, when X = Gr2(A ) one has m 2m 3m 4m 5m 6m 7m 8m Nm(X)=1+ p +2p +2p +3p +2p +2p + p + p . Now, the above examples are extremely trivial—for reasons we will explain below—but we can still use them to demonstrate the general idea of the Weil conjectures. Recall that the rational singular cohomology groups of the space CP d (with its classical topology) are given by Q if i is even and 0 i 2d, Hi(CP d; Q)= ≤ ≤ (0 otherwise. 6 Likewise, the odd-dimensional cohomology groups of Gr2(C ) all vanish and the even-dimensional ones are given by i 0 1 2 3 4 5 6 7 8 H2i(X; Q) Q Q Q2 Q2 Q3 Q2 Q2 Q Q Note that in both examples the rank of H2i(X; Q) coincides with the coefficient of im p in the formula for Nm(X). This is the kind of phenomenon predicted by the Weil conjectures: relations between a formula for Nm(X) and topological invariants of the corresponding complex algebraic variety. 1. A FIRST LOOK 7 d d In the cases of P and Gr2(A ) (as well as all other Grassmannians), there is a very easy explanation for this coincidence. Consider the sequence of subvarieties 0 1 d 1 d P P P − P ∅⊆ ⊆ ⊆···⊆ ⊆ i i 1 i Each complement P P − is isomorphic to A , and we can calculate the points of Pd by counting the− points in all the complements and adding them up. As the i m i number of points in A defined over Fpm is just (p ) , this immediately gives N (Pd)=1+ pm + p2m + + pdm m · · · just as we found earlier. But the same sequence of subvarieties—now considered over the complex numbers—gives a cellular filtration of CP d in which there is one cell in every even i i 1 dimension. The cells are the complements CP CP − . Of course this filtration d − is precisely what let’s one calculate H∗(CP ; Q). The same kind of argument applies to Grassmannians, as well as other flag varieties. They have so-called “algebraic cell decompositions” given by the Schubert varieties, where the complements are disjoint unions of affine spaces. Counting the n n Schubert cells determines both Nm(Grk(C )) and H∗(Grk(C ); Q).
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