The excess intersection formula for Grothendieck-Witt groups

Transcription

1 manuscripta mathematica manuscript No. (will be inserted by the editor) Jean Fasel The excess intersection ormula or Grothendieck-Witt roups Received: date / Revised version: date Abstract. We prove the excess intersection and sel intersection ormulae or Grothendieck-Witt roups. 1. Introduction Let X Y Y be a ibre product o (reasonable) schemes. Suppose that i and are reular embeddins and that and are o inite Tor-dimension. The excess intersection ormula or K-theory computes the deect o commutativity o the square K 0 (X ) K 0 (X) K 0 (Y ) i K 0 (Y ) where K 0 (_) denotes either the Grothendieck roup o the cateory o coherent sheaves, either the Grothendieck roup o the cateory o (coherent) locally ree sheaves on X. I N Y X denotes the normal cone to n Y and N Y X is the normal cone o X in Y then N Y X is an admissible subbundle o N Y X. I E denotes the quotient bundle, the excess intersection ormula states that or any α K 0 (X) i (α) = (e(e) α) where e(e) is the Euler class o E. This ormula makes sense in various other contexts than K-theory, such as Chow roups and motivic cohomoloy, equivariant K-theory and others (see [6], [4] and [12]). Jean Fasel: 1984 Mathematics Road, UBC, 6T 1Z2 ancouver BC, Canada

2 2 Jean Fasel Grothendieck-Witt roups are the quadratic analoues o K-theory (see [20] or instance). In this paper, we will call GW -theory the theory o Grothendieck-Witt roups. These roups share many common eatures with K-theory: There are both a GW -theory or coherent sheaves, which is a covariant unctor, and a GW -theory o locally ree sheaves which is contravariant unctor. There are various compatibility results involvin both the push-orwards and pull-backs, includin a proection ormula ([3]). But Grothendieck-Witt roups have also some undamental dierences with K- theory. First, the roup GW 0 hides in act a whole bunch o roups indexed by a number modulo 4 and by classes in the Picard roup o X modulo 2. The push-orward is then only conditionally deined, thus yieldin a lot o diiculties not present in the K-theory world. Second, the proective bundle theorem ails in the context o GW -theory ([19, Theorems ]). So the undamental splittin property is no loner true in the quadratic context. Altoether one has to be very careul when eneralizin results rom K-theory to GW -theory. However, the excess intersection ormula holds or GW -theory (Theorem 32 in the text): Theorem. Consider the ollowin ibre product Z Y o Gorenstein schemes. Suppose that i and are reular embeddins o respective codimension d and d and suppose that and are o inite Tordimension. Let N Y X (respectively N Z ) denote the normal cone to n Y (respectively to in Z) and put E = N Y X/N Z. Then or any m Z, any line bundle S over Y and any α GW m (X, det N Y X i S), we have L i (α) = (e(e) L (α)) GW m+d (Z, S) where e(e) denotes the Euler class o E. The condition on the schemes to be Gorenstein is imposed by the theory o Grothendieck-Witt roups itsel. As in the K-theory case, one o the particular case o the excess intersection ormula also proves to be useul (enouh to et a proper name): The sel-intersection ormula. This is Theorem 33 in the text: Theorem. Let Y be a Gorenstein scheme and let i : X Y be a reular closed embeddin o relative dimension d. Let N Y X be the normal cone to X in Y. Then or any line bundle S over Y and any α GW m (X, det N Y X i S) we et Li i (α) = e(n Y X) α GW m+d (X, i S).

