The Witt groups of the spheres away from two

Ivo Dell’Ambrogio^{∗}  Jean Fasel^{†}

Abstract We calculate the Witt groups of the spheres up to 2-primary torsion.

1 Introduction

Let A be a Gorenstein ring of finite Krull dimension with \frac{1}{2} ∈ A. Denote the coordinate ring of the n-sphere over A by

S_{A}^{n} := A[X_{1}, …, X_{n+1}]/(X_{1}^{2} + ⋯ + X_{n+1}^{2} − 1),

and let

ℙS_{A}^{n} := Proj A[X_{0}, …, X_{n+1}]/(X_{0}^{2} + ⋯ + X_{n}^{2} − X_{n+1}^{2})

be the corresponding projective scheme.

We are interested in determining the (total) coherent Witt groups of these varieties, in terms of the Witt groups of the base A. As a first step we obtain the following two theorems, but we hope to determine soon also the 2-primary torsion. We abbreviate:

\overline{W}^{i}(X) := W̃^{i}(X) ⊗ ℤ[1/2], \overline{W}^{tot}(X) := ⨁_{0≤i≤3} \overline{W}^{i}(X).

Theorem 1.1. We have isomorphisms

\overline{W}^{i}(S_{A}^{n}) ≃ \overline{W}^{i}(ℙS_{A}^{n}) ≃ \overline{W}^{i}(A) ⊕ \overline{W}^{i−n}(A)

for all i.

If the base A is regular, then S_{A}^{n} is also regular and its coherent and derived (locally free) Witt groups coincide. The tensor product induces a natural structure of ℤ/4-graded rings on W^{tot}(A) and W^{tot}(S^{n}) (denote it by ⋆). If q : Spec S^{n} → Spec A denotes the canonical projection, then q^{∗} makes W^{tot}(S^{n}) into a ℤ/4-graded W^{tot}(A)-algebra. All this remains true of course for \overline{W}^{tot}. Our geometric proof of Theorem 1.1 lets us easily find this multiplicative structure:

Theorem 1.2. Let moreover A be regular. Then

\overline{W}^{tot}(S_{A}^{n}) ≃ \overline{W}^{tot}(A)[α]/(α^{2}),

with α sitting in degree n.

∗ambrogio@math.ethz.ch, ETH Zentrum, 8092 Zürich, Switzerland

†jean.fasel@math.ethz.ch, ETH Zentrum, 8092 Zürich, Switzerland

Remarks. (a) Theorem 1.1 follows also from the general results of G.W. Brumfiel in [Br1] and [Br2], even in greater generality (i.e., for any commutative ring A with \frac{1}{2} ∈ A). In loc. cit. Brumfiel sketches the developement of a theory KO^{−n}(Sper A) inspired by the usual (real) topological K-theory of spaces, and which depends only on the real spectrum Sper A (equipped with its sheaf of abstract semi-algebraic functions). One of the main results of Brumfiel is that there are natural isomorphisms

W_{n}^{K}(A) ⊗ ℤ[1/2] ≃ KO^{−n}(Sper A) ⊗ ℤ[1/2] (n ≥ 0),

where W_{n}^{K} are Karoubi’s Witt groups. By identifying \overline{W}^{n}(A) ≃ W_{n}^{K}(A) ⊗ ℤ[\frac{1}{2}] on the left hand side ([HS, Lemma A.3]), and by reducing Brumfiel’s theory to the usual K-theory of some space on the right hand side, one can use topological results to calculate Balmer-Witt groups tensored with ℤ[\frac{1}{2}]. We can avoid all this, our approach here being much simpler and completely geometric in nature. Our main tools will be the (12-periodic) localization long exact sequence, homotopy invariance and dévissage (recalled in Theorem A.13).

(b) By Proposition 2.1 below, we see that our calculation produces nontrivial groups only when the base ring A has characteristic 0 (i.e., when ℤ ⊂ A).

(c) In an Appendix at the end of the paper it is shown how to construct a canonical injective resolution of O_{X}, for any Gorenstein scheme X. As a consequence, one can define coherent Witt groups which depend only on the scheme X, independently of any choice (Definition A.4). Similarly, one can define canonical transfers (push-forwards) for finite morphisms (Theorem A.10) . Moreover if X is regular, we will have canonical isomorphisms W^{i}(X) ≃ W̃^{i}(X) between derived and coherent Witt groups (Theorem A.8).

Acknowledgements. The authors sincerely thank Paul Balmer, Stefan Gille and Burt Totaro for careful reading and useful comments.

2 Proof of Theorem 1.1

The case n = 0 is trivial. We have isomorphisms

Spec S^{0} ≃ Proj A[X_{0}, X_{1}]/(X_{0}^{2} − X_{1}^{2}) ≃ Spec(A × A)

and thus W̃^{i}(S^{0}) ≃ W̃^{i}(ℙS^{0}) ≃ W̃^{i}(A) ⊕ W̃^{i}(A).

For the rest of the proof we can assume n ≥ 1. The following Proposition is the main component of our calculation.

Proposition 2.1. Let X be a finite dimensional Gorenstein scheme with \frac{1}{2} ∈ O_{X}(X), such that none of its residue fields admits a total ordering (i.e., it has no ‘real points’). Let L be a line bundle over X. Then all the coherent Witt groups W̃^{i}(X, L) of X with values in L are 2-primary torsion groups.

