STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY


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1 STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY ALEXANDER MERKURJEV 1. Introduction Let p be a prime integer. For a pair of topological spaces A X we write H i (X, A; Z/pZ) for the ith singular cohomology group with coefficients in Z/pZ. A cohomological operation of degree r is a collection of group homomorphisms H i (X, A; Z/pZ) H i+r (X, A; Z/pZ) satisfying certain naturality conditions. In particular, they commute with the pullback homomorphisms. The operations form the Steenrod algebra A p modulo p. It is generated by the reduced power operations P k, k 1, of degree 2(p 1)k and the Bockstein operation of degree 1. The idea of the definition of P k is as follows. Modulo p, the pth power operation α α p is additive. It can be described as follows. Let d : X X p be the diagonal embedding. The composition H (X; Z/pZ) H (X p ; Z/pZ) d H (X; Z/pZ), where the first map is the pth exterior power, takes a class α to the power α p. Note that the composition is a homomorphism, although the first map is not. Let G be the symmetric group S p. It acts on X p and we can define the composition H (X; Z/pZ) HG(X p ; Z/pZ) d HG(X; Z/pZ) = H (X; Z/pZ) HG(pt; Z/pZ). The ring HG (pt; Z/pZ) is isomorphic to the polynomial ring (Z/pZ)[t]. Thus, the image of α is a polynomial with coefficient in H (X; Z/pZ). The coefficients are the reduced powers P k (α) of a. Let a space X be embedable into R n. The group Hi BM (X; Z/pZ) = H n i (R n, R n X; Z/pZ) is independent on the embedding and is called the BorelMoore homology group with coefficients in Z/pZ. An operation S of degree r defines an operation Hi BM (X; Z/pZ) = H n i (R n, X; Z/pZ) S H n i+r (R n, X; Z/pZ) = Hi r BM (X; Z/pZ) on the BorelMoore homology groups. These operations commute with pushforward homomorphisms with respect to proper morphisms. We can them operations of homological type. Date: May,
2 2 A. MERKURJEV V. Voevodsky has defined operations on the motivic cohomology groups H i,j (X, U; Z/pZ) for a smooth variety X and an open subset U X. The reduced power operation P k has bidegree ( 2(p 1)k, (p 1)k ) and the Bockstein operation has bidegree (1, 0). If d = dim X we have H 2i,i (X, U; Z/pZ) CH d i (X U)/p. In particular, the reduced power operations (but not the Bockstein operation) act on the Chow groups modulo p. Our aim is to give an elementary definition of these operations. We follows P. Brosnan s approach. Notation: A scheme is a quasiprojective scheme over a field. A variety is an integral scheme. 2. Chow groups 2.1. Chow groups of homological type. Let X be a scheme over a field F and let i be an integer. The Chow groups of idimensional cycles CH i (X) is the factor group of the free group generated by dimension i closed subvarieties of X modulo rational equivalence. In particular, CH i (X) is trivial if i < 0 or i > dim X. Properties: 1. For a projective morphism f : X Y there is the pushforward homomorphism f : CH i (X) CH i (Y ). 2. For a flat morphism f : Y X of relative dimension d there is the pullback homomorphism f : CH i (X) CH i+d (Y ). 3. (Homotopy invariance) For a vector bundle f : E X of rank r the pullback homomorphism f : CH i (X) CH i+r (E) is an isomorphism. 4. For two schemes X and Y over F there is an exterior product homomorphism CH i (X) CH j (Y ) CH i+j (X Y ), α β α β. Remark 2.1. We can view Chow groups CH (X) as a BorelMoore type theory associated to the motivic cohomology groups of bidegree (2i, i) Chern classes. Let E X be a vector bundle over a scheme X. For every i 0 there is a group homomorphism c i (E) : CH k (X) CH k i (X), α c i (E) α called the ith Chern class of E. We can view c i (E) as an endomorphism of CH (X). The class c = c 0 + c 1 + c is called the total Chern class. Properties:
