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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.
##### MSC:
 13D25 Complexes (MSC2000) 14A05 Relevant commutative algebra
##### Keywords:
$$n$$-singular set of a module; complexes of modules
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