Proof. For any field K, the classical Pfister local-global principle implies that W(K) is 2-primary torsion iff K is nonreal (i.e., iff K doesn’t admit a total ordering, iff −1 is a sum of squares in K). Therefore, with the above hypothesis the first page of the Gersten-Witt spectral sequence for (X, L) (see Balmer-Walter [BW, Thm. 7.2], Gille [Gi1, Thm. 3.14]) contains only 2-primary torsion groups, because the latter are sums of Witt groups W(k(x)) of the residue fields of X. But the spectral sequence converges to the groups W̃^{i}(X, L) because X is finite dimensional.

Corollary 2.2. Let X be a finite dimensional Gorenstein scheme, and let X_{ℚ} := X ×_{Spec ℤ} Spec ℚ. Then the canonical projection X_{ℚ} → X induces isomorphisms \overline{W}^{i}(X) ≃ \overline{W}^{i}(X_{ℚ}) for all i.

Proof. Let first X = Spec R be affine. Notice that a ring R satisfies the nonreality hypothesis of Proposition 2.1 if −1 is a sum of squares in R. Thus we can assume that the characteristic of R is 0, the other case being trivial. Consider for m ≥ 2 the localization long exact sequence

⋯ ⟶ W̃_{(m)}^{i}(A) ⟶ W̃^{i}(A) ⟶ W̃(A[1/m]) ⟶ · · · .

By dévissage and Proposition 2.1 we have \overline{W}_{(m)}^{i}(A) ≃ \overline{W}^{i}(A/m) ≃ 0, so we

obtain \overline{W}^{i}(A) ≃ \overline{W}^{i}(A[1/m]). Now A ⊗ ℚ ≃ colim_{m} A[1/m] and Witt groups commute with filtering colimits ([Gi2, Thm. 1.6]), therefore \overline{W}^{i}(A) ≃ \overline{W}^{i}(A⊗ℚ). The global case is obtained with Mayer-Vietoris, or can be done directly with the same reasoning as above.

Notation. In the following, we will write

C^{n} = C_{A}^{n} := \frac{A[X_{1}, …, X_{n}]}{(1 + X_{1}^{2} + ⋯ + X_{n}^{2})}.

An immediate corollary of Proposition 2.1 is that \overline{W}^{i}(C^{n}) ≃ 0 for all i.

Now, the intersection of the projective sphere ℙS^{n} = V (Σ_{k}X_{k}^{2} − X_{n+1}^{2}) ⊂ ℙ^{n+1} with the affine open D(X_{n+1}) = {X_{n+1} ≠ 0} ⊂ ℙ^{n+1} is isomorphic to the affine sphere Spec S^{n}. We have

Lemma 2.3. The open immersion Spec S^{n} ≃ D(X_{n+1}) ↪ ℙS^{n} induces isomorphisms \overline{W}^{i}(ℙS^{n}) ≃ \overline{W}^{i}(S^{n}).

Proof. We have a localization sequence

\xymatrix{⋯ \ar[r] & W̃_{(X_{n+1})}^{i}(ℙS^{n}) \ar[r] & W̃^{i}(ℙS^{n}) \ar[r] & W̃^{i}(S^{n}) \ar[r] & · · · .}  (1)

Denote by ℙC^{n} the closed projective subscheme V (X_{n+1}) ⊂ ℙS^{n}, i.e. ℙC^{n} = V (X_{0}^{2} + ⋯ + X_{n}^{2}) ⊂ ℙ^{n}. By dévissage we have isomorphisms

W̃_{(X_{n+1})}^{i}(ℙS^{n}) ≃ W̃^{i−1}(ℙC^{n}, L_{n}),  (2)

where L_{n} is some line bundle over ℙC^{n} (n ≥ 1). On the affine opens D(X_{i}) = {X_{i} ≠ 0} (i = 0, …, n), the scheme ℙC^{n} is isomorphic to C^{n}; thus ℙC^{n} clearly satisfies the hypothesis of Proposition 2.1 and therefore \overline{W}^{i}(ℙC^{n}, L_{n}) = 0. Then the exactness of (1) implies the claim.

Lemma 2.4. Write f := 1 + X_{1}^{2} + ⋯ + X_{n}^{2} ∈ A[X_{1}, …, X_{n}], and denote by 𝔸_{f}^{n} the corresponding open in the affine plane. Then we have isomorphisms S_{1−X_{n+1}}^{n} ≃ 𝔸_{f}^{n}.

Proof. The isomorphism is given by the stereographic projection. Indeed, writing U := S_{1−X_{n+1}}^{n} and V := A[Y_{1}, …, Y_{n}]_{1+Y_{1}^{2}+···+Y_{n}^{2}} for the localized rings, we define a morphism V → U by

Y_{ℓ} ↦ \frac{X_{ℓ}}{1 − X_{n+1}} (ℓ = 1, …, n).

Its inverse is given by

X_{n+1} ↦ \frac{Σ_{k}Y_{k}^{2} − 1}{Σ_{k}Y_{k}^{2} + 1} , X_{ℓ} ↦ \frac{2Y_{ℓ}}{Σ_{k}Y_{k}^{2} + 1} (ℓ = 1, …, n).

(Recall that 2 is invertible in A.)