3 STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY 3 1. c 0 (E) is the identity. 2. c i (E) = 0 for i > rank E. 3. (Cartan formula) For an exact sequence 0 E E E 0, c (E) = c (E ) c (E ). 4. The operation c 1 (L) for a line bundle L X is the intersection with the Cartier divisor associated with L. The Chern classes uniquely determined by the properties 14. If E X is a vector bundle of rank r, the class e(e) = c r (E) is called the Euler class of E. For an exact sequence as above we have e(e) = e(e ) e(e ). Let A be a commutative ring. A characteristic class over A is a power series w over A in the Chern classes c i. For a vector bundle E X the operation w(e) on CH (X) A is well defined. Let f(x) = 1 + a 1 x + a 2 x be a power series over A. A characteristic class w f over A is called a multiplicative class associated with f if the following holds: 1. For a line bundle L X, w f (L) = f ( c 1 (L) ) = id + a 1 c 1 (L) + a 2 c 1 (L) ; 2. For an exact sequence 0 E E E 0 of vector bundles, w f (E) = w f (E ) w f (E ). The class w f exists and unique for every f. If E has a filtration by subbundles with line factors L i. Then w f (E) = i f ( c 1 (L i ) ). Example 2.2. If f = 1 + x, the class w f coincides with the total Chern class c. Note that the operation w f (E) is invertible. We write w f ( E) for w f (E) Chow groups of cohomological type. If X is a smooth variety of dimension d we write CH i (X) = CH d i (X). Properties: 1. For a morphism f : Y X of smooth varieties there is the pullback homomorphism f : CH i (X) CH i (Y ). 2. For a smooth variety X, the graded group CH (X) has a structure of a graded ring with identity 1 X = [X] via the pullback with respect to the diagonal morphism d : X X X: α β = d (α β).
4 4 A. MERKURJEV Remark 2.3. The group CH i (X) for a smooth X coincides with the motivic cohomology group H 2i,i (X) = H 2i,i (X, ) and the latter has obvious contravariant properties. We can view the assignment X CH (X) as a cohomology theory. Example 2.4. Let h CH 1 (P n ) be the class of a hyperplane section of the projective space P n. Then { CH i (P n Zh i if 0 i n, ) = 0 otherwise. In other words, CH (P n) = Z[h]/(h n+1 ) Refined Gysin homomorphisms. Let i : M P be a closed embedding of smooth varieties of codimension d. Let X be a closed subscheme of P and set Y = M X. Thus we have a pullback diagram Y X f M i P. The pullback homomorphism i (called also a Gysin homomorphism) has a refinement, called the refined Gysin homomorphism i! : CH k (X) CH k d (Y ). In the case X = P the homomorphism i! coincides with i. Remark 2.5. The refined Gysin homomorphism coincide with the pullback homomorphism H 2i.i (P, P X) H 2i.i (M, M Y ). for i = dim P k. Suppose we are given two diagrams as above: Y X f l M l i l Pl (l = 1, 2) with i l : M l P l closed embeddings of smooth varieties of codimension d l. Let N l be the normal bundle of the closed embedding i l over M l. It is a vector bundle on M l. The following proposition is a variant of socalled excess formula. Proposition 2.6. For an element α CH k (X), in CH k d1 d 2 (Y ). e(f 2 N 2 ) i! 1(α) = e(f 1 N 1 ) i! 2(α)
5 STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY 5 3. Equivariant Chow groups In topology: Let G be a group and X be a Gspace. For a (co)homology theory H we can define an equivariant (co)homology group H G (X) = H ( (X EG)/G ) where EG is a contractible Gspace with free Gaction. In algebra we don t have a scheme representing EG. Instead, we consider certain approximations of EG. Let G be an algebraic group of dimension d G over F and let X be a G scheme. Let V be a Gvector space (a representation of G) of dimension d V such that G acts regularly on a nonempty open subscheme U V. Then for every i Z, the group CH G i (X) = CH i+dv d G ( (X U)/G ) is independent of V as long as V U has sufficiently large codimension in V. This group is the ith equivariant Chow group of X. A closed Gequivariant subvariety Z X has its class [Z] G in CH G i (X) defined as [(Z U)/G] CH i+dv d G ( (X U)/G ) = CH G i (X). If G is the trivial group we have CH G i (X) = CH i (X). If H is a subgroup of G, for a Gscheme X there is a natural restriction homomorphism res G/H : CH G i (X) CH H i (X). For a smooth Gvariety X of dimension d we write CH i G(X) = CH G d i(x). The equivariant Chow groups have similar functorial properties as the ordinary ones (projective pushforwards, flat pullbacks, refined Gysin homomorphisms, homotopy invariance). In particular, the graded group CH G(X) has a structure of a commutative ring. Example 3.1. Let G = G m the multiplicative group. Consider the natural Gaction on the affine space A r+1. Clearly G acts regularly on A r+1 {0} with the factor variety P r. Hence CH i G(pt) = CH G i(pt) = CH r i (P r ) = Zh i and therefore, CH G(pt) = Z[h] is a polynomial ring over Z. Note that this ring is a domain contrary to the ordinary Chow rings having only nilpotent elements in positive degrees. Let p be a prime integer. We introduce a new notation: Ch (X) = CH (X)/p, Ch G (X) = CH G (X)/p.
