# Deriving DG categories

@article{Keller1994DerivingDC, title={Deriving DG categories}, author={Bernhard Keller}, journal={Annales Scientifiques De L Ecole Normale Superieure}, year={1994}, volume={27}, pages={63-102} }

— We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a "triangulated analogue" (4.3) of a theorem of Freyd's [5], Ex. 5.3 H, and Gabriel's [6], Ch. V, characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a "Morita theorem" (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG…

## 801 Citations

Perfect Derived Categories of Positively Graded DG Algebras

- Mathematics
- 2008

Author’s abstract: We investigate the perfect derived category dgPer( A ) of a positively graded diﬀerential graded (dg) algebra A whose degree zero part is a dg subalgebra and semisimple as a ring.…

Derived categories of N ‐complexes

- MathematicsJ. Lond. Math. Soc.
- 2017

We study the homotopy category KN(B) of N ‐complexes of an additive category B and the derived category DN(A) of an abelian category A . First we show that both KN(B) and DN(A) have natural…

Smoothness of equivariant derived categories

- Mathematics
- 2012

We introduce the notion of (homological) G‐smoothness for a complex G‐variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses a…

On the construction of triangle equivalences

- Mathematics
- 1998

We give a self-contained account of an alternative proof of J. Rickard's Morita-theorem for derived categories [135] and his theorem on the realization of derived equivalences as derived functors…

Geometric aspects of representation theory for DG algebras: answering a question of Vasconcelos

- MathematicsJ. Lond. Math. Soc.
- 2017

A version of a theorem of Voigt is proved by exhibiting an isomorphism between the Yoneda Ext group $\operatorname{YExt}^1_U(M,M)$ and a quotient of tangent spaces coming from angebraic group action on an algebraic variety.

Spherical DG-functors

- Mathematics
- 2013

For two DG-categories A and B we define the notion of a spherical Morita quasi-functor A -> B. We construct its associated autoequivalences: the twist T of D(B) and the co-twist F of D(A). We give…

Uniqueness of dg enhancements for the derived category of a Grothendieck category

- MathematicsJournal of the European Mathematical Society
- 2018

We prove that the derived category of a Grothendieck abelian category has a unique dg enhancement. Under some additional assumptions, we show that the same result holds true for its subcategory of…

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