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An Invariant of Quadratic Forms Over Schemes Doc Math J DMV An Invariant of Quadratic Forms over Schemes Marek Szyjewski Received September Communicated by Alexander S Merkurjev 0 Abstract A ring homomorphism e W X EX from the Witt ring of a scheme X into a prop er sub quotient EX of the Grothendieck ring K X 0 is a natural generalization of the dimension index for a Witt ring of a eld 0 In the case of an even dimensional pro jective quadric X the value of e on the Witt class of a bundle of an endomorphisms E of an indecomp osable comp onent V of the Swan sheaf U with the trace of a pro duct as a bilinear 0 form is outside of the image of comp osition W F W X E X Therefore the Witt class of E is not extended Introduction An imp ortant role in the quadratic form theory is played by the rst dimensional 0 cohomological invariant the dimension index e W F ZZ which maps a Witt class of a symmetric bilinear space V over a eld F onto dim V mo d A straightforward generalization of this map for symmetric bilinear spaces over rings or schemes which assigns to a Witt class the rank of its supp orting mo dule or bundle is 0 commonly used We dene a b etter invariant e in Section b elow It is a variant of 0 the construction used in and The map e dened in Section assigns to a Witt class of a symmetric bilinear space V a class V of V in the group EX attached functorially to a scheme X The group EX consist of the selfdual ie stable under dualization elements of the Grothendieck group K X up to the split selfdual ones 0 ie sums of a class and its dual Thus the rank mo d may b e obtained by passing to the generic stalk The group EX carries much more information on the Witt 0 group W X than ZZ and so do es the map e dened here when compared to the rank mo d In particular we use it here to show that certain Witt classes are not extended ie are not of the form V O for a symmetric bilinear space X V over a base eld In the Section basic facts on dualization in the Grothendieck group denition 0 and elementary prop erties of the group EX and map e are given Theorem describ es EX for a smo oth curve X In the geometric case algebraically closed base eld the group EX app ears to coincide with the Witt group W X of curve X itself Documenta Mathematica Marek Szyjewski Moreover it is shown that Witt classes of line bundles of order two in Picard group are not extended from the base eld Section contains a number of examples to show that EX may b e actually computed the ane space Prop osition the pro jective space over a eld Prop osition the pro jective space over a scheme Prop osition The main ob jective of this pap er is to prove that on the pro jective quadric of even dimension d dened by a hyperb olic form there exist nonextended Witt classes For this purp ose a close lo ok at the Swan computation of the K theory of a quadric hypersurface is needed Section contains all needed facts on Cliord algebras and mo dules the construction of the Swan bundle its b ehavior under dualization and how to nd a canonical resolution of a regular bundle In Section we develop a combinatorial metho d for op erations with resolutions using generating functions Next we use the classical computation of the Chow ring of a split quadric X to establish the ring structure of K X Theorem gives the 0 description of EX for a split quadric Thus in Section we show in Theorem that in case of even dimension d of a quadric the bundle of endomorphisms of each indecomp osable comp onent of the Swan bundle carries a canonical symmetric bilinear form whose Witt class is not 0 extended from the base eld since its invariant e has a value outside the image of the comp osite map W F W X EX The rst version of this pap er contained only an explicit computation for a quadric of dimension The referee made several suggestions for simplication of pro ofs and computations These remarks led author to the present more general re sults The author would like to thank very much the referee for generous assistance The author is glad to thank Prof W Scharlau for helpful discussions and Prof K Szymiczek who suggested several improvements of the exp osition 0 The group EX and the invariant e Notation If X is a scheme with the structural sheaf O and M N are coherent lo cally free X sheaves of O mo dules vector bundles on X M N is a morphism then we X write M H om M O and N M O X X for the duals A symmetric bilinear space M consists of a coherent lo cally free sheaf M which is selfdual ie and a morphism M M For a subbundle a subsheaf which is lo cally a direct summand N M dene its