zbMATH — the first resource for mathematics

The singular sets of a complex of modules. (English) Zbl 0926.13009
From the introduction: A. Grothendieck and J. Dieudonné [Éléments de géométrie algébrique, Publ. Math., Inst. Hautes Étud. Sci. 4 (1960), 9 (1961), 11 (1962; Zbl 0118.36206)] defined the singular sets of a module. Let \(R\) be a Noetherian ring and \(M\) be a finitely generated \(R\)-module. Then for any \(n\in\mathbb{N}\) the set \(S^*_n (M)= \{{\mathfrak p} \in\text{Spec}(R): \text{depth} M_{\mathfrak p} +\dim R/{\mathfrak p}\leq n\}\), is called the \(n\)-singular set of \(M\). They showed that when \(R\) is a homomorphic image of a biequidimensional regular ring then for any \(n\in\mathbb{N}\) the \(n\)-singular set is closed in the Zariski topology of \(\text{Spec} R\). – The extension of homological algebra from modules to complexes of modules was started already in the last chapter of the book by H. Cartan and S. Eilenberg, “Homological algebra” (1956; Zbl 0075.24305) and pursued later.
The aim of this paper is to introduce the concept of the singular set of a complex of modules and to show in some special cases that the singular set is closed in the Zariski topology.
13D25 Complexes (MSC2000)
14A05 Relevant commutative algebra
Full Text: EMIS EuDML