3-DIMENSIONAL REGULAR LOCAL RING AND A
PURITY THEOREM
M. OJANGUREN, R. PARIMALA, R. SRIDHARAN V. SURESH 0. Introduction
Let A be a regular local ring with quotient field K. Assume that 2 is invertible in
A. Let W(A),- W(K) be the homomorphism induced by the inclusion A K, where W( ) denotes the Witt group of quadratic forms. If dim A 4, it is known that this map is injective [6, 7]. A natural question is to characterize the image of W(A) in
W(K ). Let Spec"(A) be the set of prime ideals of A of height 1. For P ? Spec"(A), let πP be a parameter of the discrete valuation ring AP and k(P)l AP\PAP. For this choice of a parameter πP, one has the second residue homomorphism cP: W(K ),- W(k(P)) [9, p. 209]. Though the homomorphism cP depends on the choice of the parameterπP, its kernel and cokernel do not. We have a homomorphism
c l (cP) : W(K ),-
G
P?Spec"(A)
W(k(P)).
A part of the so-called Gersten conjecture is the following question on ‘ purity ’. Is the sequence
W(A),- W(K) ,-c
G
P?Spec"(A)
W(k(P))
exact ? This question has an affirmative answer for dim(A) 2 [1; 3, p. 277]. There have been speculations by Pardon and Barge-Sansuc-Vogel on the question of purity. However, in the literature, there is no proof for purity even for dim(A)l 3. One of the consequences of the main result of this paper is an affirmative answer to the purity question for dim(A)l 3.
We briefly outline our main result. For any scheme X let Wε(X ) denote the Witt group of ε-symmetric spaces on X, ε lp1 (W+"(X) l W(X) being the usual Witt
group of symmetric spaces over X ). Let A be a regular local ring of dimension 3 with maximal ideal m and Yl Spec(A)Bomq. We associate (§3) to an ε-symmetric space over Y a (kε)-symmetric space over a finite-length A-module. This assignment leads to a homomorphism Wε(Y ),- W−ε
fl(A), where W
ε
fl(A) is the Witt group of
ε-symmetric spaces of finite-length A-modules (cf. §1). Then we prove (§4) that the sequence
0,- Wε(A),- Wε(Y ),- W−ε
fl(A),- 0
is exact, where the map Wε(A),- Wε(Y ) is induced by the restriction. Since
Wεfl(A)# Wε(A\m), it follows that W−"
fl(A)l 0. Thus the map W(A) ,- W(Y) is an
isomorphism. This leads to the purity theorem for the Witt groups. On the other hand, since every skew-symmetric space over A is hyperbolic, W−"(A) l 0 and we get
an isomorphism W−"(Y) # W(A\m). We observe the curious fact that if A is
complete, Wp"(Y) is isomorphic to W(A\m).
Received 15 August 1996.
1991 Mathematics Subject Classification 19G12.
A crucial result used in our proof of the main theorem is a theorem of Horrocks [2] on vector bundles on the punctured spectrum Yl Spec(A)Bomq, where A is a regular local ring of dimension 3 and m is its maximal ideal. We use his theorem on the equivalence of the category of ‘Φ-equivalence’ classes of vector bundles on Y with the category of finite-length A-modules.
We would like to remark parenthetically that purity for dimension 3 was used in [8] while establishing the equivalence of the finite generation of Witt groups of affine real 3-folds and the finite generation of Chow groups of codimension 2 cycles modulo 2.
1. ε-symmetric spaces reminisced
Let A be a regular local ring of dimension 3 in which 2 is invertible. We recall the definition of ε-symmetric spaces on finite-length A-modules and their Witt groups. For A-modules M, N and i 0, let Exti(M, N ) denote the group of congruence classes
of i-fold extensions of N by M [4, p. 84]. For any homomorphism f : M,- Mh of A-modules, let Exti(N, f ) : Exti(N, M ),- Exti(N, Mh) be the induced homomorphisms
defined as follows. Let
ζ l 0 ,- M ,-α Zi,-ci Zi−" ,- ( ,-Z#,-c# Z",-β N,- 0
be an i-fold extension of N by M. Let Zl (Zi&Mh)\(o(α(x), f(x)) Q x ? Mq) be the push-out of Figure 1 [4]. α M M f Zi F 1. Then Exti(N, f ) (ζ) l 0 ,- Mh ,-αh Z,-ch Z i−" ,-ci−" (,-Z # ,-c# Z",-β N,- 0 whereαh and ch are the natural homomorphisms induced by the push-out. Similarly, we define Exti( f, N ) as the pull-back under f of an i-fold extension of N by Mh. Let Mbe a finite-length A-module and MGl Ext$(M, A). If M, Mh are two finite-length
A-modules and f : M,- Mh is an A-linear map, then we denote Ext$( f, A) by fG. Let
l 0 ,- P$,-c$ P#,-c# P",-c" P!,-θ M,- 0
be a projective resolution of M. Since Exti(M, A)l 0 for i l 0, 1, 2 [5, Theorem 18.1],
by dualizing the above exact sequence we see that
* l 0 ,- P!,-c" P",-c# P#,-c$ P$,-θh MG,- 0
is a projective resolution of MG, where Pi l HomA(Pi, A),ci is induced byciand for any f? P$,
Throughout this paper, for any surjectionθ: P!,-M as above, θh denotes the map defined as above. We define a canonical homomorphism: M ,- MGGas follows. Let x? M. Choose y ? P! such that θ(y)lx. We define
(x) l Ext$(key, MG) (*) ? MGG,
where, for f? P!, ey( f )l f(y). Then it is easy to see that (x) is independent of the choice of y and Figure 2 is commutative, where : Pi,- Pi are the canonical isomorphisms. P**3 P3 0 0 P**2 ∂3 ∂2 ∂2** ∂3** P**0 ∂1 θ ∂1** P2 P**1 P1 P0 θ M∨∨ M – 0 0 F 2.
