Manuel Saorín

On exact categories and applications to triangulated adjoints and model structures

Abstract We show that Quillenʼs small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coherent sheaves leads to a streamlined and generalized version of recent results obtained by Estrada, Gillespie, Guil Asensio, Hovey, Jorgensen, Neeman, Murfet, Prest, Trlifaj and possibly others.

The Group of Outer Automorphisms and the Picard Group of an Algebra

The primary goal of this work is to develop a strategy to compute PicK(A) and OutK(A) where A is a finite-dimensional algebra over a field K. The basic idea is to put the normal subgroup Inn*(A) of the inner automorphisms of A induced by elements of 1 + J(A), as a common denominator in the ‘fraction’ AutK(A) / Inn(A). The new numerator AutK(A)/Inn*(A) and the new denominator Inn(A)/Inn*(A) are much easier to deal with than the original AutK(A) and Inn(A). The main ingredient to study Out(A) is now the appearance of an Abelian group Ch(Γ, K), the group of acyclic characters of the quiver Γ of A, that can be completely calculated. We show how to apply these results to compute the Picard group of a split finite-dimensional algebra in several cases.

Compactly generated t-structures on the derived category of a Noetherian ring

Abstract We study t -structures on D ( R ) the derived category of modules over a commutative Noetherian ring R generated by complexes in D fg − ( R ) . We prove that they are exactly the compactly generated t -structures on D ( R ) and describe them in terms of decreasing filtrations by supports of Spec ( R ) . A decreasing filtration by supports ϕ : Z → Spec ( R ) satisfies the weak Cousin condition if for any integer i , the set ϕ ( i ) contains all the immediate generalizations of each point in ϕ ( i + 1 ) . If a compactly generated t -structure on D ( R ) restricts to a t -structure on D fg ( R ) then the corresponding filtration satisfies the weak Cousin condition. If R has a pointwise dualizing complex the converse is true. If the ring R has dualizing complex then these are exactly all the t -structures on D fg b ( R ) .

Lifting and Restricting Recollement Data

We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated categories. This, together with the parametrizations of TTF triples given in Nicolás and Saorín (Parametrizing recollement data for triangulated categories. To appear in J. Algebra), allows us to show that many well-known recollements of right bounded derived categories of algebras are restrictions of recollements in the unbounded level, and leads to criteria to detect recollements of general right bounded derived categories. In particular, we give in Theorem 1 necessary and sufficient conditions for a right bounded derived category of a differential graded (=dg) category to be a recollement of right bounded derived categories of dg categories. Theorem 2 considers the case of dg categories with cohomology concentrated in non-negative degrees. In Theorem 3 we consider the particular case in which those dg categories are just ordinary algebras.

Endomorphism rings and category equivalences

The use of category equivalences for the study of endomorphism rings stems from the Morita theorem. In a sense, this theorem can be viewed as stating that if P is a finitely generated projective generator of R-mod and S = End( RP), then properties of P correspond to properties of S through the equivalence between the categories R-mod and S-mod given by the functor Hom,(P, -). Generalizations of this theorem were given in [4, 51, In [S], P is only assumed to be finitely generated and projective, and Hom,(P, -) gives in this case an equivalence between S-mod and a quotient category of R-mod, while in [S] it is shown that if P is a finitely generated quasiprojective self-generator, then the equivalence induced by the same functor is now defined between the category a[P] of all the R-modules subgenerated by P and S-mod. Later on, other category equivalences were constructed, in an analogous way to those already mentioned, by replacing S-mod by a certain quotient category of S-mod. Thus, in [14] Morita contexts are used to obtain a category equivalence between quotient categories of both R-mod and S-mod for an arbitrary R-module M. On the other hand, if M is a C-quasiprojective module, then it is shown in [8 3 that the functor Hom,(M, -) induces an equivalence between quotient categories of o[M] and S-mod, and the latter quotient category coincides with S-mod when M is finitely generated. All the above constructions can be considered as particular cases of the following: if V is a locally finitely generated Grothendieck category and M is an object of V with S = End,(M), then the class of the M-distinguished objects of g (in the sense of [lo]) is the torsionfree class of a torsion

On semiregular rings whose finitely generated modules embed in free modules

We consider rings as in the title and find the precise obstacle for them not to be Quasi-Frobenius, thus shedding new light on an old open question in Ring Theory. We also find several partial affirmative answers for that question. This paper was finished while Juan Rada was preparing his Ph.D. at the Universidad de Murcia. Manuel Saorı́n was partially supported by D.G.I.C.Y.T. (PB93-0515, Spain) and the Comunidad Autónoma de Murcia (PIB 94-25). Received by the editors October 24, 1995. AMS subject classification: Primary: 16D10, 16L60; Secondary: 16N20. c Canadian Mathematical Society 1997.

GENERALIZED TILTING THEORY

Given small dg categories A and B and a B-A-bimodule T, we give necessary and sufficient conditions for the associated derived functors of Hom and the tensor product to be fully faithful. Special emphasis is put on the case when RHom

Automorphism groups of trivial extensions

_\mathrm{A}