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Witt group of Hermitian forms over a noncommutative discrete valuation ring. (English) Zbl 1081.19003

Let \(R\) be a commutative discrete valuation ring and \(k\) the residue class field of \(R\). Then the Witt group \(WT(R)\) of regular symmetric bilinear forms on finitely generated torsion \(R\)-modules is isomorphic to the Witt group \(W(k)\) of the field \(k\) [W. Scharlau, “Quadratic and Hermitian forms”, Grundlehren Math. Wiss. 270 (1985; Zbl 0584.10010)]. The present author proves a similar result in the case when \(R\) is a noncommutative discrete valuation ring and bilinear forms are replaced by Hermitian forms.
Reviewer’s remarks: 1. The author offers a very short proof of the well known result that finitely generated torsion free modules over a PID are free. However, the proof works only in the case of discrete valuation rings.
2. In the proof of Theorem 3.6 the author uses weakly metabolic spaces instead of metabolic spaces in the definition of the Witt group of a commutative ring which is incorrect. The author invokes Lemma 1.2 from the paper [M. Knebusch, A. Rosenberg and R. Ware, Am. J. Math. 94, 119–155 (1972; Zbl 0248.13030)], where metabolic is proved to be equivalent with the existence of a self-orthogonal subspace, and subspace means direct summand. The author, however, interprets subspace as a submodule.

MSC:

19G12 Witt groups of rings
11E81 Algebraic theory of quadratic forms; Witt groups and rings
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