Thus: M ,- MGGis an isomorphism and it is obvious that it is independent of the choice of the projective resolution. We use this isomorphism to identify M with
MGG. The choice of the negative sign at eyin the definition of is explained in the following. Let ml (x",x#,x$) be the maximal ideal of A and
ζ l 0 ,- A ,-δ$ A$ ,-δ# A$ ,-δ" A,-η A\m ,- 0
be the Koszul resolution of A\m with respect to (x",x#,x$). With respect to the standard basisoe",e#,e$q of A$, we have
δ"l(x" x# x$), δ#l
I
J
kx#x" 0 0 kx$ x" kx$ 0 x#K
L
, δ$lI
J
kx# x$ x"K
L
and η: A ,- A\m is the natural homomorphism. Let M be a finite-dimensional vector space over A\m. Then M is a finite-length A-module. Let Mg l Hom(M, A\m). The assignment f Ext$( f, A) (ζ) ? MGinduces a homomorphism
ΦM: Mg,-MG.
The following lemmas are well known, but for the sake of completeness we give their proofs here.
L 1.1. The homomorphism ΦM is an isomorphism and Figure 3 is
com-mutatie, where ι: M ,- Mgg is the canonical isomorphism.
M M M∨∨ $ " (M )∨ ι ΦM∨ ΦM" F 3.
Proof. Since M#
G
n"A\m, MG#
G
n"(A\m)Gand Mg#G
n"A a\m, it is enough to prove the lemma in the case when Ml A\m. In this case it is easy to see that ΦM 0. Since MG# A\m [5, Theorem 18.1] and Mg # A\m, ΦM is an
isomor-phism. We now prove the commutativity of Figure 3. For all x? M and f ? Mg we have
ι(x) ( f ) l f(x) and
ΦMg(ι(x)) l 0 ,- A ,-δ$ A$ ,-δ# A$ ,-δ" A
,,-ι(x)−"η
A\m ,- 0.
Let y? A be such that η(y) l x. Then we have
(x) l 0 ,- A ,,-kδ"e−y"A$* ,-δ# A$* ,-δ$ A*,-ηh (A\m)G,- 0. SinceΦMis an isomorphism, we have
ΦG
M((x)) l 0 ,- A ,,-kδ"e −"
y A$* ,-δ# A$* ,-δ$ A*,,-Φ−M"ηh (A\m)G,- 0. Letoe",e#,e$q be the dual basis of A$*. For il1,2, let θi: A$ ,- A$* be given by the following matrices, with respect to the basesoe",e#,e$) and oe",e#,e$q.
θ"l
I
J
0 0 ky−" y−" 0 0 0 ky−" 0K
L
, θ#lI
J
0 0 y−" ky−" 0 0 0 y−" 0K
L
.Letθ$:A,-A* be the homomorphism defined by θ$(1)lly, where ly(a)l ay for all
a? A. It is easy to see that Figure 4 is commutative. Thus ΦMg ι l ΦGM.
id A 0 0 A3* δ3 A* θ1 A3 A3* A3 A (A/m)∨ 0 0 A –δ1 * e y–1 δ2 δ2* θ2 δ1 δ3* θ3 id (A/m)∨ ι(x)–1η ΦM –1η F 4.
L 1.2. Let ψ: M ,- Mg be a homomorphism and ψg: Mgg ,- Mg be the
induced homomorphism. Then Figure 5 is commutatie.
M ψ ψ∨ $ " M" (M )∨ M∨ " ΦM " ΦM F 5.
Proof. Let f? Mgg. Then Φ
Mg( f ) is the pull-back of the Koszul resolutionζ under f: Mg,- A\m and ψG(ΦMg( f )) is the pull-back of the extensionΦMg( f ) underψ. Thus
ψG(Φ
Mg( f )) is the pull-back of the Koszul resolution under the homomorphism fψ: M ,- A\m. Since ψg( f ) l fψ, ΦM(ψg) ( f ) is the pull-back of the Koszul
resolution under fψ. Thus ΦMψGl ψGΦMg.
Lemmas 1.1 and 1.2 enable us to embed the category ofε-symmetric spaces on finite-dimensional A\m-vector spaces into the category of ε-symmetric spaces on finite-length A-modules (cf. Corollary 1.3)).
Let ε lp1. An ε-symmetric space of finite length is a pair (M, ψ) where M is a finite-length A-module andψ: M ,- MGl Ext$(M, A) is an isomorphism with
ψG
l εψ. Let ψ" and ψ# be two ε-symmetric spaces on finite-length A-modules M"
and M# respectively. We say that ψ" is isometric to ψ# if there exists a homomorphism
θ: M",-M# such that ψ"lθG
ψ#θ. An ε-symmetric space ψ on M is called metabolicif there exists a submodule N of M such that
0,- N ,-i M,-i
Gψ
NG,- 0
is exact, where i : N,- M is the inclusion. The Witt group of ε-symmetric spaces of finite-length A-modules is defined as the quotient of the Grothendieck group of isometry classes ofε-symmetric spaces with the orthogonal sum as addition, modulo the subgroup generated by metabolic spaces. It is denoted by Wεfl(A).