6 6 A. MERKURJEV Example 3.2. Let H = µ p. We consider H as a subgroup of G = G m. The restriction ring homomorphism Ch G(pt) Ch H(pt) is an isomorphism, so that Ch H(pt) = (Z/pZ)[h]. More generally, if H acts trivially on a scheme X, we have a canonical isomorphism Ch H (X) = Ch (X) Z/pZ[h] = Ch (X)[h]. Example 3.3. Let G be the semidirect product of µ p and Aut(µ p ) = (Z/pZ). Since the order of (Z/pZ) is prime to p, we have for a Gscheme X: Ch G (X) = [Ch µ p (X)] (Z/pZ). A class i+pz (Z/pZ) acts on h by h h i. The restriction homomorphism Ch G(pt) Ch µ p (pt) = Z/pZ[h] is injective and identifies the ring Ch G(pt) with Z/pZ[h p 1 ]. More generally, if G acts trivially on a scheme X, we have a canonical isomorphism Ch G (X) = [Ch µ p (X)] Aut(µ p ) = Ch (X)[h p 1 ] Equivariant Chern classes. Let X be Gscheme and let E X be a Gvector bundle over X. For every i 0 there is a group homomorphism c G i (E) : CH G k (X) CH G k i(x), α c G i (E) α called the equivariant ith Chern class of E. The equivariant Chern classes satisfy similar properties as the ordinary ones. Example 3.4. If X = pt, a Gvector bundle over X is a representation V of G. Let G = G m or µ n and let V be the canonical 1dimensional representation of G. Then c G 1 (V ) is the multiplication by h in Ch G(pt). We will simply write c G 1 (V ) = h. Example 3.5. Let p be a prime integer. Let G be the semidirect product of µ p and Aut(µ p ) = (Z/pZ). Consider the algebra F p = F [x]/(x p 1). It is an étale F algebra of degree p. If F contains all pth roots of unity, F p is a product of p copies of F. The group G acts naturally on F p by algebra automorphisms: the group µ p acts on x by multiplication and an automorphism i+pz (Z/pZ) by x x i. In particular, F p is a Gmodule having a (trivial) submodule F. We claim that the mod p equivariant Euler class e G ( F ), where F = F p /F, is the multiplication by h p 1. Indeed, since Ch G(pt) is contained in Ch µ p (pt) it is sufficient to replace G by µ p. Then F is a direct sum of 1dimensional modules F x i V i for i = 1, 2,..., p 1. Hence p 1 p 1 e G ( F ) = e G (V i ) = (ih) = (p 1)! h p 1 = h p 1. i=1 i=1 We set t = h p 1 so that CH G(pt)/p = Z/pZ[t] and t = e G ( F ) modulo p.
