Extendability of quadratic modules with sufficient Witt index
Let K be a field of characteristic different from two. A well-known theorem of Harder states that any quadratic space over the polynomial ring K[X] in one variable is extended from K and hence is decomposable. It is also known that isotropic quadratic spaces over \(K[X_ 1,...,X_ m]\) are extended from K for any m and hence these are again decomposable. On the other hand, if K admits a quaternion division algebra, then there do exist indecomposable anisotropic quadratic spaces of ranks 3 and 4 over K[X,Y] (Parimala's counter-examples to the quadratic analogue of Serre's conjecture). Suslin and Kopeiko considered the corresponding question for a general commutative ring R with characteristic different from two. They proved: Any quadratic space q over \(R[T_ 1,...,T_ m]\) with Witt-index \(Witt(\bar q)\geq \dim R+1\) which is stably extended from R is actually extended from R. Here the bar denotes \(''modulo\quad(T_ 1,...,T_ m)''.\) In the paper under review some improvements of this result for a regular ring R are given: (1) If R is an equicharacteristic complete regular local ring, then every quadratic space q over \(R[T_ 1,...,T_ m]\) with \(Witt(\bar q)\geq 1\) is extended from R. (2) If \(\dim(R)=d\), \(d=2\) or 3, and q is a quadratic \(R[T_ 1,...,T_ m]-space\) with \(Witt(\bar q)\geq d,\) then q is extended. (3) If R is a regular ring of essentially finite type over an infinite field K of dimension \(d\geq 1\), and q is a quadratic \(R[T_ 1,...,T_ m]-space\) with \(Witt(\bar q)\geq d,\) then q is extended. Here a ring R is of essentially finite type over K if \(R=S^{- 1} C\), where S is a multiplicatively closed subset of a finitely generated K-algebra C.
scientific article; zbMATH DE number 3853184
scientific article