## Abstract

Let CDG_{cont} be the category whose objects are pairs ((Aa¯)), where A is a commutative DG-algebra and a¯ ⊆ H (A) is a finitely generated ideal, and whose morphisms f(Aa¯) → (Bb¯) are morphisms of DG-algebras A → B, such that (H (f) (a¯)) ⊆ b¯. Letting Ho(CDG_{cont}) be its homotopy category, obtained by inverting adic quasi-isomorphisms, we construct a functor LΛ: Ho(CDG_{cont}) → Ho(CDG_{cont}) which takes a pair ((Aa¯)) into its non-abelian derived 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 = 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 k is a commutative ring, and A is a commutative k-algebra which is a-adically complete with respect to a finitely generated ideal a⊆ A, then the derived Hochschild cohomology modules ExtA⊗kLAn(AA) and the derived complete Hochschild cohomology modules ExtA⊗^kLAn(AA) coincide, without assuming any finiteness or noetherian conditions on k, A or on the map k→ A.

Original language | English |
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Pages (from-to) | 531-588 |

Number of pages | 58 |

Journal | Israel Journal of Mathematics |

Volume | 232 |

Issue number | 2 |

DOIs | |

State | Published - 1 Aug 2019 |

## ASJC Scopus subject areas

- General Mathematics