Lemma 2.5. The projection p : 𝔸_{f}^{n} → Spec A induces an isomorphism p^{∗} :

\overline{W}^{i}(A) ≃ \overline{W}^{i}(𝔸_{f}^{n}).

Proof. We use the localization long exact sequence:

\xymatrix{… \ar[r] & \overline{W}_{(f)}^{i}(𝔸^{n}) \ar[r] & \overline{W}^{i}(𝔸^{n}) \ar[r] & \overline{W}^{i}(𝔸_{f}^{n}) \ar[r] & …} .

Since f = 1 + X_{1}^{2} + ⋯ + X_{n}^{2} is a regular element, by dévissage we have

\overline{W}_{(f)}^{i}(𝔸^{n}) ≃ \overline{W}^{i}(C^{n}).

Because of Proposition 2.1, the latter is zero. Now we use homotopy invariance of coherent Witt groups to conclude.

Lemma 2.6. There are split short exact sequences

\xymatrix{0 \ar[r] & \overline{W}_{1−X_{n+1}}^{i}(S^{n}) \ar[r] & \overline{W}^{i}(S^{n}) \ar[r] & \overline{W}^{i}(A) \ar[r] & 0.}

Proof. Consider the long exact sequence associated to the localization S^{n} → S_{1−X_{n+1}}^{n}:

\xymatrix{… \ar[r] & \overline{W}_{(1−X_{n+1})}^{i}(S^{n}) \ar[r] & \overline{W}^{i}(S^{n}) \ar[r] & \overline{W}^{i}(S_{1−X_{n+1}}^{n}) \ar[r] & \overline{W}_{(1−X_{n+1})}^{i+1}(S^{n}) \ar[r] & …} ,

and consider the projection q : Spec S^{n} → Spec A. We then have a commutative diagram

\xymatrix{Spec S^{n} \ar[dr]_{q} & Spec S_{1−X_{n+1}}^{n} \ar[l]_{i} \ar[d]^{p} \\  & Spec A}

where i is the inclusion. Using Lemma 2.5 and the fact that Spec S_{1−X_{n+1}}^{n} ≃ 𝔸_{f}^{n} (Lemma 2.4), we obtain a commutative diagram

\xymatrix{\overline{W}^{i}(S^{n}) \ar[r]^{i^{∗}} & \overline{W}^{i}(𝔸_{f}^{n}) \\  & \overline{W}^{i}(A) \ar[u]^{p^{∗}}_{≃} \ar@{-->}[ul]^{q^{∗}}}

The above long exact sequence yields finally the split sequences

\xymatrix{0 \ar[r] & \overline{W}_{1−X_{n+1}}^{i}(S^{n}) \ar[r] & \overline{W}^{i}(S^{n}) \ar[r] & \overline{W}^{i}(A) \ar[r] & 0.}

Corollary 2.7. For any i, we have \overline{W}^{i}(S^{n}) ≃ \overline{W}^{i}(A) ⊕ \overline{W}_{1−X_{n+1}}^{i}(S^{n}).

Next we compute \overline{W}_{(1−X_{n+1})}^{i}(S^{n}). In order to do this, we introduce some more notation:

Notation.

B^{n} = B_{A}^{n} := \frac{A[X_{1}, …, X_{n}]}{(X_{1}^{2} + ⋯ + X_{n}^{2})}.

Thus we have isomorphisms S^{n}/(1 − X_{n+1}) ≃ B^{n}. For n ≥ 1, the element 1 − X_{n+1} ∈ S_{n} is regular and we can use dévissage to obtain:

W̃_{(1−X_{n+1})}^{i}(S^{n}) ≃ W̃^{i−1}(B^{n}).  (3)

Notice en passant that the rings B^{n} are singular. This makes the use of coherent (rather that derived) Witt groups necessary for our calculation, even when A is regular.

Lemma 2.8. For n ≥ 1 and any i, we have \overline{W}^{i}(B^{n}) ≃ \overline{W}^{i−n+1}(A).

Proof. We will exploit the recursive property B^{n}/(X_{n}) ≃ B^{n−1}. Consider the long exact sequence

\xymatrix{⋯ \ar[r] & W̃_{(X_{n})}^{i}(B^{n}) \ar[r] & W̃^{i}(B^{n}) \ar[r] & W̃^{n}(B_{X_{n}}^{n}) \ar[r] & ⋯}

associated to the localization B^{n} → B_{X_{n}}^{n}. Notice that B_{X_{n}}^{n} ≃ C^{n−1}[X_{n}, X_{n}^{−1}]. It is a result of Gille that W̃^{i}(R[T, T^{−1}]) ≃ W̃^{i}(R) ⊕ W̃^{i}(R) for R a finite dimensional Gorenstein ring (see [Gi1, Thm. 5.6]). Thus we have

\overline{W}^{i}(B_{X_{n}}^{n}) ≃ \overline{W}^{i}(C^{n−1}) ⊕ \overline{W}^{i}(C^{n−1}) = 0,

where the vanishing due to Proposition 2.1. For n ≥ 2, the element X_{n} ∈ B^{n} is regular, so dévissage yields

W̃_{(X_{n})}^{i}(B^{n}) ≃ W̃^{i−1}(B^{n}/X_{n}) = W̃^{i−1}(B^{n−1}).