C 1.3. Let M be a finite-dimensional ector space oer A\m. Let
ψ: M ,- M be an ε-symmetric space, that is, ψgι l ψ and ψ is an isomorphism. Then
ΦMψ: M ,- MGis anε-symmetric space.
Proof. By Lemma 1.1, we have (ΦMψ)G l ψGΦGM l ψGΦMg ι. Using Lemma 1.2, we get that (ΦMψ)G l ΦMψgι l εΦMψ. Thus ΦMψ is an ε-symmetric
space.
We need the following lemma.
L 1.4. Let M be a finite-length A-module and ψ: M ,- MG be an
ε-symmetric space. If (M, ψ) is stably metabolic, then it is metabolic.
Proof. If M is an A\m-module, then the result follows from the corresponding result forε-symmetric spaces over the field A\m. We reduce the general case to the above case by induction on the length of M. Assume that the length of M is at least 2. Let V be a maximal submodule of M which is an A\m-module. Suppose that ψ restricted to V is singular. Then there exists a non-zero submodule L of V such that
L9 LUl ker(M ,-i Gψ
LG)
and ψ induces an ε-symmetric form ψ` on LU\L which is Witt equivalent to (M,ψ). Suppose that (M, ψ) is stably metabolic. Then (LU\L, ψ`) is stably metabolic. By induction there exists a submodule N" of LU\L such that
0,- N",-i LU\L ,-i
Gψ`
NG
is exact. Let N be the submodule of M containing L such that N\L l N". Then it is easy to see that the sequence
0,- N ,-i M,-i
Gψ
NG,- 0
is exact and (M,ψ) is metabolic. We may therefore assume that ψ restricted to V is non-singular. Then (M,ψ) # (V, ψ QV)U(M",ψ"). If M"0, then M" contains a non-zero submodule which is an A\m-module, contradicting the maximality of V. Thus
M"l0 and MlV is an A\m-module. This completes the proof of the lemma.
Let X be a scheme such that 2 is invertible inΓ(X). Let be a vector bundle over
X of finite rank. An ε-symmetric space on is an isomorphism q: ,- * l Hom(, X) such that q* l εq, where : ,- ** is the canonical identification. Let Wε(X ) be the Witt group of ε-symmetric spaces on vector bundles over X [3, p. 144]. If Xl Spec(A), then we denote Wε(X ) by Wε(A).
Throughout this paper, by an A-module we mean a finitely generated A-module. We call an ε-symmetric space simply a quadratic space if ε lj1 and a symplectic
space ifε lk1. We also denote W+"(X) by W(X). For a vector bundle over X,
we denote the hyperbolic space on by () [3, p. 130]. 2. Reflexie modules
Let A be a regular local ring of dimension 3 with 2 invertible. An A-module E is said to be reflexie if it is finitely generated and the canonical homomorphism
E,- E** is an isomorphism. For a reflexive A-module E we use the canonical
isomorphism to identify E ** with E. It is well known that a reflexive module over a regular ring of dimension 3 has projective dimension at most 1. Let E be a reflexive
A-module and Ml Ext"(E*, A), where E* l HomA(E, A). Since reflexive modules over regular rings of dimension at most 2 are projective, M is a finite-length A-module. We define a homomorphism βE: Ext"(E, A) ,- MGl Ext$(M, A) as follows. Let
0,- P",-c" P!,-c! E*,- 0 (2.1) be a projective resolution of E *. Then by dualizing, we get an exact sequence
0,- E ,-c! P!,-c" P",-δ Ext"(E*, A) l M ,- 0
whereδ is defined by push-outs. We have the following lemma.
L 2.1. The Yoneda composition [4, p. 82] βE: Ext"(E, A) ,- MGgien by βE(0,- A ,-η Z,-ηh E,- 0) l (0 ,- A ,-η Z,-c!ηhP!,-c" P",-δ M,- 0) is an isomorphism and is independent of the choice of the projectie resolution (2.1) of E*.
Proof. Consider the long exact sequence of cohomology associated to the short exact sequences
Since Exti(M, A)l 0 for i 2 and P" is a projective module, Ext"(ker(δ),A)l0 and
the connecting homomorphisms
Ext"(E, A) ,- Ext#(ker (δ), A) and Ext#(ker (δ), A) ,- Ext$(M, A) induced by the above short exact sequences are isomorphisms. SinceβE, up to sign, is the composition of these two connecting homomorphisms [4, Theorem 9.1, p. 97],
β is an isomorphism.
Suppose that
0,- F",-c" F!,-c! E*,- 0
is another projective resolution of E *. Then by lifting the identity map on E *, we get homomorphisms Pi,- Fi, il 0, 1, such that Figure 6 is commutative.
id P1 0 0 ∂1 E* 0 0 F1 ∂0 ∂1 F ∂0 0 E* P0 F 6.
By dualizing this diagram we get a commutative diagram (Figure 7) where δh is defined by push-outs. id E 0 0 F1* M δ E ∂ * 0 P* 0 ∂ * 1 F* 0 ∂* 0 P 1 * δ 0 0 id M ∂* 1 F 7.