3 The excess intersection ormula or Grothendieck-Witt roups 3 These two theorems also hold or Witt roups, as well as or Chow- Witt roups. One act worth mentionin is that the Euler classes are not very interestin or Witt roups, at least when the vector bundle is o odd rank ([5, Proposition 14]). So an obvious corollary o the sel-intersection ormula is that Li i is always zero i the embeddin X Y is o relative odd dimension. The oranization o this article is as ollows: Section 2 reviews the basic material needed in the proo o the excess intersection ormula. It starts rom the deinitions o GW -theory or coherent sheaves and or locally ree (coherent) sheaves ollowin [20], and then reminds a ew stu on the respective unctorial properties o those two theories (see [3]). A quick survey on Euler classes then ollows, culminatin at the computation o the homomorphism s associated to the zero section s : X E associated to a vector bundle E over X. Then we deal with the deormation to the normal cone, eneralizin a bit the results o Nenashev ([15]). This leads to the computation o or any reular embeddin. All the pieces all toether to produce a proo o the excess intersection ormula in Section 3. We irst prove a simple lemma, and then reduce to that case usin the deormation to the normal cone. This is classic, and the reader amiliar with the proo o the ormula in other theories will not ind any new strikin idea here. Finally, the sel intersection ormula is stated and proved at the end o the paper. Conventions All schemes are separated, noetherian over Spec(Z[ 1 2 ]). 2. Grothendieck-Witt roups 2.1. Deinitions and basic acts Let X be a scheme and let D b (P(X)) be the derived cateory o bounded complexes o locally ree (coherent) O X -modules. For any line bundle S over X, the usual unctor _ S = Hom OX (_, S) induces a duality on D b (P(X)) in the sense o [1, Deinition 1.4.1] and we call derived Grothendieck-Witt roups the Grothendieck-Witt roups o that trianulated cateory endowed with the duality _ S (see [20] or more inormation on those roups). We denote them by GW i (X, S). Observe that the total roup GW (X, _) is a rin ([9]) whose product we denote by. I Z s a closed subscheme, we denote by DZ b (P(X)) the subcateory o Db (P(X)) o complexes whose homoloy is supported on Z. Aain, the unctor _ S induces a duality on this cateory and one obtains Grothendieck-Witt roups which we denote by GWZ i (X, S) and call derived Grothendieck-Witt roups with support on Z.

4 4 Jean Fasel Let D b c(x) be the derived cateory o complexes o O X -modules whose homoloy is coherent and bounded. I s a Gorenstein scheme (see [10, Chapter, 9]), then any line bundle S over ves a dualizin obect in the sense o [3, Deinition 2.1] (and conversely all dualizin obects are shited line bundles). This means that the unctor RHom(_, S) is a duality on D b c(x) and thereore we can deine the coherent Grothendieck-Witt roups o X to be the Grothendieck-Witt roups o that cateory endowed with the duality RHom(_, S). We denote them by GW i (X, S). I Z s as above, we can in the same way deine coherent Grothendieck-Witt roups with support on Z, which we denote by GW i Z(X, S). I s a Gorenstein scheme, the tensor product o sheaves induces a unctor D b (P(X)) D b c(x) D b c(x) which ives a product or any i, Z and any line bundles S, S over X ([9, 3.2]) GW i (X, S) GW (X, S ) GW i+ (X, S S ). For any α GW i (X, S) and any β GW (X, S ) we will denote by α β the above product. Remark that this product endows GW (X, _) with a structure o a let GW (X, _)-module. It is clear that the product deined above respects the supports: I Z and Z are closed subscheme o X, then we et a product GW i Z(X, S) GW Z (X, S ) GW i+ Z Z (X, S S ) Transers and pull-backs Let X and Y be Gorenstein schemes and let : X Y be a morphism. Then : P(Y ) P(X) induces a unctor : D b (P(Y )) D b (P(X)) which ives homomorphisms : GW i (Y, S) GW i (X, S) or any i Z and any line bundle S over Y (use the canonical isomorphism Hom(A, S) Hom( A, S)). I is o inite Tor-dimension (see or instance [10, Chapter II, 4]), it ives a unctor L : D b c(y ) D b c(x) ([10, Proposition II.4.4]) which ives homomorphisms L : GW i (Y, S) GW i (X, S) or any i Z and any line bundle S over Y ([3, Theorem 4.1]). I is exact (e.. i is lat), we will simply denote by (instead o L ) the induced homomorphisms on Grothendieck-Witt roups.

