Lecture 4 Cohomological operations on theories of rational type.

Lecture 4 Cohomological operations on theories of rational type. 4.1 Main Theorem The Main Result which permits to describe operations from a theory of rational type elsewhere is the following: Theorem 4.1 Let A be a theory of rational type, and B be any theory. Fix n, m Z. Then there is 1-to-1 correspondence between additive operations A n G B m on Sm k and additive transformations commuting with the pull-backs for: (i) the action of S l ; A n ((P ) l ) G B m ((P ) l ), for l Z 0 (ii) (iii) (iv) the partial diagonals; the partial Segre embeddings; (Spec(k) P ) (P ) r, r. In Topology an analogous result was proven by T.Kashiwabara in [1, Theorem 4.2] The condition on A is really necessary. For example, the identification CHalg ((P ) l ) = CH ((P ) l ) can not be extended to an operation. Let us interpret the conditions (i) (iv) algebraically. The module A ((P ) l ) has a topological basis consisting of monomials in zi A = c A 1 (O(1) i ). Due to partial diagonals, the transformation A n G B m on (P ) l is defined uniquely by it s action on elements of the form α ( l i=1 za i ), where α A n l. Then l G(α ( zi A )) = G l (α)(z1 B,..., zl B ). i=1 for some G l (α)(z B 1,..., z B l ) B[[z B 1,..., z B l ]] (m). Conditions (i), (iii) and (iv) impose certain restrictions on these power series. Namely, (i) implies (a i ), (iii) implies (a iii ), and (iv) implies (a ii ): 1

(a i ) G l is symmetric with respect to S l ; (a ii ) G l (α) = l i=1 z i F l (α), for some F l (α) B[[z 1,..., z l ]] (m l). (a iii ) G l (α)(x + B y, z 2,..., z l ) = i,j G l+i+j 1 (α a A i,j)(x i, y j, z 2,..., z l ), where a A i,j are the coefficients of the formal group laws of A. The idea of the proof is to start with the above algebraic setting and extend G to X (P ) l, for all l, by induction on the dimension of X. For each smooth quasi-projective variety X we denote as G(X) = {G l, l Z 0 } the following data: G l Hom Z lin (A n l (X), B (X)[[z 1,..., z l ]] (m) ) satisfying the same conditions (a i ), (a ii ), (a iii ) above (with evident modifications). We show inductively on d Z 0 that G(X) can be defined for all varieties of dimension d, and it commutes with all pull-backs for morphisms between such varieties. It appears though that for the inductive proof to proceed one needs to carry along not only this compatibility with pull-backs, but also some Riemann-Roch type condition describing compatibility with pushforwards. As a bonus, in the end, we get the respective additive Riemann- Roch style result. The conditions we carry are the following: Definition 4.2 Let d be some natural number. We say that G(d) is defined, if, for all X smooth quasi-projective of dimension d, G(X) is defined, and these satisfy: (b i ) For any f : X Y with dim(x), dim(y ) d, and any α A n l (Y ), G l (f A(α)) = f BG l (α). (b ii ) For any regular embedding j : X Y of codimension r with normal bundle N j with B-roots µ B 1,..., µ B r, for any α A n l r (X), one has: F l (j (α))(z 1,..., z l ) = j (F l+r (α)(µ B 1,..., µ B r, z 1,..., z l )) (due to condition (a i ) the latter expression depends only on the B-Chern classes of the normal bundle, but not on roots). 2

