On the local cohomology of powers of ideals in idealizations

Ngày đăng: 15/11/2023

Let $(R, fm)$ be a Noetherian local ring and $M$ a finitely generated $R$-module. Denote $A=Rltimes M$ is the idealization of $M$ over $R.$ Let $Q$ be an ideal in $A.$ Set $fq= ho(Q)$, where $ ho: Rltimes M ightarrow R$ is the canonical projection defined by $ ho(a,x)=a$. We show that $H^i_{fm imes M}(A/Q^{n+1}A)simeq H^i_{fm}(R/fq^{n+1})oplus H^i_{fm}(M/fq^{n+1}M)$ for all $ngeq 0.$ From this result, we prove that the length functions $ell(H^0_{fm imes M}(A/Q^{n+1}A))$ is a polynomial when $Q$ is the principal ideal or $Q$ is the ideal generated by part of an almost p-standard system of parameters-a very strict subclass of d-sequences on the module. Furthermore, we give formulas for the coefficients of this polynomial through the usual multiplicities and the length of the local cohomology modules.