7 STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY 7 4. Definition of reduced powers We fix a prime integer p. We assume that char F p Scheme R p (X). Consider again the algebra F p = F [x]/(x p 1). Let X be a scheme over a field F. Write R p (X) for the scheme R Fp/F (X F F p ) representing the functor S X(S F F p ). If F contains all pth roots of unity, X is a product of p copies of X. In particular, dim R p (X) = p dim X. Consider the semidirect product G of µ p with Aut(µ p ) = (Z/pZ). The action of G on F p extends to an action of G on R p (X). The embedding X(S) X(S F F p ) gives rise to the closed embedding of X into R p (X). In fact, X is identified with the subscheme of Ginvariant elements of R p (X). We have a well defined map (not a homomorphism!) P X : CH k (X) CH G pk( R p (X) ), [Z] [R p (Z)] G. Now assume that X is a smooth variety. We would like to compute the normal cone N of the closed embedding X R p (X). One checks that hence T R p (X) X = T X F F p, N = T X F F, where as above F = F p /F. The group G acts on N via F. Let us compute the Euler class e G (N). Consider the multiplicative characteristic class w = w f corresponding to the power series f(x) = 1 + x p 1. Recall that for every line bundle L, we have w(l) = 1 + c 1 (L) p 1. We write so that for a line bundle L, w(y) = 1 + w p 1 y + w 2(p 1) y , w(l, y) = 1 + c 1 (L) p 1 y. Proposition 4.1. For a vector bundle E X of rank r we have e G (E F F ) = t r w(e, 1/t) Ch G (X) = Ch (X)[t]. Proof. Since Ch G (X) injects into Ch µ p (X) we may replace G by µ p. We also may assume that E = L is a line bundle. Then
8 8 A. MERKURJEV e G (L F p 1 F ) = c G 1 (L V i ) i=1 p 1 ( = c G 1 (L) + c G 1 (V i ) ) i=1 p 1 ( = c G 1 (L) + ih ) i=1 = c G 1 (L) p 1 h p 1 = t ( 1 + c G 1 (L) p 1 /t ) = t w(l, 1/t). Remark 4.2. It follows from the proposition that the Euler class e G (E F F ) is invertible if we invert t Reduced power operations. Let X be a scheme over F. We embed X as a closed subscheme into a smooth variety M of dimension d. The commutative diagram X R p (X) M i R p (M) gives rise to a refined Gysin homomorphism ( i! : CH G r R p (X) ) CH G r (p 1)d(X), where G be the semidirect product of µ p and (Z/pZ). Note that the composition D M X : Ch k (X) P X Ch G pk is a group homomorphism. Since G acts trivially on X, we have ( R p (X) ) i! Ch G pk (p 1)d(X) Ch G (X) = Ch (X)[t] = i 0 Ch (X)t i. The map DX M changes the degree by a multiple of p 1. Therefore, for an element α Ch k (X) we can then write D M X (α)(t) = i S M i (α)t d k i for some elements S M i (α) Ch k (p 1)i (X).
9 STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY 9 In other words, S M (α)(t) def = i S M i (α)t i = t d k D M X (α)(1/t). Thus, for any scheme X over F and a smooth variety M containing X as a closed subscheme we get the reduced power operations for the pair (M, X): S M i : Ch k (X) Ch k (p 1)i (X). Note that Si M lowers the dimension by (p 1)i. We also have the total reduced power operation S M = S M (t) t=1 = S M i : Ch (X) Ch (X). Remark 4.3. On the motivic cohomology groups the operation Si M with the reduced power S i : H 2k,k (M, M X) H 2k 2(p 1)i,k (p 1)i (M, M X). coincides Properties: 1. Let Y i X M be closed embeddings with M smooth. Then for every α Ch (Y ). S M (i α) = i (S M α) 2. (Cartan formula) Let X M and Y N be closed embeddings with M and N smooth. Then for α Ch (X) and β Ch (Y ). S M N (α β) = S M α S N β 4.3. Steenrod operations of cohomological type. Let X be a smooth variety over F. Set S = S X. Thus, S i is an operation S i : Ch k (X) Ch k+(p 1)i (X), called the reduced power of cohomological type. Properties: 1. Let f : Y X be a morphism of smooth varieties over F. Then for every α Ch (X). f S (α) = S f (α) 2. (Cartan formula) S (αβ) = S (α) S (β) for α, β Ch (X), S (1 X ) = 1 X, i.e., S is a ring endomorphism of Ch (X). 3. For every α Ch k (X), S i α = { α p, if i = k, 0, if i > k or i < 0.
