An Invariant of Quadratic Forms over Schemes
Marek Szyjewski
Received: September 29, 1996 Communicated by Alexander S. Merkurjev
Abstract. A ring homomorphisme0: W(X)!EX from the Witt ring of a schemeX into a proper subquotientEX of the Grothendieck ringK0(X) is a natural generalization of the dimension index for a Witt ring of a eld.
In the case of an even dimensional projective quadricX, the value ofe0 on the Witt class of a bundle of an endomorphisms E of an indecomposable component V0of the Swan sheafU with the trace of a product as a bilinear form is outside of the image of composition W(F) ! W(X) ! E(X).
Therefore the Witt class of (E;) is not extended.
Introduction
An important role in the quadratic form theory is played by the rst (0-dimensional) cohomological invariant, the dimension index e0 : W(F) ! Z=2Z, which maps a Witt class of a symmetric bilinear space (V;) over a eldF onto dimV mod 2. A straightforward generalization of this map for symmetric bilinear spaces over rings or schemes, which assigns to a Witt class the rank of its supporting module or bundle, is commonly used. We dene a better invariante0in Section 1 below. It is a variant of the construction used in [8] and [9]. The mape0dened in Section 1 assigns to a Witt class of a symmetric bilinear space (V;) a class [V] ofV in the groupEX, attached functorially to a schemeX. The groupEX consist of the self-dual (i.e., stable under dualization) elements of the Grothendieck groupK0(X) up to the split self-dual ones (i.e., sums of a class and its dual). Thus the rank mod 2 may be obtained by passing to the generic stalk. The group EX carries much more information on the Witt groupW(X) thanZ=2Z, and so does the mape0dened here when compared to the rank mod 2. In particular, we use it here to show that certain Witt classes are not extended, i.e., are not of the form (V OX;1) for a symmetric bilinear space (V;) over a base eld.
In the Section 1 basic facts on dualization in the Grothendieck group, denition and elementary properties of the group EX and map e0 are given. Theorem 1.1 describesEX for a smooth curveX. In the geometric case (algebraically closed base eld) the groupEXappears to coincide with the Witt groupW(X) of curveX itself.
Moreover, it is shown that Witt classes of line bundles of order two in Picard group are not extended from the base eld.
Section 2 contains a number of examples to show that EX may be actually computed: the ane space - Proposition 2.1.1, the projective space over a eld - Proposition 2.1.3, the projective space over a scheme - Proposition 2.1.5.
The main objective of this paper is to prove that on the projective quadric of even dimensiond6= 2 dened by a hyperbolic form, there exist nonextended Witt classes.
For this purpose, a close look at the Swan computation of theK-theory of a quadric hypersurface is needed. Section 3 contains all needed facts on Cliord algebras and modules, the construction of the Swan bundle, its behavior under dualization, and how to nd a canonical resolution of a regular bundle.
In Section 4, we develop a combinatorial method for operations with resolutions using generating functions. Next we use the classical computation of the Chow ring of a split quadricX to establish the ring structure ofK0(X). Theorem 4.3 gives the description ofEX for a split quadric.
Thus, in Section 5, we show in Theorem 5.1 that, in case of even dimensiond >2 of a quadric the bundle of endomorphisms of each indecomposable component of the Swan bundle carries a canonical symmetric bilinear form, whose Witt class is not extended from the base eld, since its invariante0 has a value outside the image of the composite mapW(F)!W(X)!EX.
The rst version of this paper contained only an explicit computation for a quadric of dimension 4. The referee made several suggestions for simplication of proofs 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 exposition.
1 The groupEX and the invariante0
1.1 Notation.
IfX is a scheme with the structural sheafOX, andM; N are coherent locally free sheaves ofOX-modules (vector bundles on X),: M!N is a morphism, then we write
M
^=HomOX(M;OX) and ^: N^!M^ for the duals.
A symmetric bilinear space (M;) consists of a coherent locally free sheaf M and a morphism: M!M^, which is self-dual, i.e. ^=.
For a subbundle (a subsheaf which is locally a direct summand) : N ! M dene its orthogonal complementN? as a kernel of composition^ :
N
?= Ker(M ! M^ !^ N^): Thus induces an isomorphismN? = (M=N)^.
There are two important special cases: the rst, whenN has trivial intersection withN? or is non-singular, then induces an isomorphism N =N^ ; the second, whenN =N?, and in this caseN is said to be a Lagrangian subbundle.
