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Functorial construction of Cousin complexes. (English) Zbl 0814.13011
Bruns, Winfried (ed.) et al., Commutative algebra: Extended abstracts of an international conference, July 27 - August 1, 1994, Vechta, Germany. Cloppenburg: Runge. Vechtaer Universitätsschriften. 13, 94-96 (1994).
A complex \({\mathcal M}^ \bullet\) of sheaves of \({\mathcal O}_ X\)-modules on a locally Noetherian scheme \(X\) is called a Cousin complex with respect to a partition \(\{{\mathcal I}_ n\}_{n\in\mathbb{Z}}\) of (the underlying topological space of) \(X\) and \({\mathcal O}_{X,{\mathfrak p}}\)-modules \(M({\mathfrak p})\) with zero dimensional support if \({\mathcal M}^ n=\bigoplus_{{\mathfrak p}\in{\mathcal I}_ n}(i_{\mathfrak p})_ *\widetilde{M({\mathfrak p})}\), where \(\widetilde{M({\mathfrak p})}\) is the quasi-coherent module on \(\text{Spec} {\mathcal O}_{X,{\mathfrak p}}\) associated with \(M({\mathfrak p})\) and furthermore \(i_{\mathfrak p}:\text{Spec} {\mathcal O}_{X,{\mathfrak p}}\to X\) is the canonical map, and if the map \((i_{\mathfrak p})_ *\widetilde{M({\mathfrak p})}\to(i_{\mathfrak p})_ *\widetilde{M({\mathfrak q}})\) vanishes unless \({\mathfrak q}\) is an immediate specialization of \({\mathfrak p}\). If furthermore each \(M({\mathfrak p})\) is an injective \({\mathcal O}_{X,{\mathfrak p}}\)-module, \({\mathcal M}^ \bullet\) is bounded below and has coherent cohomologies, we call \({\mathcal M}^*\) a residual complex. Let \(X,Y\) be locally Noetherian schemes and \(f:Y \to X\) be a morphism locally of finite type. Given a partition \(\{{\mathcal I}_ n\}_{n \in \mathbb{Z}}\) of \(X\), let \(\{{\mathcal I}_ n\}_{n \in \mathbb{Z}}\) be the partition of \(Y\) induced by \(\{{\mathcal I}_ n\}_{n \in \mathbb{Z}}\) via the map \(f\). There is a functor \(f^ !\) from the category of Cousin complexes with respect to the partition \(\{{\mathcal I}_ n\}_{n \in \mathbb{Z}}\) on \(X\) to the category of Cousin complexes with respect to the partition \(\{{\mathcal J}_ n\}_{n \in \mathbb{Z}}\) on \(Y\) which preserves residual complexes.par From the paper.
For the entire collection see [Zbl 0799.00021].

13D25 Complexes (MSC2000)
14A15 Schemes and morphisms
32C38 Sheaves of differential operators and their modules, \(D\)-modules