5 The excess intersection ormula or Grothendieck-Witt roups 5 Suppose now that : X Y is proper, that X and Y are Gorenstein, and that is o constant relative dimension d = dim X dim Y. Then induces a unctor R : D b c(x) D b c(y ) which in turn ives homomorphisms R : GW i+d (X,! S) GW i (Y, S) or any i Z and any line bundle S over Y, where! S is a suitable line bundle over X ([3, Theorem 6.2]). Observe that R actors throuh the Grothendieck-Witt roups with support on (X). We still denote by R the homomorphisms R : GW i+d (X,! S) GW i (X)(Y, S). Moreover, we will write instead o R i is exact (e.. i is inite) Homotopy invariance One o the basic properties one would expect rom a ood theory is the so called homotopy invariance. We prove that coherent Grothendieck-Witt roups satisy this property. Proposition 21. Let X be a Gorenstein scheme and S be a line bundle over X. Let p : E X be a vector bundle over X. Then is an isomorphism or any i Z. p : GW i (X, S) GW i (E, p S) Proo. Let K 0 (X) (respectively K 0 (E)) be the Grothendieck roup o the cateory o complexes o O X -modules with coherent and bounded homoloy (respectively o O E -modules). By deinition, there is a commutative diaram with exact rows K 0 (X) H GW i (X, S) W i (X, S) 0 p p p K 0 (E) H GW i (E, p S) W i (E, p S) 0. Now the riht and let p are both isomorphisms by [7, Corollary 4.2] and [16, 6, Theorem 8]. This shows that the middle p is surective. I s : X E is the zero section, then s is obviously o inite Tor-dimension and then Ls is deined. But ps = Id and then Ls p = Id, showin that p is also inective.

6 6 Jean Fasel 2.4. Base chane in a simple case Let X Y X Y be a ibre product o Gorenstein schemes. Under certain hypotheses, this diaram yields a commutative diaram o homomorphisms o Grothendieck- Witt roups ([3, Theorem 5.5]). However, to prove the excess intersection ormula, we will need only an easy version: Lemma 22. Consider the ollowin ibre product o Gorenstein schemes: D Y where i is a reular embeddin o codimension d, D is a principal eective Cartier divisor iven by the lobal section t o O Y. Suppose moreover that D meets X properly. Then the ollowin diaram commutes or any m Z and any line bundle S over Y GW m (, i! S) L GW m+d (D, S) L GW m (X, i! S) i GW m+d (Y, S). Proo. In view o [3, Theorem 5.5], we only have to prove that the diaram D Y is Tor-independent in the sense o [13, Deinition ]. Let y Y be a point and Spec(A) Y be a neihborhood o y such that i 1 (Spec(A)) = Spec(A/M) where M is a complete intersection ideal o heiht d. Since D and ntersect transversally, the sequence (M, t) is reular and thereore we et Tor A i (A/M, A/t) = 0 or i > 0 by [14, Theorem 43]. Remark 23. In the statement o the lemma, we have used the act that there is a canonical isomorphism i! S! S ([3, 5]).

7 The excess intersection ormula or Grothendieck-Witt roups Euler classes Let X be a scheme and let p : E X be a vector bundle o constant rank r. The zero section s : p E O E ives the Koszul complex K(E): 0 det p E r 1 p E... p E s O E 0 which can be seen as an obect o D b (P(X)). It turns out that it carries a symmetric orm or the d-th shited duality. Indeed, we have or any 0 r canonical isomorphisms ϕ : p E Hom OE ( r p E, p det E ) which ive, up to some sin convention, a symmetric isomorphism ([2, 4] or [15, 2.2]) θ(e) : K(E) T r Hom(K(E), p det E ), where T is the translation unctor in the trianulated cateory D b (P(X)). Thereore we can see κ(e) := (K(E), θ(e)) as an element o GW r (E, p det E ). The Euler class o E is the pull-back s κ(e) in GW r (X, det E ). We denote it by e(e). This class satisies several unctorial properties that we survey in the next proposition: Proposition 24. Let X, Y be schemes and let p : E X be a vector bundle o constant rank r. Then 1. I : Y X, then e( E) = e(e). 2. I 0 E E E 0 is an exact sequence o vector bundles over X, then e(e) = e(e ) e(e ). Proo. The irst assertion ollows rom κ( E) = κ(e) ([15, Proposition 2.2]) and the act that pull-backs are unctorial. The second assertion is proven in [5, Theorem 21]. I s a Gorenstein scheme, observe that the zero section s yields homomorphisms (see [3, 7.2]): s : GW i (X) GW i+r (E, p det E ) which are not hard to describe. Indeed, let 1 X : O X RHom(O X, O X ) be the obvious (symmetric) isomorphism and let 1 X also denote its class in GW 0 (X). Then: Proposition 25. Let X be a Gorenstein scheme. Let p : E X be a vector bundle o constant rank r and let s : X E be the zero section. Then we have s (1 X ) = κ(e) 1 E in GW r (E, p det E ).