Altogether, we obtain the formula \overline{W}^{i}(B^{n}) ≃ \overline{W}^{i−1}(B^{n−1}) (n ≥ 2). We finish the proof by remarking that W̃^{j}(B^{1}) ≃ W̃^{j}(A) for all j. (For example, one can use the generalization of affine dévissage to zero dimensional ideals, see [Gi2, Thm. 3.5]: W̃^{j}(A[X]/X^{2}) = W̃_{(X)}^{j}(A[X]/X^{2}) ≃ W̃^{j}((A[X]/X^{2})/X) = W̃^{j}(A).)

Finally, we have \overline{W}^{i}(S^{n}) ≃ \overline{W}^{i}(A) ⊕ \overline{W}_{1−X_{n+1}}^{i}(S^{n}) by Corollary 2.7 and

\overline{W}_{1−X_{n+1}}^{i}(S^{n}) ≃ \overline{W}^{i−n}(A) by (3) and the last Lemma. Together with Lemma 2.3, this ends to proof of Theorem 1.1.

3 Proof of Theorem 1.2

From now on, the base A and thus also S_{A}^{n} will be assumed to be regular rings. We will still denote by q : Spec S^{n} → Spec A the structure morphism.

For any ring R, we will denote by e_{R} ∈ W^{0}(R) the multiplicative unit of W^{tot}(R); this is just the diagonal form < 1 >= [id : R → R] (we make the usual identification R = R^{∨}). In this proof we will also abbreviate

P := S^{n}/(X_{1}, …, X_{n})
𝔫 := (X_{n+1} − 1) 𝔰 := (X_{n+1} + 1),

so that P/𝔫 ≃ A and P/𝔰 ≃ A are the North Pole and the South Pole of the sphere. Write i_{P}, i_{N} and i_{S} for the corresponding closed immersions. We will further write

α_{N} := i_{N}_{∗}(e_{A/𝔫}) ∈ W^{n}(S^{n}), α_{S} := i_{S}_{∗}(e_{A/𝔰}) ∈ W^{n}(S^{n}),

and we will keep the same notation for the images in \overline{W}^{n}(S^{n}) of these forms. The next Lemma is just a corollary of the proof of Theorem 1.1.

Lemma 3.1.

\overline{W}^{tot}(S^{n}) = \overline{W}^{tot}(A) · e_{S^{n}} ⊕ \overline{W}^{tot}(A) · α_{N}

Proof. From the proof of Lemma 2.6 we see that q^{∗} : \overline{W}^{tot}(A) → \overline{W}^{tot}(S^{n}) is injective. Since q^{∗}(e_{A}) = e_{S^{n}}, we recognize the first direct summand. By the proof of Lemma 2.8 and functoriality of the transfer (Remark A.12, see also Remark A.14), we see that the other summand is the image of i_{N}_{∗} : \overline{W}^{tot}(P/𝔫) → \overline{W}^{tot}(S^{n}). This image is \overline{W}^{tot}(A) · α_{N} by the following Lemma.

Lemma 3.2. Let i : Spec A ↪ Spec S^{n} be the closed immersion of an A-point, let i_{∗} : W^{0}(A) → W^{n}(S^{n}) be the induced transfer morphism. Then for every form β ∈ W^{0}(A):

i_{∗}(β) = q^{∗}(β) ⋆ i_{∗}(e_{A}) ∈ W^{n}(S^{n}).

Proof. This follows from the projection formula for coherent Witt groups (Gille [Gi3, Thm. 5.2]), applied to the morphism i:

i_{∗}(β) = i_{∗}(\underbrace{i^{∗}q^{∗}}_{id}(β) ⋆ e_{A}) = q^{∗}(β) ⋆ i_{∗}(e_{A}).

Proposition 3.3. The relation α_{N} = −α_{S} holds in W^{n}(S^{n}).

As an immediate consequence of this, the form α_{N}^{2} = −α_{N} ⋆ α_{S} is supported on the intersection of the North and the South Poles, which is empty: Spec P/𝔫 ∩ Spec P/𝔰 = ∅; so it is trivially equal to zero.

To prove Proposition 3.3, we first recall straight from [Gi3, § 9] some facts about forms on Koszul complexes. For any ring R and any regular sequence (x_{1}, …, x_{n}) in R, we will denote by K_{•}(x_{1}, …, x_{n}) the Koszul complex for this sequence, and we set it in (homological) degrees from n to 0. We are interested in the situation when R is an S-algebra S → R over some other ring S, and the sequence is such that R/(x_{1}, …, x_{n}) ≃ S. (Below we will specialize to the P -algebra S^{n} and the regular sequence (X_{1}, …, X_{n}) in S^{n}.) For 1 ≤ i ≤ n and any unit r ∈ R^{×}, the complex K_{•}(x_{i}) can be equipped by the following symmetric 1-form:

\xymatrix{K_{•}(x_{i}) : \ar[d]^{ℓ_{r}:=} & ⋯ \ar[r] & 0 \ar[d] \ar[r] & R \ar[r]^{·x_{i}} \ar[d]_{·r} & R \ar[r] \ar[d]^{·(−r)} & 0 \ar[d] \ar[r] & ⋯ \\ K_{•}(x_{i}) : & ⋯ \ar[r] & 0 \ar[r] & R \ar[r]^{·(−x_{i})} & R \ar[r] & 0 \ar[r] & ⋯}

For any choice of n units in R, the product (K_{•}(x_{1}), ℓ_{r}) ⋆ ⋯ ⋆ (K_{•}(x_{n}), ℓ_{r}) is a symmetric n-space on the complex K_{•}(x_{1}, …, x_{n}) ≃ K_{•}(x_{1}) ⊗ ⋯ ⊗ K_{•}(x_{n}). We denote its form by ℓ_{r_{1}·...·r_{n}}.

