Cohomological invariants for quadratic forms over local rings
The main result of the paper under discussion (Theorem 5.5) is the following variant of the Milnor conjecture for local rings: if \(A\) is a local ring in which \(1+1\) is invertible, then there is a well-defined group homomorphism \[e_n : I^n(A) \rightarrow H_{\text{et}}^n(A,\mathbb{Z}/2), \ \text{given by}\] \[\langle \! \langle a_1,\dots,a_n \rangle \! \rangle \mapsto (a_1) \cup \dots \cup (a_n),\] whose kernel is exactly \(I^{n+1}(A)\). A key tool is the proof of the Gersten conjecture for Witt groups (Theorem 3.8) for such \(A\) under the additional assumption that \(A\) be regular and unramified. In the last section (no. 6), the author provides sufficient conditions for the triviality of the intersection \(\bigcap I^n(A)\) (which in the case of fields follows the Arason-Pfister Hauptsatz), such as \(A\) being a regular ring for which the Gersten conjecture holds.
scientific article; zbMATH DE number 6837430
scientific article