Associated primes and arithmetic degrees (Q1359020)

Let \(S= k[X_0,\dots, X_n]\) be the polynomial ring over a field \(k\), \(M\) be a finitely generated graded \(S\)-module, \(P\) be a homogeneous prime ideal of \(S\). The authors define the length-multiplicity \(\text{mult}_MP\) of \(M\) at \(P\) as the length of \(H^0_P(M_P)\) (as an \(S_P\)-module) and (for an integer \(r\geq -1\)) \[ \text{arith-deg}_r(M)= \sum_{\substack{ P\in\text{Ass}(M)\\ \dim(S/P)= r+1}} \text{mult}_M(P)\cdot \deg(S/P) \] (where \(\dim\) is the Krull dimension). If \(f\) is a homogeneous element of \(S\), an expression for \[ \text{arith-deg}_{r-1}(M/fM)- \det(f)\cdot \text{arith-deg}_r(M) \] is obtained, from which some conditions for the vanishing of this difference follow, applied to the case of a homogeneous ideal \(I\) of \(S\). Examples where \[ \text{arith-deg}_{r- 1}(I+ (f))> \deg(f)\cdot \text{arith-deg}_r(I) \] are presented. A generalization of a Bertini-type theorem is obtained. A result by \textit{B. Sturmfels}, \textit{Ngô Viêt Trung} and \textit{W. Vogel} [Math. Ann. 302, No. 3 417-432 (1995; Zbl 0828.14040)] concerning the effective Nullstellensatz is improved.