Abstract
Let CDGcont be the category whose objects are pairs (\((A, \bar{\mathfrak{a}})\)), where A is a commutative DG-algebra and \(\bar{\mathfrak{a}} \subseteq {{\rm{H}}^0}\left( A \right)\) is a finitely generated ideal, and whose morphisms \(f:\left( {A,\bar{\mathfrak{a}}} \right) \to \left( {B,\bar{\mathfrak{b}}} \right)\) are morphisms of DG-algebras A → B, such that \(\left( {{{\rm{H}}^0}\left( f \right)\left( {\bar{\mathfrak{a}}} \right)} \right) \subseteq \bar{\mathfrak{b}}\). Letting Ho(CDGcont) be its homotopy category, obtained by inverting adic quasi-isomorphisms, we construct a functor LΛ: Ho(CDGcont) → Ho(CDGcont) which takes a pair (\(\left( {A,\bar{\mathfrak{a}}} \right)\)) 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 = H0(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 \(\mathbb{k}\rightarrow A\).
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The research was partially supported by the Israel Science Foundation (grant no. 1346/15).
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Shaul, L. Completion and torsion over commutative DG rings. Isr. J. Math. 232, 531–588 (2019). https://doi.org/10.1007/s11856-019-1866-6
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DOI: https://doi.org/10.1007/s11856-019-1866-6