This implies that (0,- E ,-c ! F ! ,-c " F " ,-δh M,- 0) l (0 ,- E ,-c! P!,-c" P",-δ M,- 0)
in Ext#(M, E). Thus the homomorphism βE is independent of the choice of the
projective resolution of E *.
L 2.2. (i) For any reflexie A-module E we hae
βE*lkβGE.
(ii) Let E and Eh be reflexie A-modules. Then, for any isomorphism f: E ,- Eh,
we hae
Ext"( f *)GβEhl βEExt"( f ).
Proof. Let 0,- P",-c" P!,-c! E*,- 0 and 0 ,- F",-c" F!,-c! E,- 0 be
projective resolutions of E * and E respectively. By dualizing these exact sequences, we get exact sequences
0,- E ,-c! P!,-c" P",-δ Ext"(E*, A) ,- 0 and 0,- E* ,-c ! F ! ,-c " F " ,-δ" Ext"(E, A) ,- 0.
Letζ l (0 ,- A ,-α Z,-β E*,- 0) ? Ext"(E*, A). Since P! and P" are projective, there exist homomorphisms f : P",-A and g:P!,-Z such that Figure 8 is commutative. f Z 0 0 E* A ∂1 P0 P1 0 0 id g α β ∂0 E* F 8.
By the definition ofδ we have δ( f ) l ζ. Since
0,- F",-c" F!,,-c!c! P!,-c" P",-δ Ext"(E*, A) ,- 0
is a projective resolution of Ext"(E*, A), by dualizing it we get an exact sequence
0,- P",-c" P!,,-c ! c! F ! ,-c " F " ,-δ" Ext"(E*, A)G,- 0. Thus(ζ) lkξ, where ξ l (0 ,- A ,-α Z,,-c ! β F ! ,-c " F " ,-δ" Ext"(E*, A)G,- 0). From the definitions ofδ, δh and βE, it follows that Figure 9 is commutative.
0 0 E* ∂0 * F0* 0 0 id βE δ1 E* id id ∂0* F 0 * ∂1* ∂1* F1* F1* δ1 Ext1 (E, A) Ext1 (E*, A)∨ F 9.
It follows from the definition ofβGE that
βG
E(ξ) l (0 A ,-α
Z,,-c ! β F! ,-c " F" ,-δ" Ext"(E, A) ,- 0). On the other hand, we have
βE*(ζ) l (0 A ,-α Z,,-c ! β F ! ,-c " F " ,-δ" Ext"(E, A) ,- 0) l βGE(ξ).
ThuskβGE l βE*.
Let f : E,- Eh be an isomorphism. Then 0 ,- P",-c" P!,,-f*
−"c
! Eh* ,- 0 is a
projective resolution of Eh*. By dualizing it we get an exact sequence
0,- Eh ,-c!f
−"
P!,-c" P",-δ# Ext"(Eh*, A) ,- 0
withδ#lExt"(f*) δ. Let ζl(0,-A,-α Z,-β Eh ,- 0) ? Ext"(Eh, A). Then βEh(ζ) l (0 A ,-α Z,,-c!f −"β P!,-c" P",-δ# Ext"(Eh*, A) ,- 0) and Ext"( f *)GβEh(ζ) l (0 ,- A ,-α Z,,-c!f −"β P!,-c" P",-δ Ext"(E*, A) ,- 0) sinceδ l Ext"( f *)−"δ#. On the other hand, we have
Ext"( f ) (ζ ) l (0 ,- A ,-α Z,-f −"β E,- 0) and βEExt"( f ) (ζ) l (0 ,- A ,-α Z,,-c!f −"β P!,-c" P",-δ Ext"(E*, A) ,- 0) l Ext"( f *)Gβ Eh(ζ).
This proves the lemma.
Let A be any local ring in which 2 is invertible and let m be its maximal ideal. Let
Ebe a reflexive A-module. By anε-symmetric space on E we mean an isomorphism
q: E,- E* such that q* l εq, where : E ,- E** is the canonical isomorphism. Let V be an A-module. By a unimodular element of V we mean an element x? V such that f(x)l 1 for some A-linear map f: V ,- A. For example, an element (a",…,an)? Anis unimodular if and only if ai@ m for some i. Thus, if an A-module V
has no unimodular elements andη: V ,- Anis an A-linear map, thenη(V) 9 mAn.
L 2.3. Let E be a reflexie A-module and q be an ε-symmetric space on E.
Suppose that El E!&An
with E! haing no unimodular elements. Then there exist
ε-symmetric spaces q" and q# oer E! and An respectiely such that
(E, q)# (E!,q")U(An
, q#).
