Regularity of modules over a Koszul algebra
If \(R\) is a graded Noetherian \(k\)-algebra generated in degree 1, for \(k\) a field, the regularity \(\text{reg}_ RM\) of a module \(M\) is the infimum of the integers \(r\) such that for all \(i\geq 0\), \(\text{Tor}^ R_ i(M,k)_{i+s}=0\) for all \(s>r\). A Koszul algebra is a ring \(R\) as above such that \(\text{reg}_ Rk=0\). --- In this paper it is proved that if \(R\) is a Koszul algebra and \(Q=k[R_ 1]\) is the polynomial ring mapping onto \(R\), then the regularity of any module \(M\) over \(R\) is finite and \(\text{reg}_ RM\leq\text{reg}_ QM\). This is used to derive results on the existence on linear resolutions.
scientific article; zbMATH DE number 130475
scientific article