Completion and torsion over commutative DG rings

TL;DR

This paper develops a derived functor for adic completion of commutative DG-algebras, showing it generalizes classical completion and applies to Hochschild cohomology without finiteness assumptions.

Contribution

It introduces a non-abelian derived adic completion functor for commutative DG-algebras, extending classical notions and resolving a question about Hochschild cohomology in a broad setting.

Findings

The derived adic completion functor has properties analogous to classical completion.

For ordinary noetherian rings, the functor coincides with classical adic completion.

Derived Hochschild cohomology modules coincide under adic completion without finiteness assumptions.

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 $\mathrm{L}\Lambda:\mathrm{Ho}(\operatorname{CDG}_{cont}) \to \mathrm{Ho}(\operatorname{CDG}_{cont})$ 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 $\Bbbk \to A$.