Proof. Let El E!&An
be such that E! has no unimodular elements. Then
ql
0
q"εη* q"η
1
for some q":E!,-E!, q":An,- An* and η: An,- E!. Since E! has no
unimodular elements,η*(E!)9mAn* and henceη(An)9 mE!. This implies that q
0
q"0 0
Since q is an isomorphism, q" and q" are isomorphisms. We have
0
1 kεη*q−" " 0 110
εη*q" η q"10
1 0 kq−" " η 11
l0
q"0 0 kη*q−" " ηjq"1
. Let q#lkη*q−"" ηjq":An,- An*. Since q"lεq", (E,q)#(E!,q")U(An, q#). 3. Spaces oer the punctured spectrum and on finite-length modules
We begin by recalling from a paper of Horrocks [2] an equivalence between the categoriesΦ of Φ-equivalence classes of vector bundles on the punctured spectrum of a regular local ring A of dimension 3 and the category of finite-length
A-modules. Let m be the maximal ideal of A and Yl Spec(A)Bomq. Let be a vector bundle over Y and El Γ() be the module of sections of . Then E is a reflexive
A-module [2, Theorem 4.1] and Ml Ext"(E*, A), which is isomorphic to H"(Y, ) [2,§5], is a finite-length A-module [2, Corollary 7.2.5]. The functor T: Φ ,- given by T() l Ext"(E*, A) is an equivalence of categories [2, Corollary 7.2.5]. Let M be a finite-length A-module. The construction below gives a vector bundle on Y such that T() l M. Let, in fact,
0,- P$,-c$ P#,-c# P",-c" P!,-η M,- 0
be a projective resolution of M. Let El ker (c"). Then E is an A-module of projective dimension at most 1 and Ext"(E*, A) l M. Since M is a finite-length module, for any prime ideal p of A, p m, Mpl 0 and hence Ep is free. Thus El Γ() for some vector bundle on Y [2, Theorem 4.1].
Let A be a regular local ring of dimension 3 in which 2 is invertible. Let be a vector bundle over Y and q be anε-symmetric space on . We associate to (, q) a (kε)-symmetric space ρ(q) of finite length. The ε-symmetric space q on gives rise to anε-symmetric space (E, q), where E l Γ(). Then M l Ext"(E*, A) is a finite-length A-module. The isomorphism q : E,- E* induces an isomorphism Ext"(q): M l Ext"(E*, A) ,- Ext"(E, A). Let ρ(q) l βEExt"(q). We have the follow-ing lemma.
L 3.1. ρ(q): M ,- MGis a(kε)-symmetric space.
Proof. In Figure 10, clearly, all the squares except perhaps the top left one commute.
βE*
M = Ext1(E*, A) βE Ext1(E*, A)∨ = M∨
id Ext1(E*, A)∨ Ext1(E, A)∨ Ext1(E*, A)∨∨ Ext1(E, A) Ext1(E*, A)∨ Ext1(E, A)∨ εExt1(q)∨–1 εExt1(q)∨ id id Ext1(E*, A)∨ βE εExt1(q)∨ Ext1(q) βE∨ βE∨–1 F 10.
Since q*l εq, by Lemma 2.2 this square also commutes. By Lemma 2.2, the composition of maps on the first column is equal tok. Thus
ρ(q)G l Ext"(q)GβG
E lkεβEExt"(q) lkερ(q).
L 3.2. Let M be a finite-length A-module and ψ be an ε-symmetric form on
M. Suppose that there exists an exact sequence
N,,-f M,,-f
Gψ
NG of finite-length A-modules. Then (M,ψ) is metabolic.
Proof. Since the map f factors as N,- N\ker( f ) ,-f` M, we have an exact sequence
0,,- N\ker( f ) ,,-f` M,,-f`
Gψ
(N\ker( f ))G.
Since, the dimension of A being 3, Ext%(L, A) l 0, the map f`Gψ is surjective and hence
(M,ψ) is metabolic.
L 3.3. If (, q) is metabolic, then (M, ρ(q)) is metabolic.
Proof. Suppose that (, q) is metabolic. Let be a subbundle of such that the sequence
0,- ,-i ,-i*q * ,- 0
is exact, where ,-i is the inclusion. By taking global sections and then applying the Ext functor to the following exact sequence of bundles
0,- ,-qi * ,-i* * ,- 0 we get an exact sequence
Ext"(F*, A) ,- Ext"(E*, A) ,- Ext"(F, A)
of finite-length modules, where Fl Γ( ). Let N l Ext"(F*, A). Then the canonical identification of Ext"(F, A) with NGgives an exact sequence
N,-f M,,-f
Gρ(q)
NG.
Now the lemma follows from Lemma 3.2.
L 3.4. The assignment (, q) (M, ρ(q)) induces a homomorphism
ρ: Wε(Y ),- W−ε fl(A).
Proof. Since ρ is clearly additive, it is enough to show that ρ takes stably metabolic spaces to metabolic spaces. Let (, q) be an ε-symmetric space over Y which is stably metabolic. Then there exists a metabolic space (",q") such that (,q)U(",q") is metabolic. By Lemma 3.3, ρ(q") and ρ(qUq")lρ(q)Uρ(q") are metabolic. Thus
We note that if is a trivial bundle then M l 0. Thus, if (, q) comes from an ε-symmetric space on A, then ρ(q) l 0.
The proof of the following lemma is by straightforward verification ; hence we omit it.
L 3.5. Let R be a ring. Let 0 ,- N ,-i M,-j Nh ,- 0 be an exact sequence of R-modules. Assume that the projectie dimensions of N and Nh are finite.
Let 0,- Pn,-cn Pn− " ,-cn−" (P " ,-c" P!,-α N,- 0 and 0,- Qn,-cn Qn− " ,-cn−" (,-Q " ,-c" Q!,-β Nh ,- 0
be projectie resolutions of N and Nh respectiely. Let, for l 1, φl: Ql,- Pl− "and
θ: Q!,-M be R-linear homomorphisms. Let δll
0
cl0
(k1)lφ l
cl
1
.Then Figure11 is commutatie if and only if Figure 12 is commutatie.
