On finiteness properties of local cohomology modules over Cohen-Macaulay local rings
Let \(\phi:(R,m)\to(A,n)\) be a local morphism of local rings which is module-finite and flat. Let \(I\) be an ideal of \(A\) such that, if \(J=I\cap R,\) then \(I=JA.\) Let \(i\) be an integer. The following facts are proved: \noindent a) if the set \(\mathrm{Ass}(H_J^i(R))\) is finite, then the set \(\mathrm{Ass}(H_I^i(A))\) is also finite; \noindent b) if all the Bass numbers of \(H^i_J(R)\) are finite, then all the Bass numbers of \(H^i_I(A)\) are also finite. Assume moreover that \(R\) and \(A\) are regular local rings and that \(R\) is unramified. Then: a) \(\mathrm{Ass}(H^i_I(A))\) is a finite set; b) all the Bass numbers of \(H^i_I(A)\) are finite. As a corollary, it is proved that if \(A\) is a Cohen-Macaulay local ring containing a field \(k\) such that the residue field of \(A\) is separable over \(k\) and \(I\) is an ideal of \(A\) generated by polynomials with coefficients in \(k\) in a system of parameters of \(A\), all the Bass numbers of \(H^i_I(A)\) are finite.
scientific article; zbMATH DE number 5615665
scientific article