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A Gersten-Witt spectral sequence for regular schemes. (English) Zbl 1012.19003
Let $$X$$ be a regular integral separated Noetherian scheme of finite Krull dimension, in which $$2$$ is everywhere invertible. P. Balmer [$$K$$-Theory 19, No. 4, 311-363 (2000; Zbl 0953.18003); Math. Z. 236, No. 2, 351-382 (2001; Zbl 1004.18010)] defined derived Witt groups $$W^n(X)$$ of $$X$$, which depend only on $$n\bmod 4$$, and showed that $$W^0(X)=W(X)$$ and $$W^2(X)=W^-(X)$$ can be identified with the usual Witt groups of symmetric resp. skew-symmetric vector bundles on $$X$$. The authors show that there exists a spectral sequence converging to $$E^n = W^n(X)$$, such that $$E_1^{pq}=0$$ unless $$0 \leq p \leq\dim(X)$$, and such that the horizontal line $0 \rightarrow E_1^{0q} \overset {d_1} \rightarrow E_1^{1q} \overset {d_1} \rightarrow E_1^{2q} \overset {d_1} \rightarrow \cdots \overset {d_1} \rightarrow E_1^{\dim(X),q} \rightarrow 0$ of the spectral sequence vanishes for $$q \not \equiv 0 \mod 4$$, whereas for $$q \equiv 0 \mod 4$$ this line equals the Gersten-Witt complex $\mathcal W_X: 0 \rightarrow W(K) \rightarrow \bigoplus_{x_1 \in X^{(1)}} W(k(x_1)) \rightarrow \cdots \rightarrow \bigoplus_{x_e \in X^{(e)}} W(k(x_e)) \rightarrow 0.$ Here $$K$$ is the field of rational functions on $$X$$, $$X^{(p)}$$ denotes the set of points of codimension $$p$$ in $$X$$, $$e = \dim(X)$$ and $$k(x)$$ is the residue field of $$X$$ at the point $$x$$. In fact, a more general version is proved for twisted Witt groups, where the duality is twisted by a line bundle. The periodicity of the Gersten-Witt spectral sequence and the vanishing of the horizontal lines in three out of 4 cases imply (and improve) most of the known Gersten-Witt related results in low dimension. For $$\dim X \leq 7$$ the authors establish a long exact sequence relating the Witt groups $$W^n:=W^n(X)$$ of $$X$$ and the cohohomology groups $$H^n(\mathcal W):= H^n(\mathcal W_X)$$ of the Gersten-Witt complex $$\mathcal W_X$$: $\begin{tikzcd} 0 \ar[r] & H^4(\mathcal W) \ar[r] & W^0 \ar[r] & H^0(\mathcal W) \ar[r] & H^5(\mathcal W) \ar[r] & W^1\ar[r] & H^1(\mathcal W) \ar[d]\\ 0 & H^3(\mathcal W) \ar[l] & W^3 \ar[l] & H^7(\mathcal W) \ar[l] & H^2(\mathcal W) \ar[l] & W^2\ar[l] & H^6(\mathcal W) \ar[l] \rlap{\,.} \end{tikzcd}$ Since $$H^0(\mathcal W_X)$$ is shown to be equal to the unramified Witt group $$W_{\text{nr}}(X)$$, immediate consequences are weak purity for $$\dim X \leq 4$$, i.e., the surjectivity of $$W(X) \rightarrow W_{\text{nr}}(X)$$, as well as purity for $$\dim X \leq 3$$, i.e., $$W(X) \cong W_{\text{nr}}(X)$$. The authors also show that for $$X = \operatorname {Spec}(R)$$ with $$R$$ a regular local ring of Krull dimension $$\leq 4$$ containing $$\frac{1}{2}$$ the Gersten conjecture holds, i.e., the Gersten-Witt complex augmented by the natural map $$W(X) \rightarrow W(K)$$ is exact.

##### MSC:
 19G12 Witt groups of rings 11E81 Algebraic theory of quadratic forms; Witt groups and rings 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 19E08 $$K$$-theory of schemes
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