On Witt groups with support (Q2716045)

The author studies the Witt group of a regular scheme. In the main theorems, it is established that if \(A\) is a regular ring and \(J\) an ideal generated by a regular sequence of length \(l\) such that \(A/J\) is also regular, then the Witt groups \(W^i(A/J)\) and \(W^{i+l}_J(A/J)\) are isomorphic for all \(i\in{\mathbb{Z}}\). For proving it, several results about triangular Witt groups, the derived Witt group of a scheme, coherent Witt groups and the Gersten-Witt spectral sequence are obtained. In applications the Witt groups \(W^i({\mathbb{P}}^1_A)\) of the projective line over a regular ring \(A\) of finite Krull dimension, the Witt groups of the punctured affine space over a regular local ring and the Witt groups of a Laurent ring \(R[T,T^{-1}]\) over a regular ring of finite Krull dimension are studied. NEWLINENEWLINENEWLINEIt is also proved that if \({\mathbb{S}}^n\) is the affine sphere over an algebraically closed field, then NEWLINE\[NEWLINE\text{card } W({\mathbb{S}})^n \leq 4^{\alpha(n)},\qquad\text{where}\quad \alpha(n) = \begin{cases} \left\lfloor {{n-1}\over 8} \right\rfloor+1 &\text{if \(n\) is odd},\\ \left\lfloor{{n\over 8}} \right\rfloor+1 &\text{if \(n\) is even}. \end{cases}NEWLINE\]NEWLINE Similar results are obtained for other affine quadrics.