\(q\)-torsion freeness of symmetric powers
Let \(R\) be a commutative noetherian unitary ring and \(E\) a finitely generated \(R\)-module. Let \(S_R(E)\) be the symmetric algebra of \(E\) over \(R\) and \(S_R(R^n)= R[X_1,\dots, X_n]\) the polynomial ring over \(R\). If \(E\) is of rank \(e\) and of finite projective dimension then there is an epimorphism between the cycles of the Koszul complex on the kernel of the canonical morphism \(S_R(R^n)\to S_R(E)\) and the cycles of the complex \(Z(E)\). Hence, if the Koszul complex \(K(f_1,\dots, f_m;S)\) of the ideal of relations of \(S(E)\) is exact then so is \(Z(E)\) (theorem 2.11.); here \(Z_r(E)= \text{Ker }\partial_r\) with \(\partial_R: \bigwedge^r R_1^n\to \bigwedge^{r-1} R^n\bigotimes_R E\) gives the complex \(Z(E)\): \[ 0\to Z_n(E)\otimes S[-n]\to Z_{n-1}(E)\otimes S[-n+1]\to \dots\to Z_0(E)\otimes S\to 0. \] For an integer \(q\) the module \(E\) is said to be \(q\)-torsion free if every \(R\)-regular sequence of length \(q\) is also \(E\)-regular. In the last part several results concerning the \(q\)-torsion freeness of the symmetric powers of \(E\) are proved.
scientific article; zbMATH DE number 1140075
scientific article