Witt groups of smooth projective quadrics
One way of studying quadratic forms over a field \(F\) of characteristic different from \(2\) is to consider quadratic forms over the function field \(K(Q_d)\) of the projective quadric \(Q_d\) given by a certain homogeneous quadratic equation. Let \(W(F)\) denotes the Witt group of quadratic forms over \(F\). Then a natural problem arises, that is: What is the kernel of the homomorphism \(W(F) \rightarrow W(K(Q_d))\). There have been solutions to the problem in some cases. The author of the paper under review proves that the sequence \( W(C_0(q)_{\sigma}) \rightarrow W(F) \rightarrow W(K(Q_d))\) is exact, where the map \( W(C_0(q)_{\sigma}) \rightarrow W(F)\) is the trace map (see Proposition 8.1 for the precise statement and a proof). The author also investigates the case when the quardric \(Q_d\) is defined by an isotropic quadratic form \(q\) (see Proposition 8.5 for the details). As an application of the exact sequences, the author derive the computation of the Witt group of quadrics over commutative semi-local ring (see subsection 8.3 for more details) and Witt group of quadrics defined by \(q_{m,n}=m \left\langle -1 \right\rangle \perp n \left\langle 1 \right\rangle \), assuming that \(-1\) is not a sum of two squares (see subsection 8.4 for more details).
scientific article; zbMATH DE number 7035895
scientific article