Cohomology with support of dimension \(\leq d\) (Q2796995)

The authors introduce an improvement of generalized local cohomology modules.NEWLINENEWLINELet \(R\) be a Noetherian ring and \(d \geq 0\) an integer. For two \(R\)-modules \(M\) and \(N\), the \(i\)th generalized cohomology modules with support of dimension \(\leq d\) is defined to be NEWLINE\[NEWLINE H_d^i(M, N) = \injlim_{\dim R/\mathfrak a \leq d} \text{Ext}^i_R(M/\mathfrak aM, N). NEWLINE\]NEWLINE The authors compute \(\inf \{i \mid \text{Ext}^i_d(M, N) \neq 0\}\) if \(M\) and \(N\) are finitely generated and if \(R\) is a quotient of a regular, bi-equidimensional ring. Furthermore they also give an upper bound of \(\{i \mid \text{Ext}^i_d(M, N) \neq 0\}\) when \(M\) is of finite projective dimension.