— 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 augmented categories.

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Deriving DG categories

Applications of contravariantly finite subcategories

We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in the existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, and are hence saturated. In contrast, the similar category for a smooth compact analytic surface with no curves is not saturated. 2000 Math. Subj. Class. Primary 18E30.

Generators and representability of functors in commutative and noncommutative geometry

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is M. Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim $X\geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

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Braid group actions on derived categories of coherent sheaves

We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study [8] with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also [16]) and a question by Hideto Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel [29], whose triangulated structure goes back to Auslander-Reiten’s work [6], [44], [7].

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On triangulated orbit categories.

Morita Theory for Derived Categories

Derived categories and stable equivalence

Derived Equivalences As Derived Functors

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Thick subcategories of the stable module category

We introduce the notion of a splendid equivalence between blocks of finite groups, which is a strengthening of the notion of an equivalence of derived categories. Such an equivalence is always induced by a two-sided tilting complex whose terms are summands of permutation modules, a feature that is compatible with the belief that these complexes should be related to the /-adic cohomology of Deligne-Lusztig varities in the case of reductive groups over finite fields. We show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence. In particular, it gives, for principal blocks, a structural explanation of Brou^'s concept of an isotypy (a compatible family of perfect isometries). We give several examples, including examples of Morita equivalences that are splendid in an unexpectedly non-trivial way.

Jeremy Rickard

Papers

Morita Theory for Derived Categories

Derived categories and stable equivalence

Derived Equivalences As Derived Functors

Thick subcategories of the stable module category

Splendid Equivalences: Derived Categories and Permutation Modules