Suppose G(d 1) is defined, and X p X Spec(k) is a smooth quasiprojective variety of dimension d. We would like to define G(X). Since A is a theory of rational type, we have a natural identification A (X) = A H(c), where c is the bi-complex from previous lecture. For the constant part, we define: For the A -part, recall that G l (p X(α))(z 1,..., z l ) := p XG l (α)(z 1,..., z l ). c 0,0 = (Z z X ρ X) (A (V ) im(ρ! )), where the sum is taken over all closed subschemes Z z X and projective birational morphisms ρ, isomorphic outside V = ρ 1 (Z), which is a divisor with strict normal crossing on X. If γ A n l 1 (V ) im(ρ! ), then the respective element α A n l (X) is ρ v (γ) ρ (1. Let us define: A ) where F l (γ V )(z 1,..., z l ) := i F l (α)(z 1,..., z l ) := ρ F l (γ V )(z 1,..., z l ), ρ (1 B ) (v i ) F l+1 (γ i )(λ B i, z 1,..., z l ) B ( X)[[z 1,..., z l ]] (m), ˆv i where V i V (and vi = v ˆv i : V i X), γ = i (ˆv i) (γ i ) A n l 1 (V ), and λ B i = c B 1 (O(V i )). Notice, that dim(v i ) d 1, so G(V i ) is defined. It follows from (b ii ) that, for dim(x) (d 1), F l (γ V )(z 1,..., z l ) = F l (v (γ))(z 1,..., z l ). And since γ im(ρ! ), we have v (γ) = ρ (α), and so by (b i ), the new definition of F l agrees with the old one. To check that F l (α) is well-defined one needs to show that it is trivial on d c 1,0 the image of c 1,0 c dc 0,1 0,1 c 0,0. The key role here is played by the following statement: Suppose, we have a cartesian square: E e Y f f D d X. 3

with X and Y smooth, and D and E - divisors with strict normal crossing on them. Proposition 4.3 Suppose, X and Y have dimension d. Then: f F l (γ D)(z) = F l (f (γ) E)(z). Here f is just the combinatorial version of f!. For D smooth (and it is the only case really needed), it can be expressed as follows: Let E = s j=1 m j E j, µ A j = c A 1 (O Y (E j )), and f J : E J D, for all J {1,..., s}, (µ A ) J = j J µa j, and CJ A := F m 1,...,m s J be the coefficients defining the divisor class ( J {1,...,s} (µa ) J F m 1,...,m s J = m 1 A µ 1 + A... + A m s A µ s ). Then f (γ) = (ê J ) fj(γ) CJ A. J {1,...,s} The meaning of the above statement is that: f F l (d (γ)) = F l (f d (γ)). The point is that this follows from (b i ) and (b ii ) for varieties of dimension (d 1) (notice, that we are in the process of defining F l for varieties of dimension d). More precisely, the combination of (b i ) and (b ii ) used here is the following: Proposition 4.4 Let G(d 1) be defined, Y be smooth quasi-projective variety of dimension (d 1), and L be a linear bundle on Y with η A = c A 1 (L), η B = c B 1 (L). Then, for any α A n l 1 (Y ), G l (α η A )(z) = G l+1 (α)(η B, z). The proof of the Proposition 4.3 uses the fact that the coefficients CJ A I J can be chosen as ( 1) J I µ A I, where µ A (µ A ) J I = A j I [m j] A µ A j. Then one can move the first Chern classes back and forth using Proposition 4.4 (of course, to make sense of the above formula with denominators one has to use residues). Other ingredients of the proof of the fact that F l is trivial on im(d c 1,0 d c 0,1) are resolution of singularities results of Hironaka, and the following blow up result: 4

Proposition 4.5 Let A be any theory, and π : Ṽ V be the permitted blow up of a smooth variety with smooth centers R i and the respective components ε of the exceptional divisor E i i Ri. Then one has exact sequence: 0 A (V ) π A (Ṽ ) i Coker(A (R i ) (ε i) A (E i )) As soon as we know that F l (α) is well-defined, conditions (a i ), (a ii ) and (a iii ) are straightforward, and we obtain that G(X) is defined, for all varieties X of dimension d. To show that G(d 1) extends to G(d) we need to check that the conditions (b i ) and (b ii ) are satisfied. (b i ): Let f : Y X be some map of smooth quasi-projective varieties of dimension d. For constant elements the statement is trivial. For the A -part, we can assume that α = (ρ X) (v X ) (γ) (ρ X ) (1), where X ρ X X is a projectivebirational map, which is an isomorphism outside the strict normal crossing v divisor V X X X, and γ A n l 1 (V X ) ρ! X. Then, by the results of Hironaka, there exists a projective-birational map Ỹ ρ Y Y, which fits into the commutative diagram with the left square cartesian, and V Y v Y Ỹ - a divisor with strict normal crossing: f V v Y V Y Ỹ ρ Y Y f V X v X X ρx X. Then (b i ) can be proven using the fact that F l (γ V X ) im(ρ X ), Proposition 4.3, Multiple Points Excess Intersection Formula, and the following simple fact: f Lemma 4.6 Let T j S q T j S be commutative diagram with p and q - projective bi-rational. Let x im(p ). Then: ( ) q ( j (x)) = j p (x). q (1) p (1) p 5

