ON THE RIEMANN ROCH THEOREM WITHOUT DENOMINATORS

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1 Algebra i analiz St. Petersburg Math. J. Tom. 18 (2006), 6 Vol. 18 (2007), No. 6, Pages S (07) Article electronically published on October 2, 2007 ON THE RIEMANN ROCH THEOREM WITHOUT DENOMINATORS O. B. PODKOPAEV AND E. K. SHINDER Abstract. A proof of the Riemann Roch theorem without denominators is given. It is also proved that Grothendiec s ring functor CH mult is not an oriented cohomology pretheory. The Riemann Roch formula without denominators for a closed embedding i : Y X of codimension d expresses the Chern class c d (i O Y )intermsoftheclass[y ] CH d (X). In the present paper, we give a proof of this formula in the spirit of Verdier [Ve] but without using local Chern classes. The proof consists of several reductions. First, we use excision of singularities and deformation to the normal cone to reduce the proof to the case of an embedding of a smooth variety Y in a projective bundle P(E) overy.at the next step, we reduce the proof to the simplest case of an embedding of a rational point in a projective space. Finally, we present two simple arguments that prove the formula in this case: one involves the Koszul complex, and the other involves the ring functor CH mult, which arises naturally under the study of the behavior of the Chern classes under direct images. In the second part of the paper, we present some properties of the functor CH mult and prove that this functor is not an oriented cohomology pretheory. The authors are thanful to I. A. Panin and K. I. Pimenov for generous guidance and support and to J. L. Colliot-Thélène (Orsay University, Paris) and B. Erez (University of Bordeaux I) for inspiring discussions and hospitality. In the present paper, we use the following notation. 1) X is a quasiprojective variety over a field. 2) CH (X) = CH i (X) is the Chow ring of classes of rationally equivalent cycles on X. 3) K 0(X) is the Grothendiec K-group of the category of coherent sheaves on X. If F is a coherent sheaf on X, then: 4) c i (F) istheith Chern class of the sheaf F (see [Fu]); 5) s i (F) istheith Segre class of the sheaf F (see [Fu]). If f : Y X is a flat morphism of varieties, then: 6) f CH : CH (X) CH (Y ) is the inverse image homomorphism for the Chow theory (see [Fu]); 7) f K : K 0(X) K 0(Y ) is the inverse image homomorphism for K-theory (see [Qu]). If f : Y X is a proper morphism of varieties, then: 8) f CH : CH (X) CH (Y ) is the direct image homomorphism for the Chow theory (see [Fu]); 9) f K : K 0(X) K 0(Y ) is the direct image homomorphism for K-theory (see [Qu]) Mathematics Subject Classification. Primary 14C40. Key words and phrases. Riemann Roch formula without denominators, deformation to the normal cone, Koszul complex, Chern classes, oriented cohomology pretheory. Partially supported by CNRS, France c 2007 American Mathematical Society

2 1022 O. B. PODKOPAEV AND E. K. SHINDER 1. Riemann Roch theorem without denominators Theorem. Let X be a nonsingular variety over a field. Let i : Y X be a closed embedding of an irreducible subvariety Y of codimension d. Then the following relation is true in CH d (X): (1) c d (i O Y )=( 1) d 1 (d 1)![Y ]. Proof. The proof is given in Subsections Reduction to the case of the zero section of a projective bundle. 1. Reduction to the case of a nonsingular Y. From now on, we write i K O Y for i K [O Y ]=[i O Y ]. Let Z be the subvariety of singularities of Y.Leti : Y Z X Z be the regular embedding of the nonsingular variety Y Z. We consider the exact sequence of localization, CH d (Z) CH d (X) CH d (X Z) 0, where CH d (Z) = 0 because dim Z<d. Thus, we have an isomorphism CH d (X) CH d (X Z). Under this isomorphism, the classes c d (i K O Y )and( 1) d 1 (d 1)![Y ] are mapped to the classes c d (i K O Y Z) and( 1) d 1 (d 1)![Y Z], respectively. This reduces the proof to the case where Y is nonsingular. Lemma 1 (Properties of the classes c d (i K O Y )and[y] CH d (X)). 1. The classes c d (i K O Y ) and [Y ] are functorial with respect to transversal squares (see [Pa]); i.e., for a transversal square we have X i Y φ X i ψ Y φ CH (c d (i K O Y )) = c d (i K O Y ), φ CH ([Y ]) = [Y ]. 2. The classes c d (i K O Y ) and [Y ] lie in the ernel of the homomorphism j CH, where j is the open embedding X Y X. Proof. 1. From the commutative diagrams K 0 (X ) i K φ K K 0 (X) i K and K 0 (Y ) CH d (X ) i CH ψ K K 0 (Y ) φ CH CH d (X) i CH CH 0 (Y ) ψ CH CH 0 (Y ),