β 0 Pn Pn ⊕Qn Qn 0 0 Pn–1 Pn–1 ⊕Qn–1 Qn–1 P1 P1 ⊕Q1 Q1 . . . . . . . . . P0 P0 ⊕Q0 Q0 N M N 0 0 0 ∂n ∂n δn ∂1 ∂1 δ1 α (iα, θ) i j F 11. β Pn 0 Pn–1 Qn P1 Q2 . . . . . . P 0 j id M N 0 0 ∂n ∂2 iα θ 0 φn Pn–2 Qn–1 φn–1 ∂n–2 φ2 Q1 φ1 Q0 N ∂n–1 ∂1 ∂1 ∂n–1 ∂n F 12.
P 3.6. Let (, q) be an ε-symmetric space oer Y. Suppose that E l Γ() has no unimodular elements and that ρ(q) is metabolic. Then there exist ε-symmetric spaces q" and q# on and n
Yrespectiely such that q"Uq# is metabolic and ρ(q) # ρ(q").
Proof. Let Ml Ext"(E*, A) and ρ(q) be the ε-symmetric space on M. Since ρ(q) is metabolic, there exists an exact sequence
0,- N ,-i M,,-i
Gρ(q)
NG,- 0.
Let
0,- Q$,-c$ Q#,-c# Q",-c" Q!,-η N,- 0
be a projective resolution of N. By dualizing this resolution, we get a projective resolution
0,- Q!,-c" Q",-c# Q#,-c$ Q$,-ηh NG,- 0
of NG. By lifting the identity map of NG, we obtain a commutative diagram (Figure 13). Q0* Q3 id M N∨ 0 0 ∂3 ∂3* iη θ 0 φ3 φ2 ∂1* η Q2 Q1* Q1 ∂2 ∂2* ∂1 φ1 Q2* Q0 Q3* N∨ i∨ρ(q) F 13. Let δ"l
0
c"0 kφ"c$1
, δ#l0
c#0 φ#c#1
, δ$l0
c$0 kφ$c"1
. By Lemma 3.5, Figure 14 is commutative.Q0* N N∨ ∂3 η Q3 ∂2* i∨ρ(q) 0 0 0 0 0 0 0 0 0 0 0 Q3 ⊕ Q0* Q1* Q2 Q2 ⊕ Q1* Q2* Q1 Q3* Q0 Q1 ⊕ Q2* Q 0 ⊕ Q3* 0 0 0 0 0 η i δ3 ∂1* ∂2 δ2 M (iη,θ) ∂3* ∂1 δ1 F 14.
Since the first row, the last rows and all the columns in Figure 14 are exact and the diagram is commutative, from the long exact homology sequence [4, Theorem 4.1, p. 45] we get that the middle row is also exact. By dualizing Figure 14, we get the commutative diagram in Figure 15, with exact rows and columns.
Q0* N∨∨ N∨ ∂3 η Q3 ∂2* i∨ 0 0 0 0 0 0 0 0 0 0 0 Q3 ⊕ Q0* Q1* Q2 Q2 ⊕ Q1* Q2* Q1 Q3* Q0 Q1 ⊕ Q2* Q 0 ⊕ Q3* 0 0 0 0 0 η δ1* ∂1* ∂2 δ2* M∨ (µ,ν) ∂3* ∂1 δ3* ρ(q)∨i∨∨ F 15. Q0* N∨ ∂3 η′ Q2 i∨ 0 Q1 Q0 Q1* Q 2 * Q 3 * 0 0 φ1* ∂2 M∨ ∂1* ρ(q)∨i∨∨η″ Q3 φ2* φ3* ∂1 ∂2* ∂3* ν N∨ id F 16.
By Lemma 3.5, Figure 16 is commutative. From the definition ofηd and : N ,- NGG (cf. Figure 2) it follows that η lkηd. Since ρ(q)G lkερ(q), we have the commutative diagram in Figure 17, whereνh l ερ(q)−"ν.
Q0* N∨ ∂3 η Q2 i ∨ ρ(q) 0 Q1 Q0 Q1* Q 2 * Q 3 * 0 0 φ1* ∂2 M ∂1* iη Q3 φ2* φ3* ∂1 ∂2* ∂3* ν N∨ εid F 17.
From Figure 13 and Figure 17 we get maps s":Q#,-Q" and s#:Q",-Q# such thatφ#kεφ#lc#s#ks"c#. Let φlφ#kεc#s". Then we have
c"φlc"φ# l c"(φ#kεφ#)jεc"φ# l c"(c#s#ks"c#)jεc"φ# l εc"(φ#kεs"c#) l εc"φ*. Let δ l
I
J
c#0 φjεφ* 2 εc#K
L
.It is easy to see that Figure 18 commutes.
Q0* N N∨ ∂3 η Q3 ε∂2* i∨ρ(q) 0 0 0 0 0 0 0 0 0 0 0 Q3 ⊕ Q0* Q1* Q2 Q2 ⊕ Q1* Q2* Q1 Q3* Q0 Q1 ⊕ Q2* Q 0 ⊕ Q3* 0 0 0 0 0 η i δ1* ∂1* ∂2 δ M (η,θ) ∂3* ∂1 δ1 F 18.