8 8 Jean Fasel Proo. This is [3, Proposition 7.1]. Remark 26. The homomorphism GW i (X, S) GW i (X, S) iven by α α 1 s an isomorphism i s reular. Thereore the results obtained below are valid also or derived Grothendieck-Witt roups when the schemes are reular. A simple eneralization o Proposition 25 would be the next result, which ives a complete computation o s (see also [8, Lemma 2.8]): Proposition 27. Let X be a Gorenstein scheme and let S be a line bundle over X. Let p : E X be a vector bundle o constant rank r and let s : X E be the zero section. Then s (α) = p (e(e) α) in GW i+r (E, p det E p S) or any α GW i (X, S). Proo. We prove irst that or any α, we have s (α) = κ(e) p α. Observe irst that the homomorphism o O E -modules O E s O nduced by s ives a quasi-isomorphism K(E) s O X (here the Koszul complex K(E) is seen as an element o D b c(e)). For any P D b c(x) we et a composition o isomorphisms (use [11, Exercise II.5.1.(d)]) K(E) OE p P (s O X ) OE p P s (P ). I (P, ϕ) is a symmetric pair, we claim that this isomorphism is an isometry between κ(e) p (P, ϕ) and s (P, ϕ). This can be checked locally and inductively on the rank o E, so we can suppose that s the spectrum o a Gorenstein rin and E = X A 1. The result ollows then rom [8, Lemma 2.8]. Now κ(e) = p e(e) by homotopy invariance o the derived Grothendieck-Witt roups and p e(e) p α = p (e(e) α) by [9, Theorem 3.4]. So s (α) = κ(e) p α = p e(e) p α = p (e(e) α). In some sense, the push-orward or reular embeddins can be computed by usin this proposition as we will see in the next section Deormation to the normal cone Let : X Y be a reular embeddin o Gorenstein schemes. Let D(X, Y ) be the deormation to the normal cone space and N Y X the normal cone to n Y (see [6, Chapter 5] or the deormation over P 1, [17, 10] or [15, 3] or the deormation over A 1 that we use here). Let moreover s : X N Y X be the inclusion as the zero section. We have an embeddin µ : X A 1 D(X, Y ) and a commutative diaram whose squares are ibre product: X δ 0 X A 1 δ 1 X s µ N Y X D(X, Y ) Y. i0 i 1