10 10 A. MERKURJEV 4.4. Steenrod operations of homological type. Proposition 4.4. Let X be a scheme and let M 1 and M 2 be two smooth varieties containing X. Then w(t M2 X ) S M 1 = w(t M1 X ) S M 2. Proof. The normal bundle N l of the closed embedding M l R p (M l ) is equal to T Ml X F F, (l = 1, 2). Let dl = dim M l. For an element α Ch k (X), we have w(t M2 X, 1/t) S M 1 (1/t) = t d2 e G (N 2 ) t k d 1 D M 1(α)(t) (Prop. 4.1) = t k d 1 d2 e G (N 2 ) i! 1( PX (α) ). X It follows from Proposition 2.6 that the latter is equal to The result follows. t k d 1 d2 e G (N 1 ) i! 2( PX (α) ). Let X be a subscheme of a smooth variety M. For every α Ch k (X) set S (α) = w(t M X ) 1 S M (α). By Proposition 4.4, S (α) is independent of M. The operations S i : Ch k (X) Ch k (p 1)i (X) are called the reduced powers of homological type. Properties: 1. If i : Y X is a projective morphism of schemes over F, then S (i α) = i (S α) for every α Ch (Y ). In fact, this is a variant of a RiemannRoch theorem by I. Panin and A. Smirnov. 2. S 0 is the identity. 3. Let a scheme X be embedable in a smooth scheme of dimension d and let α Ch k (X). Then S i (α) = 0 for i / [0, d k]. If X is smooth, we have the operations S and S on the Chow groups of X modulo p. For every α Ch k (X) we have In particular, S (α) = w(t X ) 1 S (α) = w( T X ) S (α). S (1 X ) = w( T X ) 1 X, i.e., S i (1 X ) = w (p 1)i ( T X ) 1 X.
11 STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY 11 Remark 4.5. The homological operation S i is a special case of the cohomological operation Si M if X is affine. Let i : X A n be a closed embedding. Since the tangent bundle of A n is trivial, we have S i = S An i. 5. Reduced Steenrod algebra We fix a prime integer p in this section. Consider the polynomial ring H = (Z/pZ)[b] = (Z/pZ)[b 1, b 2,... ] in infinitely many variables b 1, b 2,... as a graded ring with deg b i = p i 1. The monomials b R = b r 1 1 b r , where R ranges over all sequences (r 1, r 2,... ) of nonnegative integers such that almost all of the r i s are zero, form a basis of H over Z/pZ. We set R = i 1 r i (p i 1). Clearly, deg b R = R. Denote the scheme Spec H by G. For a commutative Z/pZalgebra A the set of Apoints G(A) = Hom rings ( H, A ) can be identified with the set of sequences (a 1, a 2,... ) of the elements of A and, therefore, with the set of power series of the form x + a 1 x p + a 2 x p2 + a 3 x p3 + A[[x]]. The composition law (f 1 f 2 )(x) = f 2 (f 1 (x)) makes G a group scheme over Z/pZ and H a graded Hopf Z/pZalgebra. The comultiplication morphism c : H H H is given by the rule c(b k ) = b pj i b j. i+j=k We write A p for the graded Hopf Z/pZalgebra dual to H. Thus, A p is the graded Hopf algebra such that the dcomponent of A p is the Z/pZspace dual to the dcomponent of H. The algebra A p is the reduced Steenrod Z/pZalgebra, i.e., the Steenrod Z/pZalgebra modulo the ideal generated by the Bockstein element. It can be also considered as the subalgebra of the Steenrod algebra A p generated by the reduced power operations. Let {P R } be the basis of A p dual to the basis {b R } of H. We write P i for the basis element P R with R = (i, 0,... ). The element P 0 is the identity of A p and the algebra A p is generated by the P i, i 1. These generators are subject to the set of defining Adem relations: [b/p] ( ) (p 1)(c i) 1 P b P c = ( 1) b+i P b+c i P i. b pi i=0
