An application of a theorem of E. Davis to Koszul complexes
Let \(x_ 1,...,x_ r\) be a multiplicity system on a Noetherian R-module M of dimension d, where R is a commutative ring, and let \(H_ iK(x_ 1,...,x_ r| M)\) be the corresponding i-th homology module of the Koszul complex \(K(x_ 1,...,x_ r| M)\). In an earlier note [J. Lond. Math. Soc., II. Ser. 34, 411-416 (1986; Zbl 0608.13005)], the author proved that the function \({\mathbb{N}}\to {\mathbb{N}}_ 0\) given by \(n\to \text{length}_ RH_ iK(x^ n_ 1,...,x^ n_ r| M)\) is bounded above by a polynomial of degree at most min(d,r-i). In this note a shorter proof is given using that theorem of Davis which states that when R is a local ring there exists \(a_ 2,...,a_ r\) in R such that \(x_ 1+\sum^{r}_{2}a_ rx_ r\) is in no non-maximal prime ideal of Ass(M).
scientific article; zbMATH DE number 4063237