A symmetric bilinear space (M;) is said to be metabolic if it possesses a La- grangian subbundle, i.e., if there exists an exact sequence
0!N ! M ^! N^!0 (1.1.1)
for some subbundleN.
Direct sum and tensor product are dened in the setB(X) of isomorphism classes of symmetric bilinear spaces, and in its Grothendieck ring G(X) the set M(X) of dierences of classes of metabolic spaces forms an ideal.
The Witt ring W(X) of X is the factor ring G(X)=M(X). The Witt class of a symmetric bilinear space (M;) is its coset in W(X). Two symmetric bilinear spaces (M1;1) and (M2;2) are Witt equivalent, (M1;1)(M2;2) i their Witt classes are equal, or - equivalently - i (M1M2;1+ ( 2)) is metabolic. Each Witt class (an element ofW(X)) contains a symmetric bilinear space andX 7!W(X) is a contravariant functor on schemes, namely for arbitrary morphismf : Y !X of schemes the inverse image functorf 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 = SpecR; Y = SpecS; f# : R ! S a ring homomorphism, f : W(X) ! W(Y) is simply the scalar extensionSR : W(R) ! W(S). Important special cases are localization or taking a stalk at a point x 2X, i.e., the inverse image for SpecOX;x !X, and the extension, i.e., taking the inverse image for the structure mapf : X !SpecF for a varietyX over a eldF. In the latter case a Witt class of the form (fM;f) = (MFOX;1) for genuine bilinear space (M;) over F is said to be extended or induced from the base eldF.
1.2 Rank mod 2
In the ane caseX = SpecR, we write as usualW(R) instead ofW(SpecR). The classical situation is ifR = F is a eld of characteristic dierent from two. In this case there is a ring homomorphism
e0: W(F)!Z=2Z; e0(M;) = dim M mod 2;
known as dimension index. One may put the denition of e0 into a K-theoretical framework as follows:
The mape: (M;)7![M] induces a ring homomorphism G(F) !e K0(F) != Z
which is surjective, since each vector space overF carries a symmetric bilinear form.
Any metabolic form (M;) is hyperbolic, i.e., the sequence 1.1.1 splits, and (M;)= (N N;0 11 0):
Since each vector space is self-dual,e(M(F)) = 2K0(F)= 2Z, soe0is the induced ring homomorphism
W(F) e!0 K0(F)=2K0(F)=Z=2Z:
In general the forgetful functor (M;) 7! [M] induces a ring homomorphism which in general neither is surjective nor mapsM(X) into 2K0(X) . We shall show below how to handle this using a proper subquotient ofK0(X) .
1.3 The involution^ and the group E(X)
Denote byP(X) the category of locally free coherentOX-modules. The dualization functor^ is an exact additive functor^ : P(X)!P(X)op, which preserves tensor products and commutes with inverse image functors. Since
K(P(X)) =K(P(X)op) =K(X);
the functor^ induces a homomorphism on K-groups, known also as the Adams op- eration 1. We shall denote it by ^:
Definition 1.3.1. ^: K(X)!K(X) is the homomorphism induced by the exact functor^: P(X)!P(X)op .
Proposition 1.3.2. The homomorphism^: K(X)!K(X) enjoys the following properties:
i) ^ is an involution,^^= 1;
ii) ^ is a graded ring automorphism ofK(X) : ()^=^^ ; iii) iff : Y !X is a morphism of schemes, thenf^=^f;
iv) ifi: Z !X is a closed immersion andX is regular of nite dimension, then (i(K0(Z)))^=i(K0(Z)).
Proof. iv) Consider a nite resolution ofi(M) by vector bundles for a bundleMon Z. The stalk of this resolution at any point outsideZ is exact, so its dual is exact.
Hence the class of the alternating sum of the members of the resolution vanishes outsideZ.
We focus our attention on the Grothendieck group K0(X). The main object of this paper are the homology groups of the following complex:
!K0(X) 1+!^ K0(X) 1 !^ K0(X) 1+!^ K0(X) 1 !^ (1.3.1) Definition 1.3.3.
