A generalization of the dualizing complex structure and its applications
Let \((A, {\mathcal M})\) be a local commutative noetherian ring with has a fundamental dualizing complex \(I^\bullet\), let \(M\) be a finitely generated \(A\)-module satisfying \((S_2)\) and the property that the minimal elements of \(\text{Supp}_A(M)\) coincide with the set of prime ideals \(p\in \text{Supp}_A(M)\) with \(\dim(A/p)= \dim M\). The authors show that \[ 0\to H^{d-s}( \text{Hom}_A(M,I^\bullet))\to \text{Hom}_A(M,I^{d-s})\to \text{Hom}_A(M,I^{d-s+1})\to \dots \] is the Cousin complex of \(H^{d-s}(\text{Hom}_A(M,I^\bullet))\) with respect to the dimension filtration of \(\text{Supp}_A(M)\) where \(d=\dim A\) and \(s= \dim_A(M)\). As an application it is shown that all cohomology modules of the Cousin complex of modules satisfying the preceding conditions are finitely generated. The relation between the Serre condition of the canonical module and the local cohomology of the ring is also investigated. In the last section, it is shown that if \(A\) is an \((S_2)\)-local ring, with Cohen-Macaulay formal fibres, and admits a canonical module, then it has a dualizing complex.
scientific article; zbMATH DE number 1562999
scientific article