9 The excess intersection ormula or Grothendieck-Witt roups 9 The ollowin lemma is a eneralization o [15, Proposition 3.1]: Lemma 28. The homomorphisms in the diaram GW m X(N Y X, i 0S) Li 0 m GW X A1(D(X, Y ), S) Li 1 GW m X(Y, i 1S) are isomorphisms or any line bundle S over D(X, Y ) and any m Z. Proo. We ollow the lines o [15, Proposition 3.1]. Consider irst the special situation when Y is the total space o a vector bundle E over X, and : X Y is the zero section. Then D(X, Y ) is isomorphic to Y A 1 and i is the inclusion o Y {} in Y A 1 or = 0, 1. We then can use homotopy invariance (Proposition 21) to conclude in that case. In eneral, usin a suitable Mayer-ietoris sequence (which is a consequence o the localization sequence constructed in [18]), it suices to prove that or any y Y there exists a (Zariski) open subscheme U containin y such that Li is an isomorphism when restricted to any open subscheme o U or = 0, 1. I y X, then Y X does the ob so we can ocus on points o X. Now there exists or any x X a Zariski neihborhood U o x in Y and étale morphisms : U U and : U X U A n (where X U = X U) which induce isomorphisms X : 1 (X U ) X U and X : 1 (X U ) X U {0}. Usin étale excision property or coherent Grothendieck-Witt roups (which is a straihtorward consequence o [7, Theorem 2.4]), we see that we are reduced to the case Y = X A n already treated. In the sequel, the composition Li 1(Li 0) 1 is denoted by d(x, Y ). Proposition 29. Let Y be a Gorenstein scheme and let : X Y be a reular embeddin o codimension d. Let D(X, Y ) be the deormation to the normal cone space and N Y X the normal cone to n Y. Let moreover s : X N Y X be the inclusion as the zero section and b : D(X, Y ) Y be the composition o the blow-down map with the proection Y A 1 Y. The ollowin diaram commutes GW m (X, det N Y X S) X (Y, S) d(x,y ) s GW d+m X (N Y X, i 0b S). or any m Z and any line bundle S over Y. Proo. Both squares o the diaram GW m+d X δ 0 X A 1 δ 1 X s N Y X D(X, Y ) Y i0 i 1 µ

10 10 Jean Fasel ulill the hypothesis o Lemma 22. Moreover, Lδ 1(Lδ 0) 1 = Id since both maps are inverse to the map induced by the proection p : X A 1 X. 3. Excess intersection and Sel-intersection ormulae We irst prove the excess intersection ormula in a simple situation, and then we deduce the eneral statement rom that case. Let : X be a morphism o Gorenstein schemes and let p : H X be a vector bundle o constant rank r. Suppose that is o inite Tor-dimension and consider the ibre product H H p p X. Let 0 F i H π E 0 be an exact sequence o vector bundles over. Let : F H be the composition i and let q : F be the proection. We then et a commutative square q F H X p and, i t : F and s : X H are the zero sections, a commutative diaram Lemma 31. We have t F H s X. (L )s (α) = t (e(e) L (α)) in GW m+r (F, N) or any m Z, any line bundle N over H and any element α GW m (X, det H s N). Proo. Proposition 27 yields s (α) = Lp (e(h) α). Usin unctoriality o the pull-back we et (L )s (α) = (L )p (e(h) α) = q (L )(e(h) α) = q ( e(h) L α). Usin Proposition 24, we et q ( e(h) L α) = q ((e(e) e(f )) L α) = q (e(e)) q (e(f ) L α).

11 The excess intersection ormula or Grothendieck-Witt roups 11 But Proposition 27 shows that the latter is equal to q e(e) t (L (α)), which in turn is equal to t (e(e) L (α)) by the proection ormula ([3, Theorem 5.7]). From this lemma, we easily et the excess intersection ormula in eneral. Observe irst that i Z Y is a ibre product o schemes where i and are reular embeddins o respective constant codimensions d and d, then N Z is a sub-bundle o N Y X and the quotient E = N Y X/N Z is locally ree o rank d d ([6, Appendix B.6]). We et: Theorem 32 (Excess intersection ormula). Consider the ollowin ibre product Z Y o Gorenstein schemes. Suppose that i and are reular embeddins o respective constant codimensions d and d and suppose that and are o inite Tor-dimension. Let N Y X (respectively N Z ) denote the normal cone to n Y (respectively to in Z) and put E = N Y X/N Z. Then or any m Z, any line bundle S over Y and any α GW m (X, det N Y X i S), we have L i (α) = (e(e) L (α)) GW m+d (Z, S) where e(e) denotes the Euler class o E. Proo. Let D(X, Y ) (resp. D(, Z)) be the deormation to the normal cone space associated to i : X Y (resp. : Z). From the Cartesian square Z Y we et a commutative diaram t N Z i 0 D(, Z) i 1 Z N D X s N Y X D(X, Y ) Y i0 i 1