EX = Ker(1 ^)=Im(1 +^) E X = Ker(1 +^)=Im(1 ^):
We shall dene a natural homomorphism e0 : W(X) ! EX. The group E X will play only a technical role here, although one may consider a natural map L2k+1(X)!E X . TheEX is the group of "symmetric" or "self-dual" elements in K0(X) modulo "split self-dual" elements, i.e., elements of the form [M]+[M^]:The following observations are obvious:
Proposition 1.3.4. i) Ker(1 ^) is a subring ofK0(X) and the groups Im(1+^), Ker(1 +^), Im(1 ^) are Ker(1 ^)-modules;
ii) EX is a ring andE X is anEX-module;
iii) an arbitrary morphismf : Y !X of schemes induces a ring homomorphism f: EX !EY and anEX-module homomorphismf: E X!E Y ; iv) for a regular NoetherianX, EX andE X carry a natural ltration, induced
by the topological ltration ofK0(X) =K00(X) ; v) 2EX = 0 and 2E X= 0.
Note that the forgetful functor (M;) 7! [M] induces a ring homomorphism G(X)7!K0(X) which admits values in Ker(1 ^) and mapsM(X) onto Im(1 +^), since for a metabolic space (M;) there is exact sequence 1.1.1, i.e., the equality [M] = [N] + [N^] holds inK0(X).
Definition 1.3.5. e0 : W(X) ! EX is the ring homomorphism induced by the forgetful functor (M;)7![M] .
This notion enjoys nice functorial properties.
Proposition 1.3.6. Letf : X !Y be a morphism of schemes. Then the following diagram commutes:
W(X) e0! EX
fx?? x??f
W(Y) e0! EY
Example 1.3.7. LetX be an irreducible scheme with the function eld F(X), and let j : SpecF(X) ! X be the embedding of the generic point. Then there is a commutative diagram
W(F(X)) e0! E(F(X)) =Z=2Z
jx?? x??j
W(X) e0! EX
and the compositionje0=e0jis rank mod 2, usually used instead ofe0. Since
OX carries the standard symmetric bilinear form<1>, the surjectionj : EX !
Z=2Zsplits canonically. The kernel of the mapj : EX !Z=2Zhas been used in [9]. It is easy to see that this kernel is a nilpotent ideal of a ring EX for a regular NoetherianX of nite dimension.
Example 1.3.8. Retain the notation of example 1.3.7, and assume in addition that X is a variety over a eldF, charF 6= 2. Letf :X !SpecF be the structure map.
Thus we have a commutative diagram:
e0
W(F(X)) - Z=2Z
@
@
@
@
@
@
@
@
I
e0
W(X) - EX id
6
e0 @@
@
@
@
@
@
@ I
6
W(F) - Z=2Z
The values of e0f are inside the direct summand Z=2Z[OX] of EX. If we produce a varietyX with nontrivial (i.e., having more than two elements) EX, and a symmetric bilinear space with a nontrivial value ofe0, then the Witt class of this space must be non-extended.
1.4 Curves
The case dimX = 1 is exceptional for several reasons, so we treat it here as an illustration. The following theorem covers the classical case of (spectra of) Dedekind rings.
Theorem 1.1. LetX be an irreducible regular Noetherian scheme of dimension one.
Then
i) EX =Z=2Z[OX]I, where II = 0 and I is canonically isomorphic to the group 2Pic(X) of the elements of order2 in the Picard group;
ii) E X is canonically isomorphic to Pic(X)=2Pic(X);
iii) the mape0: W(X)!EX is surjective.
Proof. The rank map (i.e., the restriction to the generic point) yields the splitting K0(X) =Z[OX]F1K0(X)
where 0 F1K0(X) K0(X) is the topological ltration on K0(X). The map ^ maps each direct summand onto itself.
Under assumptions onX the map V: F1K0(X)!Pic(X), induced by taking the highest exterior power of a bundle, is an isomorphism. An arbitrary elementof the group F1K0(X) may be expressed as a dierence of the classes of two bundles of the same rankr:
= [M] [N]:
The isomorphism Vmapsonto the class of a line bundleL,
L=^r M^r N^
in Pic(X). The isomorphismVmaps [L] [OX] onto the classLin Pic(X), too. So, any element a of F1K0(X) may be expressed as a dierence of a line bundle and the trivial line bundle:
= [L] [OX]: Moreover, for arbitrary line bundlesL1; L2
([L1] [OX])([L2] [OX]) = [L1L2] [OX]:
Hence the involution ^ acts on F1K0(X) as taking the opposite, and it acts trivially onZ[OX]. Therefore
Ker(1 ^) =Z[OX]2F1K0(X) , Im(1 +^) = 2Z[OX], Ker(1 +^) = F1K0(X) ; Im(1 ^) = 2F1K0(X), and assertions i), ii) follow.