Since the first row, the last row and all the columns are exact, the middle row is also exact. Let Eh l ker (δ"). Since δ*lεδ, from the middle row of Figure 18 it is easy to see thatδ induces an ε-symmetric isomorphism qh: Eh ,- Eh*. Let (h, qh) be the ε-symmetric space over Y with Γ(h) l Eh and Γ(qh) l qh. Since Ext"(Eh, A) # M l Ext"(E*, A) and E has no unimodular elements, by [2, Corollary 7.2.5, Lemma 7.1] we have Eh l E & An. Then by Lemma 2.3, (Eh, qh) #
(E, q")U(An
, q#) for some ε-symmetric spaces q" and q# on E and Anrespectively. Let
q" be the ε-symmetric space on such that Γ(q") lq". Let Flker(c") and be the vector bundle over Y withΓ( ) l F. Then using Figure 18, it is easy to see that is a Lagrangian for (h, qh) # (, q")U(n
Y, q#), where Γ(q#)lq#. Since the map Eh ,- F* induced by Figure 18 induces iGρ(q): M ,- NG, we have iGρ(q) l iGρ(q"). Thus, by Lemma 3.7 below, we haveρ(q) # ρ(q").
L 3.7. Let ψ" and ψ# be two ε-symmetric spaces on M. Suppose there exists
a submodule N such that
0,,- N ,,-i M,,-i
G
ψ" NG,,- 0 is exact and iGψ"liGψ#. Then ψ"#ψ#.
Proof. Since iGψ"liGψ#, there exists a map θ:M,-N such that ψ−"
" ψ#k1l
iθ, that is, ψ"iθlψ#kψ"lθGiGψ". We have
(1jθGiG) 2 ψ" (1jiθ) 2 l (ψ"jθGiGψ") 2 (1jiθ) 2 l ψ"jψ"i2θj θGiG 2 ψ"j θGiG 2 ψ" iθ 2 l ψ"jψ#kψ"2 jψ#kψ"2 l ψ#.
4. The Witt groups of the punctured spectrum and purity
Let A be a regular local ring of dimension 3 with 2 invertible. Let Yl Spec(A)Bomq, where m is the maximal ideal of A.
P 4.1. Let be a ector bundle on Y and q: ,- * be an ε-symmetric
isomorphism. Suppose thatΓ() has no unimodular elements. If ρ(q) is isomorphic to
a hyperbolic space, then q is in the image of Wε(A).
Proof. Let N be a finite-length A-module such that (M,ρ(q)) is isomorphic to the hyperbolic space (N). Let be the vector bundle on Y with Γ( ) having no unimodular elements and such that H"(Y, ) # N (cf. §3). Since
H"(Y, ) # N&NG# H"(Y, & *)
with Γ() and Γ(& *) admitting no unimodular elements, by [2, Lemma 7.1, Corollary 7.2.5] we can and do identify with & *. Let ψg be an isometry of
ρ(q) with
ρ
00
0ε 1011
.Then by [2, Corollary 7.2.5] there exists an automorphism ψ of such that
H"(ψ) l ψg. By the definition of ρ we have
ρ(ψ*qψ) l βEExt"(Γ(ψ*)qΓ(ψ))
l βEExt"(Γ(ψ*)) Ext"(q) Ext"(Γ(ψ))
where El Γ() and q l Γ(q). By Lemma 2.2(ii), we have βEExt"(Γ(ψ*)) l Ext"(Γ(ψ))Gβ E, so that ρ(ψ*qψ) l Ext"(Γ(ψ))Gβ EExt"(q) Ext"(Γ(ψ)) l Ext"(Γ(ψ))Gρ(q) Ext"(Γ(ψ)) l H"(ψ)Gρ(q)H"(ψ) l ψgρ(q)ψ4. We replace q byψ*qψ and assume that
ρ(q) l
0
0ε 10
1
. Letql
0
α εδ* δβ1
withα: ,- *, β: * ,- , δ: * ,- * maps such that α* l εα, β* l εβ. Since
ρ(q) l
0
0ε 10
1
,H"(α) l 0 and H"(β) l 0. Therefore, by [2, Corollary 7.2.5], there exist f":,-n
Y, f#:*,-nY, g":nY,- * and g#:nY,-
such thatα l g"f" and βlg# f#. Let us consider the automorphism
ψ l
I
J
0 1 0 0 1 0 0 0 kg#\2 kg"\2 1 0 f# f" 0 1K
L
of& * & n Y& nY. We have qh l ψI
J
εδ*α 0 0 β δ 0 0 0 0 0 ε 0 0 1 0K
L
ψ* lI
J
εX* 0 f" kεg"\2 0 X f# kεg#\2 εf # εf " 0 ε kg#\2 kg"\2 1 0K
L
where Xl δkεf "g#\2kg"f#\2. Since Γ() and Γ(*) admit no unimodular elements, f"f#0modm. Hence X is an isomorphism. Since qh restricted to
& * is
0
0 εX*X
0
1
with X an isomorphism, qh splits as0
0 εX*X
0
1
Uqd for some qd supported on a bundle d. Since & nY& n
*
Y# & d, by [2, Corollary
7.2.5],d is a trivial bundle and hence qd is in the image of Wε(A). Since qUH(n Y)# qh #
0
0 εX* X 01
Uqdit follows that q is in the image of Wε(A).