3 ON THE RIEMANN ROCH THEOREM WITHOUT DENOMINATORS 1023 we obtain φ CH (c d (i K O Y )) = c d (φ K (i K O Y )) = c d (φ K (i K (1))) = c d (i K(ψ K (1))) = c d (i K(1)) = c d (i KO Y ) and φ CH ([Y ]) = φ CH (i CH (1)) = i CH(ψ CH (1)) = i CH(1) = [Y ]. 2. This is a special case of item 1, where we put X = X Y, Y =, andφ = j. The lemma is proved. 2. Deformation to the normal cone. In the case of a nonsingular variety Y,wehave the following commutative diagram of deformation to the normal cone [Pa]: P(1 N) j 0 j 1 Xt X s=i 0 i t i=i 1 Y 0 1 Yt Y, where both squares are transversal. For the Chow groups, we obtain the commutative diagram CH d j0 (P(1 N)) CH CH d j1 (X t ) CH CH d (X) s CH (i t ) CH i CH CH 0 0 (Y ) CH CH 0 1 (Y t ) CH CH 0 (Y ). Let j t be the open embedding X t Y t X t. The following statement was proved in [Pa, Lemma 1.4.2]. Lemma 2. Kerj CH t Kerj CH 0 =0. In our setting, we see that, since the classes occurring in formula (1) are functorial with respect to transversal squares (see Lemma 1), it suffices to prove (1) for the embedding Y t X t. By Lemma 1, the elements in question lie in Kerjt CH. Lemma2showsthat, to prove that the elements are equal in CH d (X t ), it suffices to chec that their images under j0 CH are equal, i.e., by Lemma 1, that the corresponding classes for the embedding s : Y P(1 N) are equal (the zero section). Thus, we may assume that X = P(E) for a vector bundle E/X and that i = s is the zero section Reduction to the embedding of a -rational point pt P d. Let K = (Y ) be the field of rational functions on Y. We consider the transversal square P d K s φ P(E) s ψ Spec(K) Y, where ψ is the embedding of the generic point and φ is the embedding of the generic fiber. Passing to the Chow groups, we obtain the following commutative diagram: CH d (P d K ) φ CH CH d (P(E)) (s ) CH s CH CH 0 (Spec(K)) ψ CH CH 0 (Y ).