To prove iii) note that a line bundleLwhich has order two in Pic(X) is isomorphic to its inverseL^, so is automatically endowed with a nonsingular bilinear form :
L ! L
^. This form must be symmetric locally at any point, hence is symmetric globally. Finally,e0maps the Witt class of (L;)(OX;<1>) onto the class ofL in 2Pic(X) via V.
Remark 1.4.1. If R is a Dedekind ring, X = SpecR, then Pic(X) = Pic(R) is sim- ply the ideal class group H(R); the claim on the form of element of F1K0(X) is a consequence of the structural theorem for projective modules: if rank(P) = r, then there exist fractional ideals I1;:::;Ir such that P = I1::: Ir; moreover, P =Rr 1I1:::Ir =Rr 1VrP . In this caseL1(X)= Pic(X)=2Pic(X) and L1(X) is isomorphic toE X via obvious generalization ofe0.
Remark 1.4.2. If 2Pic(X) is nontrivial, 2Pic(X)6= 0, then there exist non-extended Witt classes onX.
Corollary 1.4.3. IfX is a smooth projective curve of genusgover an algebraically closed eldF, then
i) if charF 6= 2, thenEX = (Z=2Z)1+2g;
ii) the degree map induces isomorphismE X =Z=2Z.
Remark 1.4.4. The result in Corollary 1.4.3. i) has been pointed out to author by W.
Scharlau.
Remark 1.4.5. The proposition 2.1 of [3] states that forF =C the Witt groupW(X) of a smooth projective curve X is itself isomorphic to (Z=2Z)1+2g, but the proof remains valid for an arbitrary algebraically closed eld F provided charF 6= 2. So under assumptions of Corollary 1.4.3.i) the mape0: W(X)!EXis an isomorphism.
2 The map e0: W(X)!EX for certain quasiprojectiveX. 2.1
We shall show now that the groupEX may be actually computed, and compare the result with known Witt rings. The simplest case is following:
Proposition 2.1.1. If R is a regular ring, andX =AnR, the ane space, then the inverse image functorffor the structure mapf : X !SpecRinduces isomorphisms W(R)!W(X),ER!EX,E R!E X.
Proof. By the homotopy property of K-theory, the map f : K0(R) ! K0(X) is an isomorphism and commutes with ^, so the assertion on E and E follows.
The assertion onW(X) is a consequence of the Karoubi theorem, see [6], Ch. VI.2, Corollary 2.2.2.
Now let X be a quasiprojective variety over a eld F, charF 6= 2, with the structure mapf : X !SpecF. Consider the commutative diagram
W(X) e0! EX
fx?? fx??
W(F) e0! EF
(2.1.1) We shall refer to "leftf" and "right f" in 2.1.1 for variousX.
Next, x a projective embeddingi: X !PnF and denote:
1 = [OX] - the unit element inK0(X); (2.1.2)
OX( 1) =iOPnF( 1); (2.1.3) H = 1 [OX( 1)] - the class of hyperplane section inK0(X): (2.1.4) We summarize some technicalities as follows:
Lemma 2.1.2. Ifd= dimX, then i) Hd+1= 0;
ii) [OX(1)] = (1 H) 1=Xd
i=0HiinK0(X) (hereH0= 1);
iii) H^= H
1 H =
d
X
i=1Hi; iv) (Hk)^=
H
1 H
k
= ( 1)kHkd kX
i=0
k+i 1 i
Hi; v) (Hd)^= ( 1)dHd:
Proof. H = 1 [OX( 1)], so [OX( 1)] = 1 H, [OX(1)] = (1 H) 1, H being nilpotent. ThusH^= 1 [OX(1)] = ([OX( 1)] 1)[OX(1)] = H(1 H) 1and (Hk)^= ( H)k(1 H) k.
In the case i = id, X = PdF, the family 1; H; ::: ; Hd forms a basis of a free Abelian groupK0(X), which allows us to compute EX; E X:
Proposition 2.1.3. IfX =PdF, the projective space, then:
i) both vertical arrows in the diagram 2.1.1 are isomorphisms;
ii) E X =Z=2Z[Hd] for odddandE X= 0 for evend.
