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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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