4 1024 O. B. PODKOPAEV AND E. K. SHINDER The lower horizontal arrow ψ CH is the isomorphism Z Z. We prove that the classes [Y ]andc d (O Y ) lie in Im s CH. obvious because [Y ]=s CH (1). Consider the exact sequence of localization, For the class [Y ]thisis 0 CH 0 (Y ) s CH CH d (P(E)) jch CH d (P(E) Y ) 0, where the first arrow is injective because p CH s CH =id. By Lemma 1, the class c d (i K O Y ) belongs to Kerj CH. Consequently, there exists a unique element a CH 0 (Y ) such that c d (O Y )=s CH (a). Now, the proof of formula (1) reduces to checing the identity a =( 1) d 1 (d 1)! in CH 0 (Y )=Z. Applying the isomorphism ψ CH and introducing the notation a = ψ CH (a), we obtain the equivalent identity a =( 1) d 1 (d 1)! in CH 0 (pt) =Z, wherept =SpecK(Y ). Applying the homomorphism s CH to the latter identity and taing into account the relation s CH(a )=s CH(ψ CH (a)) = φ CH (s CH (a)) = φ CH (c d (i K O Y )) = c d (φ CH (s K (O Y ))) = c d (s CH(ψ CH (O Y ))) = c d (s CH(O pt )), we obtain the equivalent identity c d (s K O pt )=( 1) d 1 (d 1)![pt] in CH d (P d ), which is precisely formula (1) for the embedding pt P d The case of an embedding i : pt P n (pt). Now, we prove formula (1) in the case where Y = pt is a -rational point, X = P n,andi: pt Pn is an embedding. By the projective bundle theorem, we have CH(P n )=Z[ζ]/(ζ n+1 ), ζ = c 1 (O(1)) CH(P n ). Moreover, s CH (1) = [pt] =ζ n. Therefore, in this case, (1) is equivalent to the relation c(i O pt )=1+( 1) n 1 (n 1)!ζ n. Consider the Koszul complex 0 n Q n 1 Q Q O pt 0. Here, Q is the vector bundle dual to the universal quotient bundle over P n.wehave i K ([O pt ]) = ( 1) [ Q ] in K 0 (P n ). Thus, c(i K ([O pt ])) = n c( Q ) ( 1) = n (1 ζ) ( 1) ( n ),

5 ON THE RIEMANN ROCH THEOREM WITHOUT DENOMINATORS 1025 where ζ = c 1 (O(1)). For brevity, we denote the latter product by Π. Taing the logarithm of Π, we obtain (the calculation below maes sense because ζ is nilpotent) log(π) = ( 1) log(1 ζ) = ( 1) (ζ) j ( n ζ ( 1) = ( 1) ) 1 j j j j. Finally, j=1 (2) log(π) = j=1 j=1 S n,j ζ j j, where S n,j = ( 1) 1 j, j 1. Since the S n,j differ from the second ind Stirling numbers by the factor ( 1) n 1 n!, we have S n,j =0, j < n, S n,n =( 1) n 1 n!. Using (2), finally we obtain c([o pt ]) = Π = exp(log(π)) = 1 + S n,n n ζn =1+( 1) n 1 (n 1)!ζ n. In the paper [Gr], Grothendiec introduced the functor CH mult (X) =Z {1+a 0 + a 1 +, where a i CH i (X)} (in [Gr], it was denoted by Ã) as the direct product of groups (the second factor is regarded as the multiplicative subgroup of invertible elements in CH(X)). The product in CH mult (X) is defined so that the natural transformation of functors c mult : K 0 CH mult, c mult (X) :E (r(e),c(e)), is a homomorphism of rings. In more detail, 1=(1, 1) CH mult (X), and multiplication is given by the formulas (1, 1+x) (1, 1+y) =(1, 1+x + y), (n, 1+x x n ) (m, 1+y y m )=(nm, F n,m (x 1,...,x n ; y 1,...,y m )), F n,m (x 1,...,x n ; y 1,...,y m )= (1 + ξ i + η j ), i,j 1+x x n =(1+ξ 1 ) (1 + ξ n ), 1+y y m =(1+η 1 ) (1 + η m ). In that way, CH mult becomes a ring functor. The functor CH mult is not an oriented cohomology pretheory in the sense of [PS], because the projective bundle theorem fails: though the ring CH mult (P n )isafreez = CH mult (pt)-module of ran n +1, ithas no generators of the form 1, ζ, ζ 2,..., ζ n+1 (see 2). The ring CH mult allows us to calculate c(o pt ) CH(P n ) without using the Koszul complex. Specifically, (0,c(O pt )) = c mult (O pt )=c mult ([O P n 1] n )=c mult (O P n 1) n = ζmult, n