Proof. The left f in the diagram 2.1.1 is an isomorphism by Arason's theorem [1].
Note that the statements onEX, E X are valid ford = 0, and - by Theorem 1.1 above - ford= 1. ConsiderY =PdF 1and a closed embeddingk:Y !X ofY as a hyperplane inX. There is an exact sequence
0!ZHd!K0(X) k! K0(Y)!0
sincekOX(i) =OY(i). Thus we have a short exact sequence of complexes:
1
^
! K0(Y) 1+!^ K0(Y) 1 !^
k
x
?
? k
x
?
?
1
^
! K0x(X) 1+!^ K0(X) 1 !^
?
?
x
?
?
1 ( 1)d
! ZHd 1+( 1)!d ZHd 1 ( 1)!d and an induced exact sequence in homology. For evendthis looks like
!0!E X!E Y !Z=2Z[Hd] !@ EX!EY !0!
and if - by induction - the proposition holds for Y, then @ maps the generator of E Y =Z=2Z[Hd 1] ontoHd mod 2ZHd, so the proposition holds forX: E X = 0,k: EX !EY is an isomorphism. In case of an odddwe have an exact sequence
!0!EX!EY !@ Z=2Z[Hd]!E X!E Y !0!
in homology. By induction EY = Z=2Z[OX], @ = 0, so k : EX ! EY is an isomorphism. ThusZ=2Z[Hd]!E X is an isomorphism, sinceE Y = 0.
Remark 2.1.4. The idea of this proof is due to the referee.
Proposition 2.1.5. For an arbitrary varietyY letX =PdFY and let p1:X !PdF,p2: X !Y be the projections. Then
EX= (p1(E(PdF))p2(EY))(p1(E (PdF))p2(E Y)) E X= (p1(E(PdF))p2(E Y))(p1(E (PdF))p2(EY)):
Proof. By the projective bundle theorem p1, p2 yield the identication K0(X) = K0(PdF)K0(Y). Denote
A= Ker(K0(PdF) 1 !^ K0(PdF)); B = (1 ^)K0(PdF):
The complex 1.3.1 forX =PdFY may be included into the short exact sequence of complexes:
1
^
! BK0(Y) 1+!^ BK0(Y) 1 !^
(1
^
)1 x
?
?
(1
^
)1 x
?
?
1
^
! K0x(X) 1+!^ K0(X) 1 !^
?
?
x
?
?
1
^
! AK0(Y) 1+!^ AK0(Y) 1 !^
Note that 1^ restricted toAK0(Y) coincides with 1(1^) and induces 1(1^) onBK0(Y). Therefore the exact hexagon in homology
EX
@
@
@
@
@
AEY BRE Y
@1x?? ??y@2
BEY AE Y
@
@
@
@
@ I
E X breaks into short split exact sequences:
0!E(PdF)E Y !E X!E (PdF)EY !0 (2.1.5) 0!E(PdF)EY !EX !E (PdF)E Y !0: (2.1.6) Example 2.1.6. Putd= 1,Y =P1F, i.e., X=P1FP1F. Then
EX =Z=2Z[OX]Z=2Z[HH] (2.1.7) E X=Z=2Z[H1]Z=2Z[1H] (2.1.8) whereis induced by operationFG=p1Fp2G. Since Witt ring is an invariant of birational equivalence in the class of smooth projective surfaces over a eld F, charF 6= 2 ([2], Theorem 3.4) andX=P1FP1F is birationally equivalent toP2F, the leftfin the diagram 2.1.1 is an isomorphism while the rightfis not. This example shows thate0: W(X)!EX need not be surjective in general.
Remark 2.1.7. Probably there exists a skew symmetric bilinear space (M;) onX =
P
1FP1F such that [M] = [HH] inEX.
Remark 2.1.8. X = P1F P1F may be embedded into P3F by Segre immersion as a quadric surfacex0x1 x2x3= 0. In fact in the preliminary version of this paper this example was given using Swan's description of theK-theory of a quadric. The idea to use inverse images for projections was pointed out to author by the referee.
Remark 2.1.9. Note that we know W(X) and EX for three quadrics of maximal index:
X equation W(X) EX E X
two points z02 z12= 0 W(F)W(F) Z=2ZZ=2Z 0
P
1F z02 z21+z22= 0 W(F) Z=2Z Z=2Z
P
1FP1F x0x1 x2x3= 0 W(F) Z=2ZZ=2Z Z=2ZZ=2Z We shall compute EX and E X for all projective quadrics of maximal index.
