INTERSECTION THEORY CLASSES 20 AND 21: BIVARIANT INTERSECTION THEORY RAVI VAKIL CONTENTS 1. What we re doin this week 1 2. Precise statements 2 2.1. Basic operations and properties 4 3. Provin thins 6 3.1. The Chow ( cohomoloy ) rin 6 4. Thins you miht want to be true 7 4.1. Poincare duality 7 4.2. Chern classes commute with all bivariant classes 8 4.3. Bivariant classes vanish in dimensions that you d expect them to 8 1. WHAT WE RE DOING THIS WEEK In this inal week o class, I ll describe bivariant intersection theory, coverin much o Chapter 20. Aain, you should notice that iven chapters 1 throuh 6, we can comortably jump into chapter 20. Suppose : Y is any morphism. Throuhout today and Wednesday s lectures, we ll use the ollowin notation. Suppose we are iven any Y Y. Deine = Y Y, so we have a iber square Y Y. Recall that the inal undamental intersection construction we came up with was the ollowin. Suppose is a local complete intersection o codimension d (or more enerally a local complete intersection morphism). Then we deined! : A k Y A k d Date: Monday, November 29 and Wednesday, December 1, 2004. 1
or all Y Y. These Gysin pullbacks were well-behaved in all ways, and in particular compatible with proper pushorward, lat pullback, and intersection products. An earlier example was that o a lat pullback; i i lat o relative dimension n, then is too, and we ot : A k Y A k n, which aain behaves well with respect to everythin else. We ll now eneralize this notion. Deine a bivariant class or any (not just lci) as ollows. It is a collection o homomorphisms A k Y A k p or all Y Y, all k, aain compatible with pushorward, pullback, and intersection products. We ll call the roup o such thins A p ( : Y). We ll see that the roup A k ( pt) will be (canonically) isomorphic to A k. We ll see that A k (id : ) is a rin, which Fulton calls the cohomoloy roup; I miht call the resultin rin the Chow rin. We ll denote this by A k. The rin structure is a product o the orm A p A q A p+q. We ll deine more enerally A p ( : Y) A q ( : Y Z) A p+q ( : Z). We ll prove Poincare duality when is smooth: A = A (as rins recall we deined a rin structure on the latter). We ll deine proper pushorward and pullback operations or bivariant roups. Basically, they ll behave the way you d expect rom homoloy and cohomoloy. This will ive a cap product A A A. Alarmin act: This rin is apparently not known to be commutative in eneral, because the arument requires resolution o sinularities. (It is known to be commutative in characteristic 0, and or smooth thins in positive characteristic, and a ew more thins.) I think it should be possible to show that the rin is commutative in eneral usin technoloy not available when this theory was irst developed, usin Johan de Jon s alteration theorem in positive characteristic. I you would like to patch this hole, then come talk to me. Okay, let s et started. Today I ll outline the results, and prove a ew thins; Wednesday I ll prove some more thins. 2. PRECISE STATEMENTS Let : Y be a morphism. For each : Y Y, orm the iber square Y Y A bivariant class c in A p ( : Y) is a collection o homomorphisms c (k) : A ky A k p 2.