12 12 Jean Fasel where s : X N Y X and t : N Z are the embeddins as the zero sections ([6, Appendix B.6.9]). The irst square on the let satisies the assumptions o the above lemma. Thereore, we et L N s (α) = t (e(e) L (α)). Now L d(x, Y ) = d(, Z)L( N ) by Lemma 28 and by unctoriality o the pull-backs. Usin Proposition 29, we et L i (α) = L d(x, Y )s (α) = d(, Z)L( N ) s (α) = d(, Z)t (e(e) L (α)) and the latter is (e(e) L (α)) by Proposition 29 aain. As a corollary, we et the sel-intersection ormula: Theorem 33 (Sel-intersection ormula). Let Y be a Gorenstein scheme and let i : X Y be a reular closed embeddin o relative dimension d. Let N Y X be the normal cone to n Y. Then or any line bundle S over Y and any α GW m (X, det N Y X i S) we et Li i (α) = e(n Y X) α GW m+d (X, i S). Proo. Apply the excess intersection ormula with X = Z and i =. Acknowledements. It is a pleasure to thank Baptiste Calmès or many useul conversations on the unctoriality properties o coherent (Grothendieck-) Witt roups. Many thanks are also due to the reeree or various useul comments. Reerences [1] Paul Balmer. Witt roups. In Handbook o K-theory. ol. 1, 2, paes Spriner, Berlin, [2] Paul Balmer and Stean Gille. Koszul complexes and symmetric orms over the punctured aine space. Proc. London Math. Soc. (3), 91(2): , [3] Baptiste Calmès and Jens Hornbostel. Push-orwards or witt roups o schemes. Preprint available at [4] Frédéric Délise. Interprétation motivique de la ormule d excès d intersection. C. R. Math. Acad. Sci. Paris, 338(1):41 46, [5] J. Fasel and. Srinivas. Chow-Witt roups and Grothendieck-Witt roups o reular schemes. Adv. Math., 221(1): , [6] William Fulton. Intersection theory, volume 2 o Erebnisse der Mathematik und ihrer Grenzebiete. 3. Fole. A Series o Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series o Modern Surveys in Mathematics]. Spriner-erla, Berlin, second edition, [7] Stean Gille. Homotopy invariance o coherent Witt roups. Math. Z., 244(2): , [8] Stean Gille and Jens Hornbostel. A zero theorem or the transer o coherent Witt roups. Math. Nachr., 278(7-8): , 2005.

13 The excess intersection ormula or Grothendieck-Witt roups 13 [9] Stean Gille and Alexander Nenashev. Pairins in trianular Witt theory. J. Alebra, 261(2): , [10] Robin Hartshorne. Residues and duality. Lecture notes o a seminar on the work o A. Grothendieck, iven at Harvard 1963/64. With an appendix by P. Deline. Lecture Notes in Mathematics, No. 20. Spriner-erla, Berlin, [11] Robin Hartshorne. Alebraic eometry. Spriner-erla, New York, Graduate Texts in Mathematics, No. 52. [12] Bernhard Köck. Das Adams-Riemann-Roch-Theorem in der höheren äquivarianten K-Theorie. J. Reine Anew. Math., 421: , [13] Joseph Lipman. Notes on derived unctors and rothendieck duality. Notes available at lipman/duality.pd, [14] Hideyuki Matsumura. Commutative alebra. W. A. Benamin, Inc., New York, [15] Alexander Nenashev. Gysin maps in Balmer-Witt theory. J. Pure Appl. Alebra, 211(1): , [16] Daniel Quillen. Hiher alebraic K-theory. I. In Alebraic K-theory, I: Hiher K-theories (Proc. Con., Battelle Memorial Inst., Seattle, Wash., 1972), paes Lecture Notes in Math., ol Spriner, Berlin, [17] Markus Rost. Chow roups with coeicients. Doc. Math., 1:No. 16, (electronic), [18] Marco Schlichtin. Hermitian k-theory, derived equivalences and karoubi s undamental theorem. Preprint available at mschlich/research/prelim.html, [19] Charles Walter. Grothendieck-Witt roups o proective bundles. Preprint available at [20] Charles Walter. Grothendieck-Witt roups o trianulated cateories. Preprint available at