More precisely, ( ) ( ) f (ρx ) (v X ) (γ) F l := f (ρx ) F l (γ V X ) (ρ X ) (1) (ρ X ) (1) ) (ρ Y ) F l (f V (γ) V Y ) (ρ Y ) (1) F l ( f ( (ρx ) (v X ) (γ) (ρ X ) (1) ( (ρy ) (v Y ) fv =: F (γ) l (ρ Y ) (1) )). = (ρ Y ) f F l (γ V X ) (ρ Y ) (1) = F l ( (ρy ) f (v X ) (γ) (ρ Y ) (1) = ) = The condition (b ii ) for the regular embedding X j Y follows from applying (b i ) (proven above), the usual Excess Intersection Formula and Proposition 4.4 to the blow up cartesian diagram: E j Ỹ ε π X j Y. This shows that G(d 1) can be extended to G(d). And so, by induction, G(0) can be extended to G( ). Now consider G 0 : A n (X) B m (X), for all X Sm k. By (b i ), this is an additive operation. It remains to see that it extends the original A n ((P ) l ) G B m ((P ) l ). From commutativity with the pull-backs for partial diagonals, it is sufficient to compare the results on α l i=1 za i A n ((P ) l ), where α A n l, and zi A = c A 1 (O(1) i ). Let j : (P ) l (P ) l be the product of hyperplane section embeddings. Then G 0 (j (α)) = G l (α)(z1 B,..., zl B ) = G(α l i=1 za i ), by (b ii ) and the definition of G(Spec(k)). Thus, G 0 extends the original transformation on products of projective spaces. The uniqueness follows from what we have proven in Lecture 2. This finishes the proof of Theorem 4.1. There is also the multiplicative version: Proposition 4.7 Suppose, in the situation of Theorem 4.1, the original transformation A ((P ) l ) G B ((P ) l ) commutes with the external products of projective spaces. Then the resulting operation G is multiplicative. Theorem 4.1 reduces the description and construction of additive operations (from a theory of rational type) to algebra. Namely: 6

Theorem 4.8 Let A be theory of rational type, and B be any theory. Then there is 1-to-1 correspondence between the set of (unstable) additive operations A n G B m and the set consisting of the following data {G l, l Z 0 }: G l Hom Z lin (A n l, B[[z 1,..., z l ]] (m) ) satisfying: (a i ) G l is symmetric with respect to S l ; (a ii ) G l (α) = l i=1 z i F l (α), for some F l (α) B[[z 1,..., z l ]] (m l). (a iii ) G l (α)(x + B y, z 2,..., z l ) = i,j G i+j+l 1(α a A i,j)(x i, y j, z 2,..., z l ), where a A i,j are the coefficients of the FGL of A. 4.2 Stable operations If A n G{n} B n+d, for n Z are additive operations as above, then the condition: Σ T G {n} = G {n+1} Σ T can be interpreted as: F {n} l (α)(z 1,..., z l ) = F {n+1} l+1 (α)(0, z 1,..., z l ), l. Denote as F (k) the lim F {n} n k lim Hom Z lin(a k, B[[z 1,..., z n k ]] S n k (k+d) ) = Hom Z lin (A k, B[[σ 1, σ 2,...]] (k+d) ), where σ i - the i-th elementary symmetric function in z j s has degree i. Combining all F (k) s together we obtain We can introduce F Hom(A, B[[σ 1, σ 2,...]] ( +d) ). G Hom(A[[x, y]], B[[σ 1, σ 2,...]][[x, y]] ( +d) ), defined by the rule (in z-coordinates): G(β x i y j )(z) := x i y j ( z) F (β)(x i, y j, z). Now we can describe stable operations. Theorem 4.9 Let A be theory of rational type, and B be any theory. Then there is 1-to-1 correspondence between the set of stable operations A G B +d and F Hom(A, B[[σ 1, σ 2,...]] ( +d) ) satisfying G(α (x + A y))(z) = G(α)((x + B y), z). 7

References [1] T.Kashiwabara, Hopf rings and unstable operations, J. of Pure and Applied Algebra 94, 1994, 183-193 [2] A.Vishik, Stable and Unstable Operations in Algebraic Cobordism, arxiv:1209.5793 [math.ag], 25 September 2012. 8