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Homological dimensions of local (co)homology over commutative DG-rings

  • Liran Shaul,
    Faculty of Mathematics, Bielefeld University, 33501 Bielefeld, Germany
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Abstract

Let $A$ be a commutative noetherian ring, let $\mathfrak{a}\subseteq A$ be an ideal, and let $I$ be an injective $A$-module. A basic result in the structure theory of injective modules states that the $A$-module $\Gamma_{\mathfrak{a}}(I)$ consisting of $\mathfrak{a}$-torsion elements is also an injective $A$-module. Recently, de Jong proved a dual result: If $F$ is a flat $A$-module, then the $\mathfrak{a}$-adic completion of $F$ is also a flat $A$-module. In this paper we generalize these facts to commutative noetherian DG-rings: let $A$ be a commutative non-positive DG-ring such that $\mathrm{H}^0(A)$ is a noetherian ring, and for each $i\lt 0$, the $\mathrm{H}^0(A)$-module $\mathrm{H}^i(A)$ is finitely generated. Given an ideal $\bar{\mathfrak{a}} \subseteq \mathrm{H}^0(A)$, we show that the local cohomology functor $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$ associated to $\bar{\mathfrak{a}}$ does not increase injective dimension. Dually, the derived $\bar{\mathfrak{a}}$-adic completion functor $\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}$ does not increase flat dimension.
Keywords: local cohomology, derived completion, homological dimension, commutative DG-ring local cohomology, derived completion, homological dimension, commutative DG-ring
MSC Classifications: 13B35, 13D05, 13D45, 16E45 show english descriptions Completion [See also 13J10]
Homological dimension
Local cohomology [See also 14B15]
Differential graded algebras and applications
13B35 - Completion [See also 13J10]
13D05 - Homological dimension
13D45 - Local cohomology [See also 14B15]
16E45 - Differential graded algebras and applications
 

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