Completion and torsion over commutative DG rings

Abstract

Let ${\mathrm{CDG}}_{cont}$ be the category whose objects are pairs $(A,\bar{\mathfrak{a}})$, where $A$ is a commutative DG-algebra and $\bar{\mathfrak{a}}\subseteq {\mathrm{H}}^0(A)$ is a finitely generated ideal, and whose morphisms $f:(A,\bar{\mathfrak{a}})\to (B,\bar{\mathfrak{b}})$ are morphisms of DG-algebras $A\to B$, such that $({\mathrm{H}}^0(f)(\bar{\mathfrak{a}}))\subseteq \bar{\mathfrak{b}}$. Letting $\mathrm{Ho}({\mathrm{CDG}}_{cont})$ be its homotopy category, obtained by inverting adic quasi-isomorphisms, we construct a functor $\mathrm{L}\Lambda :\mathrm{Ho}({\mathrm{CDG}}_{cont})\to \mathrm{Ho}({\mathrm{CDG}}_{cont})$ which takes a pair $(A,\bar{\mathfrak{a}})$ into its non-abelian derived $\bar{\mathfrak{a}}$-adic completion. We show that this operation has, in a derived sense, the usual properties of adic completion of commutative rings, and that if $A={\mathrm{H}}^0(A)$ is an ordinary noetherian ring, this operation coincides with ordinary adic completion. As an application, following a question of Buchweitz and Flenner, we show that if $\mathbb{k}$ is a commutative ring, and $A$ is a commutative $\mathbb{k}$-algebra which is $\mathfrak{a}$-adically complete with respect to a finitely generated ideal $\mathfrak{a}\subseteq A$, then the derived Hochschild cohomology modules ${\mathrm{Ext}}_{A{\otimes }_{\mathbb{k}}^{\mathrm{L}}A}^n(A,A)$ and the derived complete Hochschild cohomology modules ${\mathrm{Ext}}_{A{\hat{\otimes }}_{\mathbb{k}}^{\mathrm{L}}A}^n(A,A)$ coincide, without assuming any finiteness or noetherian conditions on $\mathbb{k},A$ or on the map $\mathbb{k}\to A$.

40 pages, final version, to appear in Israel Journal of Mathematics

References in corpus (3)

• Injective DG-modules over non-positive DG-rings
• Homological dimensions of local (co)homology over commutative DG-rings
• Adic reduction to the diagonal and a relation between cofiniteness and derived completion

Cited by in corpus (8)

• The Cohen-Macaulay property in derived commutative algebra
• Koszul complexes over Cohen-Macaulay rings
• Remarks on derived complete modules and complexes
• The derived deformation theory of a point
• Sequence-regular commutative DG-rings
• Maximal Cohen-Macaulay DG-complexes
• Categorical properties of reduction functors over non-positive DG-rings
• G-dimensions for DG-modules over commutative DG-rings