By displaying an explicit Lagrangian, it is easy to see that [K_{•}(x_{i}), ℓ_{r_{i}}] = 0 ∈ W^{n}(R), and therefore [K_{•}(x_{1}, …, x_{n}), ℓ_{r_{1}·...·r_{n}}] = 0 ∈ W^{n}(R) for all symmetric spaces as above.

Lemma 3.4. If ϕ : K_{•}(x_{1}, …, x_{n}) → Hom_{R}(K_{•}(x_{1}, …, x_{n}), R)[n] is a quasiisomorphism of complexes, then there exists a unit r ∈ R^{×} such that ℓ_{r·1·...·1} is chain homotopic to ϕ.

Proof. This is a slight generalization of [Gi3, Lemma 9.1]. The same proof goes through.

We have an isomorphism W^{0}(P) ≃ W^{0}(P/𝔫) ⊕ W^{0}(P/𝔰), induced by P ≃ P/𝔫 × P/𝔰, which identifies e_{P} with (e_{N}, e_{S}). Under this isomorphism, the transfer i_{P}_{∗} : W^{0}(P) → W^{n}(S^{n}) identifies with (i_{N}_{∗}, i_{S}_{∗}), and in particular

α_{N} + α_{S} = i_{N}_{∗}(e_{N}) + i_{S}_{∗}(e_{S}) = i_{P}_{∗}(e_{P}).

But the last term is zero in W^{n}(S^{n}). In fact, i_{P}_{∗}(e_{P}) can be represented by (F_{•}, ψ), where F_{•} is a projective resolution of the S^{n}-module P = S^{n}/(X_{1}, …, X_{n}), and where ψ is a symmetric quasi-isomorphism between F_{•} and its n-shifted dual, lying above the morphism id : A → A. Since (X_{1}, …, X_{n}) is a regular sequence in S^{n}, we can take F_{•} to be the Koszul complex for this sequence. By the above Lemma (or by direct inspection) we have i_{A}_{∗}(e_{A}) = [K_{•}(X_{1}, …, X_{n}), ℓ_{1}] = 0 ∈ W^{n}(S^{n}).

A Appendix: The canonical injective resolution

A.1 The local case

We first recall some basic facts about injective modules. Our source here is [Bou].

Let A be a ring and consider the module F = Hom_{ℤ}(A, ℚ/ℤ). Then F is an injective A-module ([Bou, Corollary 1]). If M is any A-module, then we have a canonical homomorphism

e_{M} : M → F^{Hom_{A}(M,F)}

given by e_{M}(m) = (ϕ(m))_{ϕ∈Hom_{A}(M,F)}. Observe that F^{Hom_{A}(M,F)} is injective and e_{M} is injective ([Bou, Corollary 2]). Let I_{0}(M) := F^{Hom_{A}(M,F)}, L_{0}(M) := Coker(e_{M}) and q_{M} : I_{0}(M) → L_{0}(M) be the canonical projection. By definition, we obtain an exact sequence

\xymatrix{0 \ar[r] & M \ar[r]^{e_{M}} & I_{0}(M) \ar[r]^{q_{M}} & L_{0}(M) \ar[r] & 0.}

We define inductively I_{−n}(M) by I_{0}(L_{−n+1}(M)), L_{−n}(M) by L_{0}(L_{−n+1}(M)) and homomorphisms δ_{−n}^{M} : I_{−n}(M) → I_{−n−1}(M) by

\{ \begin{matrix}δ_{−n}^{M} = 0 & if n < 0 \\ δ_{0}^{M} = e_{L_{0}(M)} ∘ q_{M} & if n = 0 \\ δ_{−n}^{M} = e_{L_{−n}(M)} ∘ q_{L_{−n+1}(M)} & if n > 0. \end{matrix}

We then obtain an injective resolution of M:

\xymatrix{0 \ar[r] & M \ar[r]^{e_{M}} & I_{0}(M) \ar[r]^{δ_{0}^{M}} & I_{−1}(M) \ar[r]^{δ_{−1}^{M}} & I_{−2}(M) \ar[r] & …}

This resolution is called canonical injective resolution of M. Suppose now that M is of (finite) injective dimension n. Then using [Bou, Proposition 11], we see that L_{−n+1}(M) is an injective module. Therefore we have the following definition:

Definition A.1. Let M be an A-module of injective dimension n. Then the injective resolution

\xymatrix{0 \ar[r] & M \ar[r]^{e_{M}} & I_{0}(M) \ar[r]^{δ_{0}^{M}} & … \ar[r]^{δ_{−n+2}^{M}} & I_{−n+1}(M) \ar[r]^{q_{L_{−n+2}}} & L_{−n+1}(M) \ar[r] & 0}

is called canonical finite injective resolution of M.