6 1026 O. B. PODKOPAEV AND E. K. SHINDER where By the binomial formula, we obtain We have ζ mult = c mult (O P n 1) =c mult (1 [O( 1)]) =1 c mult (O( 1)) = (0, 1 ζ). ζ n mult =(1 c mult (O( 1))) n = = ( n ( 1) =0 (3) c([o pt ]) = ) (1, 1 ζ) = ( 1) c mult (O( )) =0 ( n ) 0, (1 ζ) ( 1) ( n ). n (1 ζ) ( 1) ( n ) =Π. Remar. In the case of the embedding of a rational point into a projective space, the Riemann Roch formula without denominators can also be proved with the help of the Riemann Roch Grothendiec formula for this embedding, because the Chow ring of the projective space has no torsion. 2. The functor CH mult Theorem. 1. TheringCH mult (P n ) is a free CH mult (pt) =Z-module of ran n For n 3, there are no elements ζ mult such that 1, ζ mult,..., ζmult n generate the module CH mult (P n ). Proof. 1. In CH mult (X), there is a natural monotone decreasing filtration CH () mult (X), 0, where CH (0) mult (X) =CH mult(x), CH (1) mult (X) ={(0, 1+a 1 + a 2 + ),a i CH i (X)}, CH () mult (X) ={(0, a + ),a i CH i (X)}, 1 n, CH () mult (X) =0, > n. This filtration has the property CH (p) mult (X)CH(q) mult (X) CH(p+q) mult (X). For each p, there is a natural isomorphism CH (p) mult (X)/CH(p+1) mult CH p (X) (for p = 0, we assume that X is connected); i.e., the graded module GCH mult (X) associated with the filtration CH (p) mult is isomorphic to CH(X). Since CH(Pn )isafree Z-module of ran n +1,weseethatCH mult (P n )isalsoafreez-module of ran n +1. Moreover, it is easy to show that if ζ is a generator of CH 1 (P n ), then the elements e 0 =1,e 1 =(0, 1+ζ), e 2 =(0, 1+ζ 2 ),..., e n =(0, 1+ζ n ) form an additive basis of CH mult (P n ). 2. Now, let ζ mult =(m, 1+a 1 ζ + o(ζ)) be an arbitrary element of CH mult (P n ). We may assume that m =0. Ifa 1 = 0, i.e., ζ mult CH (2) mult, then the powers ζ mult, 1, also belong to CH (2) mult, and obviously, cannot form a basis of CH mult(p n ).

7 ON THE RIEMANN ROCH THEOREM WITHOUT DENOMINATORS 1027 Thus, we may assume that a 1 0. A computation similar to that in the proof of the Riemann Roch theorem without denominators shows then that ζmult =(0, 1+( 1) s a 1 ( 1)!ζ + o(ζ )). This means that, expanding the elements 1, ζ mult,..., ζmult n with respect to the basis e 0, e 1,...,e n, we obtain a lower triangular matrix with entries ( 1) s a 1 ( 1)! on the diagonal. Thus, a 1 =1,and1,ζ mult,..., ζmult n is not a basis if n 0, 1, 2. References [Fu] W. Fulton, Intersection theory, Ergeb. Math. Grenzgeb. (3), Bd. 2, Springer-Verlag, Berlin, MR (85:14004) [Gr] A. Grothendiec, La théorie des classes de Chern, Bull. Soc. Math. France 86 (1958), MR (22:6818) [Har] R. Hartshorne, Algebraic geometry, Grad. Texts in Math., No. 52, Springer-Verlag, New Yor Heidelberg, MR (57:3116) [MS] J. W. Milnor and J. D. Stasheff, Characteristic classes, Ann. of Math. Stud., No. 76, Princeton Univ. Press, Princeton, NJ, MR (55:13428) [P1] I. Panin, Push-forwards in oriented cohomology theories of algebraic varieties: II, [PS] I. Panin and A. Smirnov, Push-forwards in oriented cohomology theories of algebraic varieties, [Pa] I. Panin, Riemann Roch theorems for oriented cohomology, Axiomatic, Enriched, and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., Dordrecht, 2004, pp MR (2005g:14025) [Qu] D. Quillen, Higher algebraic K-theory. I, AlgebraicK-Theory, I: Higher K-Theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., vol. 341, Springer, Berlin, 1973, pp MR (49:2895) [Ve] J.-L. Verdier, Spécialisation des classes de Chern, Astérisque, No , Soc. Math. France, Paris, 1981, pp MR (83m:14015) address: opodopaev@gmail.com address: shinder@list.ru Received 14/JUN/2006 Translated by O. B. PODKOPAEV