Attached primes of the top local cohomology modules with respect to an ideal (II)

Abstract

For a finitely generated module $M$, over a commutative Noetherian local ring $(R,\mathfrak{m})$, it is shown that there exist only a finite number of non--isomorphic top local cohomology modules ${\mathrm{H}}_{\mathfrak{a}}^{\dim (M)}(M)$, for all ideals $\mathfrak{a}$ of $R$. We present a reduced secondary representation for the top local cohomology modules with respect to an ideal. It is also shown that for a given integer $r\ge 0$, if ${\mathrm{H}}_{\mathfrak{a}}^r(R/\mathfrak{p})$ is zero for all $\mathfrak{p}$ in $\mathrm{Supp}(M)$, then ${\mathrm{H}}_{\mathfrak{a}}^i(M)=0$ for all $i\ge r$.

9 pages