Example 1. Suppose that A is a Gorenstein ring. Then by definition, A is of finite injective dimension (equal to the Krull dimension of the ring) and therefore has a canonical finite injective resolution. Any invertible A-module L is also of finite injective dimension.

A.2 The global case

Let X be a scheme. Consider the constant sheaf (of abelian groups) ℚ/ℤ. Then F := Hom(O_{X}, ℚ/ℤ) is a sheaf of O_{X}-modules over X.

Lemma A.2. The sheaf F is injective in the category M_{qc} of quasi-coherent O_{X}-modules.

Proof. We consider a covering of X by open connected affine subscheme Spec A_{i}. Since the restriction of Hom(O_{X}, ℚ/ℤ) to Spec A_{i} is Hom(A_{i}, ℚ/ℤ) and the latter is an injective module by the above section, we see that Hom(O_{X}, ℚ/ℤ) is injective.

Let M be a quasi-coherent O_{X}-module. Consider Hom_{O_{X}} (M, F) and F as sheaves of sets. Then the sheaf Hom_{ShSets}(Hom_{O_{X}} (M, F), F) of morphisms of sheaves of sets can be seen as an O_{X}-module (since F is an O_{X}-module). We put F^{Hom_{O_{X}} (M,F)} := Hom_{ShSets}(Hom_{O_{X}} (M, F), F). Then we have a canonical homomorphism

e_{M} : M → F^{Hom_{O_{X}} (M,F)}

defined over an open U by e_{M}(U) = e_{M(U)}. As in the local case, observe that F^{Hom_{O_{X}} (M,F)} is an injective O_{X}-module and that e_{M} is injective. Mimicking the construction of the local case, we obtain an injective resolution of M

\xymatrix{0 \ar[r] & M \ar[r]^{e_{M}} & I_{0}(M) \ar[r]^{δ_{0}^{M}} & I_{−1}(M) \ar[r]^{δ_{−1}^{M}} & I_{−2}(M) \ar[r] & …}

called canonical injective resolution of M. If M is of (finite) injective dimension n, we see that L_{−n+1}(M) := Coker(δ_{−n+2}^{M}) is also injective. To avoid overloading the notations, we put I_{−j} := I_{−j}(M) and δ_{−j}^{M} := δ_{−j} for any j ≤ n − 1; we also put I_{−n} := L_{−n+1}(M) and q_{L_{−n+2}^{M}} := δ_{−n+1}. We have:

Definition A.3. Let X be a scheme and M a quasi-coherent O_{X}-module of injective dimension n. Then the injective resolution

\xymatrix{0 \ar[r] & M \ar[r]^{e_{M}} & I_{0} \ar[r]^{δ_{0}} & … \ar[r]^{δ_{−n+2}} & I_{−n+1} \ar[r]^{δ_{−n+1}} & I_{−n} \ar[r] & 0}

is called canonical finite injective resolution of M.

Example 2. If X is a Gorenstein scheme, then O_{X} has a canonical finite injective resolution. Any invertible O_{X}-module has the same property.

A.3 Coherent Witt groups

Our main references for this section are [Bal] and [Gi1].

Let X be an n-dimensional Gorenstein scheme such that 2 is a global unit over X. Let D_{c}^{b}(M_{qc}) be the triangulated category of bounded complexes of quasi-coherent O_{X}-modules whose homology sheaves are coherent. Let L be an invertible O_{X}-module and

\xymatrix{0 \ar[r] & L \ar[r]^{e_{L}} & I_{0} \ar[r] & … \ar[r] & I_{−n} \ar[r] & 0}

be the canonical finite injective resolution of L. For any M_{•} ∈ D_{c}^{b}(M_{qc}), we have a dual complex Hom_{O_{X}} (M_{•}, I_{•}) whose groups are

Hom_{O_{X}} (M_{•}, I_{•})_{i} := ⨁_{j∈ℤ} Hom_{O_{X}} (M_{−i−j}, I_{−j})

and differentials are given by f ↦ fd_{−i−j+1}^{M}+(−1)^{i+1}d_{−j}^{I}f. There is a canonical homomorphism ϖ_{M_{•}} of M_{•} to its bidual for any M_{•}, and it turns out that it is an isomorphism ([Gi1, Theorem 2.6.4]). Therefore (D_{c}^{b}(M_{qc}), Hom_{O_{X}} ( , I_{•}), 1, ϖ) is a triangulated category with duality in the sense of Balmer ([Bal, Definition 4.1]) and one can define the Witt groups of this category ([Bal, Definition 4.3]).

Definition A.4. Let O_{X} be a Gorenstein scheme, L an invertible O_{X}-module and I_{•} be the canonical finite injective resolution of L. Then the coherent Witt groups W̃^{i}(X, L) of X are defined to be the groups

W̃^{i}(X, L) := W^{i}(D_{c}^{b}(M_{qc}), Hom_{O_{X}} ( , I_{•}), 1, ϖ).

In the case where L = O_{X}, we put W̃^{i}(X) := W̃^{i}(X, O_{X}).

Remark A.5. The original definition of coherent Witt groups is due to Gille ([Gi1, Definition 2.6.10]), who chooses a finite injective resolution J_{•} of L and puts

W̃^{i}(X, L) := W^{i}(D_{c}^{b}(M_{qc}), Hom_{O_{X}} ( , J_{•}), 1, ϖ^{J_{•}}).