or all : Y Y, and all k, compatible with proper pushorwards, lat pullbacks, and intersection products. I ll make that precise in a moment, by statin 3 conditions explicitly. But irst I want to show you that you ve seen this beore in several circumstances. Example 1. I is a local complete intersection, or more enerally an lci morphism, we ve deined!. This ives some inklin as to why we want to deal with maps Y. We could have just had a class on Y, but we have more reined inormation; we have a class on, that pushes orward to the more reined class on Y. Example 2. I : Y is the identity, and V is a vector bundle on Y, then the Chern classes are o this orm: α ( c k (V)) α. Example 3 (which eneralizes urther): pseudodivisors. Let L be a line bundle on Y, and the zero-scheme o a section s o L. (s miht cut out a Cartier divisor, i.e. will contain no associated points o Y; at the other extreme, s miht be 0 everywhere.) Then we deined cappin with a pseudo-divisor : A k Y A k 1. Because pseudodivisors pull back, is a pseudodivisor on Y (with correspondin line bundle L, and correspondin section s), so we et a map A k Y A k 1, and this behaves well respect to everythin else. So we re creatin a machine that in some sense incorporates most thins we ve done so ar. Here are the conditions. (C 1 ): I h : Y Y is proper, : Y Y is arbitrary, and one orms the iber diaram (1) Y, h h Y Y then or all α A k Y, c (k) (h α) = h c (k) h α in A k p. (C 2 ): I h : Y Y is lat o relative dimension n, and : Y Y is arbitrary, and one orms the iber diaram (1), then, or all α A k Y, in A k+n p. c (k+n) h (h α) = h c (k) α 3
(C 3 ): I : Y Y, h : Y Z are morphisms, and i : Z Z is a local complete intersection o codimension e, and one orms the diaram (2) then, or all α A k Y, in A k p e. Y i i Y Y h h Z i Z i! c (k) (α) = c(k e) i (i! α) 2.1. Basic operations and properties. Here are some basic operations on bivariant Chow roups A ( Y). (P 1 ) Product: For all : Y, : Y Z, we have : A p ( : Y) A q ( : Y Z) A p+q ( : Z). It is pretty immediate to show this: iven any Z Z, orm the iber diaram (3) Y Z Y Z. I α A k Z, then d(α) A k 1 Y and c(d(α)) A k q p, so we deine c d by c d(α) := c(d(α)). (P 2 ) Pushorward: Let : Y be proper, : Y Z any morphism. Then there is a homomorphism ( proper pushorward ): : A p ( : Z) A p ( : Y Z). Aain, it s straihtorward: iven Z Z, orm the iber diaram (3). I c A p ), and α A k (Z ), then c(α) A k p ( ). Since is proper, (c(α)) A k p(y ). Deine (c) by (c)(α) = (c(α)). (P 3 ): Pullback (not necessarily lat!!): Given : Y, : Y 1 Y, orm the iber square (4) 1 1 For each p there is a homomorphism Y 1 Y. : A p ( : Y) A p ( 1 : 1 Y 1 ). 4
Aain, we just ollow our nose. Given c A p (), Y Y 1, then composin with ives a morphism Y. Thereor c(α) A k p ( ), = Y Y = 1 Y1 Y. Set (c)(α) = c(α). Here are seven more axioms, which can also be easily veriied. (A pr ) Associativity o products. I c A( Y), d A(Y Z), e A(Z W), then (c d) e = c (d e) A( W). (A p ) Functoriality o proper pushorward. I : Y and : Y Z are proper, Z W arbitrary, and c A( W), then () (c) = ( c) A(Z W). (A pb ) Functoriality o pullbacks. I c A( Y), : Y 1 Y, h : Y 2 Y 1, then (h) (c) = h (c) A( Y Y 2 Y 2 ). (A prp ) Product and pushorward commute. I : Y is proper, Y Z and Z W are arbitrary and c A( Z), d A(Z W), then (c) d = (c d) A(Y W). (A prpb ) Product and pullback commute. I c A( : Y), d A(Y Z), and : Z 1 Z is a morphism, orm the iber diaram 1 Y 1 Z 1 Y Z. Then (c d) = (c) (d) A( 1 Z 1 ). (A ppb ) Proper pushorward and pullback commute. I : Y is proper, Y Z, : Z 1 Z, and c A( Z) are iven, then, with notation as in the precedin diaram (A? ): Projection ormula. Given a diaram c = ( c) A(Y 1 Z 1 ). Y Y h Z. with proper, the square a iber square, and c A( Y), d A(Y Z), then c (d) = ( (c) d) A( Z). 5