To do this, some preparational work is required.
3 The SwanK-theory of a split projective quadric.
To computeEX andE X, we need some facts on dualization of vector bundles on quadrics. All needed information is known in fact, since indecomposable components of a Swan sheaf correspond to spinor representations. Nevertheless we give here complete proofs of the needed facts.
We shall apply the results of [11] in the simplest possible case of a split quadric:
X is a projective quadric hypersurface over a eld F, charF 6= 2, dened by the quadratic form of maximal index.
3.1 Notation
Consider a vector spaceV with basis v0; v1; ::: ; vd+1 over a eld F, charF 6= 2.
Denotez0; z1; ::: ; zd+1 the dual basis ofV^. Letqbe the quadratic form q=dX+1
i=0( 1)izi2:
Moreover, let ei=12(v2i v2i+1); fi= 12(v2i+v2i+1) for all possible values ofi. Thus ifdis even,d= 2m, thene0; f0; e1; f1; ::: ; em; fm form a basis ofV with the dual basisx0; y0; x1; y1; ::: ; xm; ym and
q=Xm
i=0xiyi:
If dis odd, d= 2m+ 1, then f0; e1; f1; ::: ; em; fm; vd+1 form a basis of V with the dual basisx0; y0; x1; y1; ::: ; xm; ym; zd+1and
q=Xm
i=0xiyi+zd2+1:
We shall computeEX andE Xfor ad-dimensional projective quadricXdened by equationq= 0 inPdF+1, i.e., for
X= ProjS(V^)=(q)= ProjF[z0; z1; ::: ; zd+1]=(q):
3.2 The Clifford algebra
In case of an odd d = 2m+ 1 the even part C0 = C0(q) of the Cliord algebra C(q) is isomorphic to the matrix algebra MN(F), where N = 2m+1. In particular, Kp(C0)=Kp(F).
In case of an even d = 2m, the algebra C0 has the center F F , where = v0v1:::vd+1 and 2 = 1. Thus 12(1 +), 12(1 ) are orthogonal central idempotents ofC0, so
C0= 12(1+)C012(1 )C0
where each direct summand is isomorphic to the matrix algebraM2m(F). For even d= 2m, consider the principal antiautomorphism=:C0!C0 :
=(w1w2:::wk) = ( 1)kwkwk 1:::w1 forw1;w2; ::: ;wk 2V:
Note that
=() = ( 1)m+1: (3.2.1)
Moreover, for every anisotropic vectorw2V, the reection7! ww 1 inV induces an automorphismw ofC0, which interchangeswith its opposite:
w() = : (3.2.2)
Regarding subscriptsi mod 2 denote
Pi = (1 + ( 1)i)C0 for evend:
Lemma 3.2.1. In case of an evend= 2m:
i) the involution = of the algebra C0 provides an identication of the left C0- modulePi^= HomF(Pi;F) with the rightC0-modulePi+m+1;
ii) for any anisotropic vectorw2V, the reection w interchangesPi's: w(Pi) = Pi+1.
3.3 SwanK-theory of a quadric
Recall some basic facts and notation of [11]. Denote byC1the odd part of the Cliord algebraC(q). We shall use mod 2 subscripts inCi. Recall the denition of the Swan bundleU. Put
=Xd+1
i=0zivi; 2 (X;OX(1)V):
The complex
!OX( n)Cn+d+1
!OX(1 n)Cn+d
!OX(2 n)Cn+d 1
!
(3.3.1) is exact and locally splits ([11], Prop. 8.2.(a)).
Definition 3.3.1.
Un = Coker(OX( n 2)Cn+d+3
!OX( n 1)Cn+d+2);
U =Ud 1:
Since the complex 3.3.1 is - up to a twist - periodical with period two, we have
Un+2=Un( 2): Consider the exact sequences
OX( n 2)Cn+d+3
!OX( n 1)Cn+d+2!Un!0
for two consecutive valuesn; twist the rst one by 1. For any anisotropic vectorw2V the isomorphism given by right multiplication by 1w ts into the commutative diagram:
OX( n 2)Cn+d+4
! OX( n 1)Cn+d+3 ! Un+1(1) ! 0
=
?
