Chow-Witt groups and Grothendieck-Witt groups of regular schemes
Let \(A\) be a noetherian commutative ring of Krull dimension \(d\) and let \(P\) be a projective \(A\)-module of rank \(r\). The well-known splitting theorem of \textit{J.-P. Serre} [Algebre Theorie Nombres, Sem. P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot 11 (1957/58), No. 23, 18 p. (1958; Zbl 0132.41202)] asserts \(P\) splits off a free factor of rank one if \(r>d.\) The authors try to give obstructions for splitting off from \(P\) a free factor of rank one. In order to do this they use higher Grothendieck-Witt groups. The construction of Grothendieck-Witt groups was introduced by \textit{M. Schlichting} [``Hermitian \(K\)-theory, derived equivalences and Karoubi's fundamental theorem'', preprint, \url{http://www.math.lsu.edu/~mschlich/research/prelim.html}] and then Balmer and Walter defined derived Grothendieck-Witt groups [\textit{C. Walter}, ``Grothendieck-Witt groups of triangulated categories'', preprint, \url{http://www.math.uiuc.edu/K-theory/0643/}]. The authors define Euler classes in the derived Grothendieck-Witt groups and show the following: Theorem. Let \(A\) be a noetherian ring of dimension \(d\) with \(1/2 \in A.\) Let \(P\) be a projective module of rank \(d.\) If \(d=2\) or \(d=3,\) then \(e(P)=0\) in \(GW^{d}(A, \det(P)^{\vee})\) if and only if \(P\simeq Q\oplus A\) for some projective module \(Q.\)
scientific article; zbMATH DE number 5550664
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