L 4.2. Let M be a finite-length A-module and ψ: M ,- MG be an
ε-symmetric isomorphism. Then there exists a ector bundle oer Y with a (kε)-symmetric isomorphism q: ,- * such that ρ(q) l ψ.
Proof. Let be a vector bundle on Y such that H"(Y, ) l M (cf. §3) and E l Γ() has no unimodular elements. Then (cf. §3) the projective dimension of E is less than or equal to 1 and Ext"(E*, A) # M. Let
0,- P",-P!,-E,-0 and
0,- Q",-Q!,-E*,-0
be projective resolutions of E and E * respectively. By dualizing the projective resolution of E* we get an exact sequence
0,- E ,- Q!,-Q",-M,-0.
By taking the Yoneda composition of this exact sequence with the projective resolution of E we get a projective resolution
0,- P",-P!,-Q!,-Q",-M,-0 of M. By dualizing this we get a projective resolution
0,- Q",-Q!,-P!,-P",-MG,- 0
of MG. By lifting theε-symmetric isomorphism ψ: M ,- MG, we get a commutative diagram of exact sequences (Figure 19).
P1 M Q1 0 Q0 P0* P0 Q0* Q 1 * 0 0 3 M∨ 0 2 1 0 P1* F 19.
Since E has no unimodular elements it is easy to see, as in the proof of Proposition 3.6, that Figure 19 induces a (kε)-symmetric space q on such that ρ(q) l ψ. T 4.3. Let A be a regular local ring of dimension 3 and let m be its maximal
ideal. Assume that 2 is inertible in A. Let Y l Spec(A)Bomq. Then the complex 0,- Wε(A),-ι Wε(Y ),-ρ W-ε
fl(A),- 0 is exact, where ι is induced by the restriction.
Proof. If ε l 1, then the injectivity of ι follows from the injectivity of the canonical homomorphism W(A),- W(K) [6, Theorem 23], where K is the quotient field of A. If ε lk1, ι is injective because W−"(A) l 0.
We now prove the exactness in the middle. As we remarked in §3, ρι l 0. Let (, q) be an ε-symmetric space over Y such that ρ(q) is zero in W−ε
fl(A). Then, by Lemma
1.4,ρ(q) is metabolic. We show that (, q) is in the image of ι. In view of Lemma 2.3, we assume thatΓ() has no unimodular elements. Then, by Proposition 3.6, there exist ε-symmetric spaces q" and q# supported respectively on and n
Y for some
integer n, such that q"Uq# is metabolic and ρ(q)#ρ(q"). Thus ρ(qUkq") is isomorphic to a hyperbolic space. Since Γ( & ) has no unimodular elements, it
follows from Proposition 4.1 that qUkq" is in the image of ι. Since q# is in the image ofι and q"Uq# is metabolic, q" and hence q is in the image of ι.
The surjectivity ofρ follows from Lemma 4.2.
Let A be any regular ring. Let Spec"(A) denote the set of prime ideals of A of height 1. Then for any P? Spec"(A), the local ring APis a discrete valuation ring. Let cP: W(K ),- W(AP\PAP) denote the second residue homomorphism with respect to some choice of a parameter of PAP, where K is the quotient field of A.
C 4.4. Let A be a regular local ring of dimension 3, m be its maximal
ideal and K be its quotient field. Assume that 2 is inertible in A. The sequence 0,- W(A) ,- W(K) ,-&cP
G
P?Spec"(A)
W(AP\PAP)
is exact.
Proof. The injectivity of W(A),- W(K) is proved in [6, Theorem 23]. Since
W−"
fl(A)# W−"(A\m) l 0, by Theorem 4.3 we have W(A) # W(Y). Thus it is enough
to prove that the complex
W(Y ),- W(K) ,,-&cP
G
P?Spec"(A)
W(AP\PAP)
is exact. Let q be a quadratic space over K such thatcP(q)l 0 for all height 1 prime ideals P of A. Since Y is a regular scheme of dimension 2, by [1, 2.5, p. 112], there exists a quadratic space (, q) over Y l Spec(A)Bomq, such that its image in W(K) under the restriction map is equal to q. This completes the proof. Using Corollary 4.4, one can prove the following theorem (cf. [8, Proposition 2.1]).
C 4.5. Let A be a regular ring of dimension 3 and K be its quotient field.
Assume that 2 is inertible in A. The sequence 0,- W(A) ,- W(K) ,,-&cP
G
P?Spec"(A)
W(AP\PAP)
is exact.
We end this paper by giving a computation of W−"(Y) using Theorem 4.3.
C 4.6. Let A be a regular local ring of dimension 3 and m be its maximal
ideal. Assume that 2 is inertible in A. Let Y l Spec(A)Bomq. Then W−"(Y) # W(A\m). Proof. Since W−"(A) l 0 and W
fl(A)# W(A\m), the result follows from
Theorem 4.3.
Acknowledgements. We thank the referee for patiently pointing out errors, obscurities and misprints in the various versions of this paper.
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M. Ojanguren R. Parimala
UniersiteT de Lausanne R. Sridharan
Section de MatheT matiques V. Suresh
CH-1015 Lausanne-Dorigny School of Mathematics
Switzerland Tata Institute of Fundamental Research
Homi Bhabha Road Bombay 400 005 India