Jun-ichi Miyachi

Localization of triangulated categories and derived categories

The notion of quotient and localization of abelian categories by dense subcategories (i.e., Serre classes) was introduced by Gabriel, and plays an important role in ring theory [6, 131. The notion of triangulated categories was introduced by Grothendieck and developed by Verdier [9, 16) and is recently useful in representation theory [8,4, 143. The quotient of triangulated categories by epaisse subcategories is constructed in [16]. Both quotients were indicated by Grothendieck, and they resemble each other. In this paper, we will consider triangulated categories and derived categories from the point of view of localization of abelian categories. Verdier gave a condition which is equivalent to the one that a quotient functor has a right adjoint, and considered a relation between tpaisse subcategories [16]. We show that localization of triangulated categories is similarly defined, and have a relation between localizations and Cpaisse subcategories. Beilinson, Bernstein, and Deligne introduced the notion of t-structure which is similar to torsion theory in abelian categories [2]. We, in particular, consider a stable t-structure, which is an epaisse subcategory, and deal with a correspondence between localizations of triangulated categories and stable r-structures. And then recollement, in the sense of [2], is equivalent to bilocalization. Next, we show that quotient and localization of abelian categories induce quotient and localization of its derived categories. In Section 1, we recall standard notations and terminologies of quotient and localization of abelian categories. In Section 2, we define localization of triangulated categories, and consider a relation between localizations and stable r-structures (Theorem 2.6). In Section 3, we show that if A + A/C is a quotient of abelian categories, then D*(A) -+ D*(A/C) is a quotient of triangulated categories, where * = + , -, or b (Theorem 3.2). Moreover,

DERIVED PICARD GROUPS OF FINITE DIMENSIONAL HEREDITARY ALGEBRAS

AbstractLet A be a finite-dimensional algebra over a field k. The derived Picard group DPick(A) is the group of triangle auto-equivalences of D> b( mod A) induced by two-sided tilting complexes. We study the group DPick(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPick, as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPick(A) on a certain infinite quiver Γirr. This representation is faithful when the quiver Δ of A is a tree, and then DPick(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPick(A). When A is hereditary, DPick(A) coincides with the full group of k-linear triangle auto-equivalences of Db( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to Db( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.

On t-structures and torsion theories induced by compact objects

Abstract First, we show that a compact object C in a triangulated category, which satisfies suitable conditions, induces a t-structure. Second, in an abelian category we show that a complex P · of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for P · to be a tilting complex. Finally, in the case of artin algebras, we give a one-to-one correspondence between tilting complexes of term length two and torsion theories with certain conditions.

Injective resolutions of Noetherian rings and cogenerators

We give new construction of injective resolutions of complexes and bimodules. Applying this construction to an injective resolution of a Noetherian ring, we construct a Σ-embedding cogenerator for the category of modules of projective dimension ≤ n. Moreover, for a Noetherian projective k-algebra R, we show that R satisfies the Auslander condition if and only if the flat dimension of every R-module M is equal to or larger than the one of the injective hull E(M). 0. Introduction Let R be a ring, and let 0 → R → E → E → . . . be a minimal injective resolution of a left free module R of one generator. Bass studied injective resolutions of left Noetherian rings, in particular, of commutative Noetherian rings [B1], [B2]. In the case of commutative Noetherian rings, he showed that for a prime ideal P of R, the injective dimension of RP is equal to the flat dimension of the injective hull E(R/P ) of R/P [B2], [X]. He also showed that R is Gorenstein if and only if E is the direct sum of indecomposable injective R-modules E(R/P ), where P are prime ideals of height n, for every n ≥ 0 [B2]. In other words, for every n ≥ 0, E is the direct sum of indecomposable injective R-modules I of which the flat dimensions are equal to n. Xu studied categorical properties of Gorenstein rings [X]. In the case of non-commutative Noetherian rings, Auslander provided the homological condition, which is called the Auslander condition, as one face of non-commutative versions of Gorenstein rings [FGR]. For a non-commutative Noetherian ring R satisfying the Auslander condition, Hoshino showed that every indecomposable injective left R-module I of flat dimension n is a direct summand of E [H1]. By using dualities of derived categories, we showed that if the injective dimensions of RR and RR are finite, then every indecomposable injective left R-module appears in some E [Mi]. In this paper, by using a new construction of injective resolutions of complexes, we provide results concerning injective resolutions of Noetherian rings, projective dimension, flat dimension of R-modules. Moreover we study cogenerators for categories of modules of finite flat dimension in order to describe categorical properties of module categories. Received by the editors June 25, 1998 and, in revised form, September 15, 1998. 1991 Mathematics Subject Classification. Primary 16D50, 16D90, 16E10, 18G35; Secondary 16D20, 18E30. c ©2000 American Mathematical Society

RECOLLEMENT AND TILTING COMPLEXES

Abstract First, we study recollement of a derived category of unbounded complexes of modules induced by a partial tilting complex. Second, we give equivalent conditions for P · to be a recollement tilting complex, that is, a tilting complex which induces an equivalence between recollements { D A/AeA (A), D (A), D (eAe)} and { D B/BfB (B), D (B), D (fBf)} , where e , f are idempotents of A, B , respectively. In this case, there is an unbounded bimodule complex Δ T · which induces an equivalence between D A/AeA (A) and D B/BfB (B) . Third, we apply the above to a symmetric algebra A . We show that a partial tilting complex P · for A of length 2 extends to a tilting complex, and that P · is a tilting complex if and only if the number of indecomposable types of P · is one of A . Finally, we show that for an idempotent e of A , a tilting complex for eAe extends to a recollement tilting complex for A , and that its standard equivalence induces an equivalence between Mod A/AeA and Mod B/BfB .

Derived categories and Morita duality theory

Abstract We define a cotilting bimodule complex as the non-cummutative ring version of a dualizing complex, and show that a cotilting bimodule complex includes all indecomposable injective modules in case of Noetherian rings. Moreover we define strong-Morita derived duality, and show that existence of a cotilting bimodule complex is equivalent to one of strong-Morita derived duality.

Extensions of rings and tilting complexes

Abstract We give conditions that extensions of rings make tilting complexes. Moreover, we show that Frobenius extensions are invariant under derived equivalences which are induced by these tilting complexes.

Cohen-macaulay approximations and noetherian algebras

The notion of Cohen-Macaulay approximations was introduced by Auslander and Buchweitz, and was studied widely [1]. In the case of commutative complete Cohen-Macaulay local rings, there exist minimal CM-approximations of all finitely generated modules. Then invariant theory with respect to this approximations was studied by several authors. Also, Hashimoto and Shida showed the existence of minimal ones without complteteness of rings [4]. In non-commutative ring theory, we studied the relation between duality for derived categories and CMapproximations, and studied cotilting bimodules of finite injective dimension as the non-commutative ring version of dualizing modules [6]. Also, we gave the condition of categories for the existence of minimal ones. In this note, we study approximations in case of Noetherian algebras. First, we study the case that modules of infinite injective dimension induce the theory of CM-approximations. In the case of commutative ring , by [3] and [5],