On the local cohomology of powers of ideals in idealizations

Abstract

Let $(R, {\mathfrak {m}})$ be a Noetherian local ring and M a finitely generated R-module. Let $A=R\ltimes M$ be the idealization of M over R and Q an ideal in A. Set ${\mathfrak {q}}=\rho (Q)$, where $\rho : R\ltimes M\rightarrow R$ is the canonical projection defined by $\rho (a,x)=a$. We show that $H^i_{{\mathfrak {m}}\times M}(A/Q^{n+1}A)\simeq H^i_{{\mathfrak {m}}}(R/{\mathfrak {q}}^{n+1})\oplus H^i_{{\mathfrak {m}}}(M/{\mathfrak {q}}^{n+1}M)$ for all $n\gg 0$. From this result, we prove that the length functions $\ell (H^0_{{\mathfrak {m}}\times M}(A/Q^{n+1}A))$ 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.