It is clear that this definition depends on the choice of a finite injective resolution of L. Since we have defined a canonical finite injective resolution, Definition A.4 does not depend on any choice.

Remark A.6. If

\xymatrix{0 \ar[r] & L \ar[r]^{j} & J_{0} \ar[r] & J_{−1} \ar[r] & … \ar[r] & J_{−n} \ar[r] & 0}

is any finite injective resolution of L, then there is a unique quasi-isomorphism φ : I_{•} → J_{•} making the following diagram commutative

[figure: 0; L; e_{L}; I_{•}; φ; 0; L; j; J_{•}.]

This quasi-isomorphism induces an isomorphism of Witt groups

φ : W̃^{i}(X, L) → W^{i}(D_{c}^{b}(M_{qc}), Hom_{O_{X}} ( , J_{•}), 1, ϖ^{J_{•}}).

Suppose now that X is a regular scheme (and 2 is a global unit over X). Consider the triangulated category D^{b}(P(X)) of bounded complexes of coherent locally free O_{X}-modules. The usual duality Hom_{O_{X}} ( , L) induces a 1-duality on D^{b}(P(X)) and there is a canonical natural isomorphism ϖ_{P_{•}} : P_{•} → Hom_{O_{X}} (Hom_{O_{X}} (P_{•}, L), L) for any P_{•} ∈ D^{b}(P(X)). One can check that (D^{b}(P(X)), Hom_{O_{X}} ( , L), 1, ϖ) is a triangulated category with duality in the sense of [Bal, Definition 4.1]. Thus we have the following definition (see [Bal]):

Definition A.7. Let X be a regular scheme, L an invertible O_{X}-module. The derived Witt groups of X are defined to be the groups

W^{i}(X, L) := W^{i}(D^{b}(P(X)), Hom_{O_{X}} ( , L), 1, ϖ).

When L = O_{X}, we simply put W^{i}(X) := W^{i}(X, O_{X}).

In particular, a regular scheme is Gorenstein. Hence there is a canonical quasi-isomorphism e_{L} : L → I_{•}. This quasi-isomorphism and the inclusion D^{b}(P(X)) ⊂ D_{c}^{b}(M_{qc}) yield isomorphisms W^{i}(X, L) → W̃^{i}(X, L) ([Gi1, Corollary 2.6.11]). Thus we get the following result:

Theorem A.8. Let X be a regular scheme and let L be a line bundle over X. Then we have canonical isomorphisms W^{i}(X, L) ≃ W̃^{i}(X, L).

A.4 Transfers

The reference here is [Gi3]. Let X and Y be Gorenstein schemes. For simplicity, we assume that they are connected. Suppose that f : X → Y is a finite morphism and let d = dim(Y) − dim(X). Let \overline{f} : (X, O_{X}) → (Y, f_{∗}O_{X}) be the morphism of locally ringed space induced by f. If M is a quasicoherent O_{Y} -module, then Hom_{O_{Y}} (f_{∗}O_{X}, M) is a quasi-coherent f_{∗}O_{X}-module and \overline{f}^{∗}Hom_{O_{Y}} (f_{∗}O_{X}, M) is a quasi-coherent O_{X}-module. In particular, if

\xymatrix{0 \ar[r] & L \ar[r]^{e_{O_{Y}}} & I_{0} \ar[r] & I_{−1} \ar[r] & … \ar[r] & I_{−n} \ar[r] & 0}

is the canonical finite injective resolution of an invertible O_{Y} -module L then \overline{f}^{∗}Hom_{O_{Y}} (f_{∗}O_{X}, I_{•}) is a complex of quasi-coherent O_{X}-modules. Moreover, for any i the module J_{−i} := \overline{f}^{∗}Hom_{O_{Y}} (f_{∗}O_{X}, I_{−i}) is injective ([Gi3, Chapter 4]). Now consider the complex (where d = dim(Y) − dim(X)):

\xymatrix{0 \ar[r] & Ker(δ_{−d}) \ar[r]^{ι} & J_{−d} \ar[r] & J_{−d−1} \ar[r] & … \ar[r] & J_{−n} \ar[r] & 0.}

It turns out that Ker(δ_{−d}) = \overline{f}^{∗}Ext_{O_{Y}}^{d} (f_{∗}O_{X}, L) and that J_{•} is a finite injective resolution of this O_{X}-module, which is invertible ([Gi3, Corollary 6.3]). The following theorem is due to Gille:

Theorem A.9 (Transfer). Let X and Y be connected Gorenstein schemes and let f : X → Y be a finite morphism of relative dimension d. Then the functor f_{∗} : D_{c}^{b}(M_{qc}(X)) → D_{c}^{b}(M_{qc}(Y)) induces homomorphisms

T r_{X/Y} : W^{i}(D_{c}^{b}(M_{qc}(X)), Hom_{O_{X}} ( , J_{•}), 1, ϖ^{J_{•}}) → W̃^{i+d}(Y, L).

Proof. See [Gi3, Theorem 6.4].