?
y1w =??y1w
OX( n 2)Cn+d+3
! OX( n 1)Cn+d+2 ! Un ! 0: Thus we have proved the following lemma:
Lemma 3.3.2.
Un+1=Un( 1) and Un=U0( n) for arbitrary integer n.
There is an exact sequence
0!U0 ! OXC0!U 1!0 (3.3.2) where an isomorphism(1w) was used to replaceOXC1byOXC0for evend. Lemma 3.3.3. EndX(Un)=C0acts onUn from the right.
Proof. [11], Lemma 8.7.
3.4
We are now ready to computeUn^.
Lemma 3.4.1. Un^=Un(2n+ 1), in particularU^=U(2d 1).
Proof. We have chosen a basisv0; v1; ::: ; vd+1ofV in 3.1 above. The set of naturally ordered products of several vi's in an even number forms a basis of C0. Dene a quadratic formQon C0 as follows: let the distinct basis products be orthogonal to each other and
Q(vi1vi2:::vik) =q(vi1)q(vi2):::q(vik):
The form Q is nonsingular and denes - by scalar extension - a nonsingular symmetric bilinear form on OXC0. Since (q(vi))2 = 1, a direct computation shows that Im(OX( 1)C1 ! OXC0) =U0=U0is a totally isotropic subspace ofOXC0. Therefore
U
0
=U0= (U0)? = ((OXC0)=(U0))^=U 1^ follows quickly from sect. 1.1 above and the exactness of 3.3.2. Thus
U
0
^
=U 1=U0(1) and, in general
Un^= (U0( n))^=U0^(n)=U0(n+ 1)=Un(2n+ 1): Remark 3.4.2. This argument was pointed out to the author by the referee.
Corollary 3.4.3. i) [U^] = [U(2d 1)] and [U(d 1)] + [U(d 1)]^= 2d+ 1 in K0(X);
ii) rankU =12dimC0= 2d.
In case of an even d = 2m the algebra EndX(U) = C0 splits into the direct product of subalgebras dened in 3.2 above:C0=P0P1.
Definition 3.4.4. In case of an evend:
U
n0 =UnC0P0; Un00=UnC0P1;
U
0=UC0P0; U00=UC0P1:
Note that Un =Un0 Un0 0, U =U0U0 0. U00 and U000 correspond to spinor repre- sentation and we shall copy here the standard argument on dualization (compare [4], sect. 4.3).
In case of an even d = 2m another property of and the quadratic form Q introduced in the proof of Lemma 3.4.1 may be veried by direct computation:
Lemma 3.4.5. In case of an evend= 2m
i) ifmis even, thenPi= (1)C0are orthogonal to each other, hence self-dual;
ii) ifmis odd, then Pi= (1)C0are totally isotropic, hence dual to each other;
iii) (1) = (1).
Corollary 3.4.6. In case of an evend= 2m
i) U0^=U0(2d 1) andU0 0^=U0 0(2d 1) for evenm; ii) U0^=U0 0(2d 1) andU0 0^=U0(2d 1) for oddm; iii) EndX(U0)= EndX(U00)=M2m(F);
iv) the exact sequence 3.3.2 splits into two exact parts 0!U00 ! OXP0!U00 0(1)!0 0!U000 ! OXP1!U00(1)!0
The standard way to determine indecomposable components is tensoring with the simple left module over an appropriate endomorphism algebra. We will use (from here onwards) superscript for the direct sum of identical objects.
Definition 3.4.7.
i) in case of an oddd= 2m+ 1 V=UC0F2m+1; ii) in case of an evend= 2m V0=U0M2m(F)F2m,
V
1=U0 0M2m(F)F2m.
For convenience we will use mod 2 subscripts in Vi. Since Mn(F) = (Fn)n as a leftMn(F)-module, indecomposable components inherit properties of the Swan bundle: we have
Proposition 3.4.8. a) In case of an oddd= 2m+ 1:
i) U =V2m+1; ii) V^=V(2d 1);
iii) EndX(V)=F and rankV = 2m; iv) [V(d 1)] + [V(d)] = 2m in K0(X).
b) In case of an evend= 2m: i) U0=V02m andU0 0=V12m; ii) Vi^=Vi+m(2d 1);
iii) EndX(Vi)=F and rankVi= 2m 1 iv) [Vi(d 1)] + [Vi+1(d)] = 2m inK0(X).
