Completion and torsion over commutative DG rings

Abstract

Let $\operatorname{CDG}_{cont}$ 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 $\mathrm{Ho}(\operatorname{CDG}_{cont})$ be its homotopy category, obtained by inverting adic quasi-isomorphisms, we construct a functor $\mathrm{L}Λ:\mathrm{Ho}(\operatorname{CDG}_{cont}) \to \mathrm{Ho}(\operatorname{CDG}_{cont})$ 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 $\Bbbk$ is a commutative ring, and $A$ is a commutative $\Bbbk$-algebra which is $\mathfrak{a}$-adically complete with respect to a finitely generated ideal $\mathfrak{a}\subseteq A$, then the derived Hochschild cohomology modules $\operatorname{Ext}^n_{A\otimes^{\mathrm{L}}_{\Bbbk} A} (A,A)$ and the derived complete Hochschild cohomology modules $\operatorname{Ext}^n_{A\widehat{\otimes}^{\mathrm{L}}_{\Bbbk} A} (A,A)$ coincide, without assuming any finiteness or noetherian conditions on $\Bbbk, A$ or on the map $\Bbbk \to A$.