On the (non)vanishing of some “derived” categories of curved dg algebras

Abstract

a b s t r a c tSince curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes, different notions of ''derived'' categories have been introduced in the literature. In this article, we show that for some concrete curved dg algebras, these derived categories vanish. This happens for example for the initial curved dg algebra whose module category is the category of precomplexes, and for certain deformations of dg algebras.

Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence

“…The quotient category of Hot(B-mod) by the thick subcategory of completely acyclic CDG-modules can be called the complete derived category of left CDG-modules over B. It was noticed in [32] that the complex Hom B (L, M) is acyclic for any completely acyclic left CDG-module M over B and any left CDG-module L over B for which the graded B # -module L # is projective and finitely generated. Therefore, the minimal triangulated subcategory of Hot(B-mod) containing all the left CDG-modules over B that are projective and finitely generated as graded B-modules and closed under infinite direct sums is equivalent to a full triangulated subcategory of the complete derived category of left CDG-modules.…”

“…This follows from Theorems 3.4.1(d)-3.4.2(d) and 3.6(a). Besides, using Theorem 3.6(b) one can check [32] that Acycl(A) = Acycl abs (A) whenever A has a zero differential. All of this assumes that A # has a finite left homological dimension; without this assumption, there are only some partial results obtained in 3.4.…”

Hochschild (co)homology of the second kind I

“…the corresponding CDG-modules over C and D (which are defined uniquely up to a unique isomorphism). By the results of 2.2 (see (7)(8)), the CDG-functors R and I induce isomorphisms In particular, we obtain natural isomorphisms HH II * (B) ≃ HH II * (C) and HH II, * (B) ≃ HH II, * (C).…”

“…The morphism F * of Hochschild complexes computes the map of Hochschild homology (16) HH II * (B, F * M) −−→ HH II * (C, M) obtained by passing to the homology in the morphism of Tor objects (7) for the…”

“…Similarly, composing the derived functor RHom B with the functor Tot ⊕ , we obtain the derived functor Notice that the derived functors ⊗ L B and RHom B assign distinghuished triangles to short exact sequences of CDG-modules in any argument, hence so do the derived functors Tor B,II and Ext II B . Given a k-linear CDG-functor F : B −→ C, a right CDG-module N over C, and a left CDG-module M over C, there is a natural morphism (7) Tor…”