Some applications of k-regular sequences and arithmetic rank of an ideal with respect to modules

‎Let R be a commutative Noetherian ring with identity‎, ‎I be an ideal of R‎, ‎and M be an R-module‎. ‎Let k\geqslant‎ -‎۱ be an arbitrary integer‎. ‎In this paper‎, ‎we introduce the notions of \Rad_M(I) and \ara_M(I)‎ ‎as the radical and the arithmetic rank of I with respect to M‎, ‎respectively‎. ‎We show that the existence of some sort of regular sequences‎ ‎can be depended on \dim M/IM and so‎, ‎we can get some information about local cohomology modules as well‎. ‎Indeed‎, ‎if \ara_M(I)=n\geq ۱ and {(\Supp_{R}(M/IM))}_{>k}=\emptyset‎ ‎(e.g.‎, ‎if \dim M/IM=k)‎, ‎then there exist n elements x_۱‎, ..., ‎x_n in I‎ ‎which is a poor k-regular M-sequence and generate an ideal‎ ‎with the same radical as \Rad_M(I) and so‎ ‎H^i_I(M)\cong H^i_{(x_۱‎, ..., ‎x_n)}(M) for all i\in \mathbb{N}_۰‎. ‎As an application‎, ‎we show that \ara_M(I) \leq \dim M+۱‎, ‎which is a refinement of the inequality \ara_R(I) \leq \dim R+۱ for modules‎, ‎attributed to Kronecker and Forster‎. ‎Then‎, ‎we prove‎ ‎\dim M-\dim M/IM \leq \cd(I‎, ‎M) \leq \ara_M(I) \leq \dim M‎, ‎if (R‎, ‎\mathfrak{m}) is a local ring and IM \neq M‎.

کلیدواژه ها:

regular sequences ، k-regular sequences ، Local cohomology modules ، arithmetic rank of an ideal with respect to modules