12 12 A. MERKURJEV P. Brosnan proved that the power operations S i and S i satisfy the Adem relations respectively. Hence, for any smooth variety X, the group Ch (X) has a structure of a left A p module (the generator P i acts as S i )), and for every scheme X, the group Ch (X) has a structure of a left A p module (the generator P i acts as S i )). Denote by S R (resp. S R ) the operation induced by P R on CH (X) if X is smooth (resp. arbitrary scheme). The operations S R commute with the pullback homomorphisms and the operations S R commute with the pushforward homomorphisms. Consider the generic power series f gen (x) = 1 + b 1 x p 1 + b 2 x p over the ring Z[b]. Write the corresponding multiplicative class w gen = w fgen in the form w gen = w R b R R for unique characteristic classes w R over Z. In particular, if R = (i, 0,... ), we have w R = w (p 1)i in the old notation. If X is a smooth variety, the equality S i (1 X ) = w (p 1)i ( T X ) 1 X generalizes as follows: S R (1 X ) = w R ( T X ) 1 X for every sequence R. 6. Degree formulas Let X be a scheme over F. For a closed point x X we define its degree as the integer deg(x) = [F (x) : F ] and set n X = gcd deg(x), where the gcd is taken over all closed points of X. Fix a prime integer p char F. Let X be a projective variety of dimension d > 0 and let q : X Spec F be the structure morphism. For every sequence R with R = d, the group Ch d (Spec F ) in the commutative diagram Ch d (X) q S R Ch0 (X) q =deg Ch d (Spec F ) S R Ch0 (Spec F ) is trivial. Hence the degree of a cycle αr X CH 0(X) representing the element S R ([X]) Ch 0 (X) is divisible by p. The class of the integer deg(αr X )/p modulo n X is independent on the choice of αr X ; we denote it by u R (X) Z/n X Z. Clearly, p u R (X) = 0. If X is a smooth variety, S R (1 X ) = w R ( T X ) 1 X, hence we can take αr X = w R( T X ) 1 X. Thus, deg ( ) w R ( T X ) 1 X is divisible by p and u R (X) = deg( ) w R ( T X ) 1 X + n X Z. p Z/p
13 STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY 13 Remark 6.1. The degree of the 0cycle w R ( T X ) 1 X does not change under field extensions. In particular, it can be computed over an algebraic closure of F. Let f : X Y be a morphism of varieties over F of the same dimension. We set { [F (X) : F (Y )], if f is dominant; deg(f) = 0, otherwise. Thus, if f is projective, then f (1 X ) = deg(f) 1 Y. Theorem 6.2. (Degree formula) Let f : X Y be a morphism of projective varieties over F of dimension d > 0. Then n Y divides n X and for every sequence R with R = d, and for any prime integer p char F, we have u R (X) = deg(f) u R (Y ) Z/n Y Z. In particular, if X and Y are smooth, then deg ( ) w R ( T X ) 1 X deg(f) deg( ) w R ( T Y ) 1 X p p Proof. It follows from the commutativity of the diagram S R Ch d (X) Ch0 (X) f f Ch d (Y ) and the equality f (1 X ) = deg(f) 1 Y S R Ch0 (Y ) that f S R (1 X ) = deg(f) S R (1 Y ) Ch 0 (Y ) and therefore, ( f αr X deg(f) αr Y mod p CH0 (Y ) ). Applying the degree homomorphism, we get whence the result. deg(α X R ) = deg(f α X R ) deg(f) deg(α Y R) (mod pn Y ), ( mod ny ). Remark 6.3. The case of sequences R = (i, 0, 0,... ) was considered by M. Rost. 7. Applications Let X and Y be varieties over a field F, d = dim(x). A correspondence from X to Y, denoted α: X Y, is a dimension d algebraic cycle on X Y. A correspondence α is called prime if α is given by a prime (irreducible) cycle. Any correspondence is a linear combination with integer coefficients of prime correspondences. Let α : X Y be a prime correspondence. Suppose that α is given by a closed subvariety Z X Y. We define multiplicity of α as the degree