3. PROVING THINGS Let S = Spec K, where K is some base ield. There is a canonical homomorphism iven by c c([s]). Proposition. This is an isomorphism. φ : A p ( S) A p () Proo. We will deine the inverse morphism. Given a A p (), deine a bivariant class ψ(a) A p ( S) as ollows: or any Y S, and any α A k Y, deine ψ(a)(α) = a α A p+k ( S Y). (Here a α is the exterior product.) Since exterior products are compatible with proper pushorward, lat pullback, and intersections, ψ(a) is a bivariant class. Let s check that this really is an inverse to φ. ψ(a)([s]) = a immediately, so φ ψ is the identity. To show that ψ φ is the identity, we have to show that c(α) = φ(c) α A k+p ( S Y) or all α A k Y. By compatibility with pushorward, we can assume α = [V], and V = Y a variety o dimension k: S V S Y V Y cl. imm. S Then α = p [S], where p : V S is the morphism rom V to S. Since c commutes with lat pullback, c(α) = c(p [S]) = p c([s]) = φ(c) [V] as desired. 3.1. The Chow ( cohomoloy ) rin. Deine A p := A p (id : ). We have a cup product. We also have an element 1 A 0, which acts as the identity. We have a cap product determined by : A p A q A q p A p ( ) A q ( S) A (q p) ( S) which makes A into a let A -module. All o this ollows ormally rom our axioms. 6
4. THINGS YOU MIGHT WANT TO BE TRUE 4.1. Poincare duality. Theorem. Let Y be a smooth, purely n-dimensional scheme (variety) not necessarily proper (compact). (a) The canonical homomorphism [Y] : A p Y A n p Y is an isomorphism. (b) The rin structure on A Y is compatible with that deined on A Y earlier. More enerally, i : Y is a morphism, β A Y, α A, then the class (β) α A coincides with that constructed earlier. We ll show somethin more eneral. Theorem. Let : Y Z be a smooth morphism o relative dimension n, and let [] A n ( : Y Z) be the bivariant class correspondin to lat pullback. Then or any morphism : Y and any p, is an isomorphism. [] : A p ( : Y) A p n ( : Z) In eneral, i : Y is a lat morphism, or a local complete intersection, or a local complete intersection morphism, the (lat or Gysin) pullback we ve deined earlier deines a bivariant class, which we ll denote []. ([ ] miht be better.) Fulton calls this bivariant class a canonical orientation. I m not sure o the motivation or this terminoloy, so I ll avoid it. Proo. We ll deine the inverse homomorphism Consider the iber diaram A p n ( : Z) A p ( : Y). Y γ Z Y δ Y Z Y q Y p Y p Z where δ is the diaonal map, and p and q are the irst and second projections. Here γ is the raph o the morphism Y over Z. Deine L : A p n ( : Z) A P ( Y) by L(c) = [γ] (c). Notice that γ and δ is a local complete intersection o codimension n, with [δ] = [γ]. (This requires a check in the case o γ.) Notice that p γ : and q δ : Y Y are both the identity morphisms (on and Y respectively). 7
Let s veriy that L and multiplication by [] are inverse homomorphisms. (This is easier to understand i you see someone pointin at diarams!) First, Second, L(c) [] = [γ] ( [c] []) (axiom (A pr )) = [γ] [p ] c (axiom (C 2 )) = [p γ] c = 1 c = c (axiom (A pr )) L(c []) = [δ] p (c) [] (axioms (A prpb ), (A pr )) = (p δ) (c) [δ][q] (axiom (C 2 )) = c [δ q] = c 1 = c (axiom (A pr )). 4.2. Chern classes commute with all bivariant classes. Put another way, any operation which commutes with proper pushorward, pullback, and intersections, automatically commutes with Chern classes. Precisely: Proposition. Let c A q ( : Y), Y Y, α A k (Y ), E a vector bundle on Y. Then where : = Y Y Y. c(c p (E) α) = c p ( E) c(α) A k q p Proo. Recall our deinition o Chern classes. They are certain polynomials in Sere classes. Sere classes are deined usin operations o the orm α p (c 1 (O(1)) i p α), and since c commutes with p and p, we just have to show that c commutes with c 1 (L), where L is a line bundle on Y. We may assume α = [V]. Because c commutes with proper pushorward, we may assume V = Y. Let L = O(D), D a Cartier divisor on V. We can replace V by V, where V V is proper and birational, so we may assume D = D 1 D 2, where D 1 and D 2 are eective. (Recall our trick in chapter 2: it isn t true that a Cartier divisor is a dierence o two eective Cartier divisors, but we can do a clever blow-up and turn it into a dierence o two eective Cartier divisors.) Let i : D V be the inclusion. Then we ve shown that c 1 (L) α = i i! α, and since c commutes with i and i!, c commutes with c(l). 4.3. Bivariant classes vanish in dimensions that you d expect them to. Proposition. Let : Y be a morphism. Let m = dim Y, and let n be the larest dimension o any iber 1 y, y Y. A p ( : Y) = 0 i p < n or p > m. (Think about why this is what you d expect!) In particular, or any, A p = 0 unless 0 p dim. (Proo omitted.) E-mail address: vakil@math.stanord.edu 8