orthogonal complement N as a kernel of comp osition ^ N KerM M N MN Thus induces an isomorphism N There are two imp ortant sp ecial cases the rst when N has trivial intersection with N or is nonsingular then induces an isomorphism N N the second when N N and in this case N is said to b e a Lagrangian subbund le Documenta Mathematica Quadratic Forms over Schemes A symmetric bilinear space M is said to b e metabolic if it p ossesses a La grangian subbundle ie if there exists an exact sequence ^ N M N for some subbundle N Direct sum and tensor pro duct are dened in the set B X of isomorphism classes of symmetric bilinear spaces and in its Grothendieck ring GX the set M X of dierences of classes of metab olic spaces forms an ideal The Witt ring W X of X is the factor ring GX M X The Witt class of a symmetric bilinear space M is its coset in W X Two symmetric bilinear spaces M and M are Witt equivalent M M i their Witt 1 1 2 2 1 1 2 2 classes are equal or equivalently i M M is metab olic Each 1 2 1 2 Witt class an element of W X contains a symmetric bilinear space and X W X is a contravariant functor on schemes namely for arbitrary morphism f Y X of schemes the inverse image functor f induces a ring homomorphism f W X W Y In fact f M f M and f is an exact functor In the ane case # X Sp ec R Y Sp ec S f R S a ring homomorphism f W X W Y is simply the scalar extension S W R W S Imp ortant sp ecial R cases are lo calization or taking a stalk at a p oint x X ie the inverse image for Sp ec O X and the extension ie taking the inverse image for the structure X x map f X Sp ec F for a variety X over a eld F In the latter case a Witt class of the form f M f M O for genuine bilinear space M over F X F is said to b e extended or induced from the base eld F Rank mo d In the ane case X Sp ec R we write as usual W R instead of W Sp ec R The classical situation is if R F is a eld of characteristic dierent from two In this case there is a ring homomorphism 0 0 e W F ZZ e M dim M mo d 0 known as dimension index One may put the denition of e into a K theoretical framework as follows The map e M M induces a ring homomorphism e = GF K F Z 0 which is surjective since each vector space over F carries a symmetric bilinear form Any metab olic form M is hyperbolic ie the sequence splits and M N N 0 Since each vector space is selfdual eM F K F Z so e is the induced 0 ring homomorphism 0 e W F ZZ K F K F 0 0 Documenta Mathematica Marek Szyjewski In general the forgetful functor M M induces a ring homomorphism which in general neither is surjective nor maps M X into K X We shall show 0 b elow how to handle this using a prop er sub quotient of K X 0 The involution and the group E X Denote by PX the category of lo cally free coherent O mo dules The dualization X op functor is an exact additive functor PX PX which preserves tensor pro ducts and commutes with inverse image functors Since op K PX K PX K X the functor induces a homomorphism on K groups known also as the Adams op 1 eration We shall denote it by Definition K X K X is the homomorphism induced by the exact op functor PX PX Proposition The homomorphism K X K X enjoys the following prop erties i is an involution ii is a graded ring automorphism of K X iii if f Y X is a morphism of schemes then f f iv if i Z X is a closed immersion and X is regular of nite dimension then i K Z i K Z 0 0 Pro of iv Consider a nite resolution of i M by vector bundles for a bundle M on Z The stalk of this resolution at any p oint outside Z is exact so its dual is exact Hence the class of the alternating sum of the members of the resolution vanishes outside Z We fo cus our attention on the Grothendieck group K X The main ob ject of 0 this pap er are the homology groups of the following complex ^ ^ ^ ^ 1+ 1 1+ 1 K X K X K X K X 0 0 0 0 Definition EX Ker Im E X Ker Im 0 We shall dene a natural homomorphism e W X EX The group E X will play only a technical role here although one may consider a natural map L X E X The EX is the group of symmetric or selfdual elements in 2k +1 K X mo dulo split selfdual elements ie elements of the form M M The 0 following observations are obvious Documenta Mathematica Quadratic Forms over Schemes Proposition i Ker is a subring of K X and the groups Im 0 Ker Im are Ker mo dules ii EX is a ring and E X is an EX mo dule iii an arbitrary morphism f Y X of schemes induces a ring homomorphism f EX EY and an EX mo dule homomorphism f E X E Y iv for a regular No etherian X EX and E X carry a natural ltration induced by the top ological ltration of K X K X 0 0 v EX and E X Note that
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