14 14 A. MERKURJEV of the projection Z X. We extend the notion of multiplicity to arbitrary correspondences by linearity. A rational morphism X Y defines a multiplicity 1 prime correspondence X Y given by the closure of its graph. One can think of a correspondence of multiplicity m as a generically mvalued morphism. We fix a prime integer p and a field F such that char F p. Let R be a nonzero sequence and let X be a projective variety over F of dimension R. The variety X is called R p rigid if u R (X) 0 Z/n X Z. We write v p for the padic valuation of Q. Theorem 7.1. Let X and Y be projective varieties over F and let R be a sequence such that dim(x) = R > 0. Suppose that (1) There is a correspondence α : X Y of multiplicity not divisible by p; (2) X is R p rigid; (3) v p (n X ) v p (n Y ). Then (1) dim(x) dim(y ); (2) If dim(x) = dim(y ), (2a) There is a correspondence β : Y X of multiplicity not divisible by p; (2b) Y is R p rigid; (2c) v p (n X ) = v p (n Y ). Proof. Suppose that m = dim(x) dim(y ) 0 and set Y = Y P m F. Clearly, n Y = n Y. We embed Y into Y as Y z where z is a rational point of P m F. We may assume that α is a prime correspondence, replacing if necessary, α by one of its prime components. Let Z X Y be the closed subvariety representing α. We have two natural morphisms f : Z X and g : Z Y Y. By assumption, deg(f) is not divisible by p. We write the degree formulas of Theorem 6.2 for the morphisms f and g: (1) u R (Z) = deg(f) u R (X) Z/n X Z, (2) u R (Z) = deg(g) u R (Y ) Z/n Y Z. The variety X is R p rigid and the degree deg(f) is not divisible by p, hence it follows from (1) that u R (Z) 0 in Z/n X Z. Since v p (n X ) v p (n Y ) and p u R (Z) = 0, we have u R (Z) 0 in Z/n Y Z and it follows from the degree formula (2) that deg(g) is not divisible by p, so that g is surjective, and u R (Y ) 0 in Z/n Y Z. The image of g is contained in Y, therefore Y = Y, i.e., m = 0 and dim(x) = dim(y ). The variety Z defines a correspondence β : Y X of multiplicity deg(g) not divisible by p. Since u R (Y ) = u R (Y ) 0 in Z/n Y Z, it follows that Y is R p rigid. Finally, p u R (Z) = 0 and u R (Z) is nonzero in both Z/n X Z and Z/n Y Z, therefore, we must have v p (n X ) = v p (n Y ). Corollary 7.2. The classes u R (X) Z/n X Z are birational invariants of a smooth projective variety X.
15 STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY Examples Let X be a complete intersection in P n of m smooth hypersurfaces of degrees d 1, d 2,..., d m. Let L = L can be the restriction on X of the canonical line bundle on P n. Then m [T X ] = [T P n X ] [L d i ] K 0 (X). We have [T P n X ] = (n + 1)[L] 1 X K 0 (X). and w gen (L) = 1 + h p 1 b 1 + h p2 1 b , where h = c 1 (L) Ch 1 (X) is the class of a hyperplane section. Therefore, wgen (L d i ) (1 + (di h) p 1 b 1 + (d i h) p2 1 b ) w gen ( T X ) = =. w gen (L) n+1 (1 + h p 1 b 1 + h p2 1 b ) n+1 Let R be a sequence such that R = dim X = n m. Since we have i=1 deg h n m = d i, deg w R ( T X ) = [ (1 ] + d p 1 i b 1 + d p2 1 i b ) d i (1 + b 1 + b ) n+1 Example 8.1. Let X be the SeveriBrauer variety of a central simple algebra A of degree p n. Then X is a twisted form of the projective space P pn 1. For R = (0,..., 0, n 1, 0,... ) we have deg w R ( T X ) = p n. Thus, X is R p rigid if n X = p n, i.e., A is a division algebra. Example 8.2. Let X be a smooth hypersurface of degree p in P pn. For a sequence R as above we have Thus, X is R p rigid if n X = p. deg w R ( T X ) = p(p pn 1 p n 1). Example 8.3. Let X be a smooth intersection of two quadrics in P 4. Then for R = (2, 0, 0,... ) and p = 2 we have Thus, X is R 2 rigid if n X = 4. deg w R ( T X ) = 4. Alexander Merkurjev, Department of Mathematics, University of California, Los Angeles, CA , USA address: R.
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