Now \overline{f}^{∗}Ext_{O_{Y}}^{d} (f_{∗}O_{X}, L) is an invertible O_{X}-module and therefore it admits a canonical finite injective resolution

\xymatrix{0 \ar[r] & \overline{f}^{∗}Ext_{O_{Y}}^{d} (f_{∗}O_{X}, O_{Y}) \ar[r]^{e} & I_{0} \ar[r] & … \ar[r] & I_{−n+d} \ar[r] & 0.}

By Remark A.6, there are canonical isomorphisms

φ : W̃^{i}(X, \overline{f}^{∗}Ext_{O_{Y}}^{d} (f_{∗}O_{X}, L)) → W^{i}(D_{c}^{b}(M_{qc}(X)), Hom_{O_{X}} ( , J_{•}), 1, ϖ^{J_{•}}).

We finally have the following result:

Theorem A.10 (Canonical transfer). Let X and Y be connected Gorenstein schemes and L an invertible O_{Y} -module. Let f : X → Y be a finite morphism and d = dim Y − dim X. Then the functor f_{∗} : D_{c}^{b}(M_{qc}(X)) → D_{c}^{b}(M_{qc}(Y)) induces canonical homomorphisms

f_{∗} : W̃^{i}(X, \overline{f}^{∗}Ext_{O_{Y}}^{d} (f_{∗}O_{X}, L)) → W̃^{i+d}(Y, L).

Proof. The canonical transfer is defined to be the composition

\xymatrix{W̃^{i}(X, \overline{f}^{∗}Ext_{O_{Y}}^{d} (f_{∗}O_{X}, O_{Y})) \ar[r]^{φ} \ar[dr]_{f_{∗}} & W^{i}(D_{c}^{b}(M_{qc}(X)), Hom_{O_{X}} ( , J_{•}), 1, ϖ^{J_{•}}) \ar[d]^{T r_{X/Y}} \\  & W̃^{i+d}(Y)}

Remark A.11. In a recent preprint, Nenashev announced a canonical transfer associated to any inclusion Y ↪ X of smooth schemes ([Ne, Definition 4.1]). It would be interesting to compare his canonical transfer with the one defined above.

Remark A.12. Let X, Y, Z be connected Gorenstein schemes and \xymatrix{X \ar[r]^{f} & Y \ar[r]^{g} & Z} be finite morphisms. Then using the fact that T r_{Y/Z} ∘ T r_{X/Y} = T r_{X/Z} for some good choices of injective resolutions (see [Gi1]) it is not hard to see that (gf)_{∗} = g_{∗}f_{∗}. Thus the canonical transfer is functorial.

The following theorem is also useful:

Theorem A.13 (Dévissage). Let Y be a connected Gorenstein scheme and j : X ↪ Y be a connected closed subscheme of Y of codimension d. Then the canonical transfer gives canonical isomorphisms

j_{∗} : W̃^{i}(X, \overline{j}^{∗}Ext_{O_{Y}}^{d} (j_{∗}O_{X}, O_{Y})) → W̃_{X}^{i+d}(Y).

Proof. See [Gi4, Theorem 3.2].

Remark A.14. It happens sometimes that \overline{j}^{∗}Ext_{O_{Y}}^{d} (j_{∗}O_{X}, O_{Y}) is isomorphic to O_{Y}, e.g. when Y = Spec A is affine and j : Spec A/I → Spec A is a closed immersion whose defining ideal I can be given by a regular sequence (a_{1}, ..., a_{d}) in A. Every choice of a sequence gives an isomorphism O_{Y} ≃ \overline{j}^{∗}Ext_{O_{Y}}^{d} (j_{∗}O_{X}, O_{Y}) and therefore an isomorphism of Witt groups

ϕ : W̃^{i}(X) → W̃^{i}(X, \overline{j}^{∗}Ext_{O_{Y}}^{d} (j_{∗}O_{X}, O_{Y})).

One would usually make this choice tacitly and will still write f_{∗} : W̃^{i}(X) ≃ W̃_{X}^{i}(Y) for the composite isomorphism f_{∗} ∘ ϕ.

References

[Bal] Paul Balmer, Witt Groups, in: Handbook of K-Theory, Springer (2005), vol. 2, pp. 539-576

[BW] P. Balmer and C. Walter, A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. École Norm. Sup. (4) 35, no. 1, pp. 127–152 (2002)

[Bou] Nicolas Bourbaki, Éléments de mathématique. Algebre, Chapitre 10, Masson, Paris (1980)

[Br1] G.W. Brumfiel, Witt rings and K-theory, Rocky Mountain J. of Math., Vol. 14, Nr. 4 (1984)

[Br2] G.W. Brumfiel, The real spectrum of an ideal and KO-theory exact sequences, K-Theory 1 (1987), pp. 211-235

[Gi1] Stefan Gille, On Witt groups with support, Math. Ann. 322, pp. 103-137 (2002)

[Gi2] Stefan Gille, Homotopy invariance of coherent Witt groups, Math. Zeitschrift 244, pp. 211-233 (2003)

[Gi3] Stefan Gille, A transfer morphism for Witt groups, J. reine angew. Math. 564, pp. 215-233 (2003)

[Gi4] Stefan Gille, The general dévissage theorem for Witt groups of schemes, preprint

[GN] S. Gille and A. Nenashev, Pairings in triangular Witt theory, J. of Algebra 261, pp. 292-309 (2003)

[HS] J. Hornbostel and M. Schlichting, Localization in Hermitian K-theory of rings, J. London Math. Soc. (2) 70, pp. 77-124 (2004)

[Ne] Alexander Nenashev, Gysin maps in Balmer-Witt theory, preprint