Periodic resolutions and self-injective algebras of finite type

We say that an algebra A is periodic if it has a periodic projective resolution as an (A,A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B→A, B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba’s classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi-Yau dimensions.     

Radical cube zero weakly symmetric algebras and support varieties

One of our main results is a classification of all the weakly symmetric radical cube zero finite dimensional algebras over an algebraically closed field having a theory of support via the Hochschild cohomology ring satisfying Dade’s Lemma. In the process we give a characterization of when a finite dimensional Koszul algebra has such a theory of support in terms of the graded centre of the Koszul dual.     

Radical cube zero selfinjective algebras of finite complexity

One of our main results is a classification of all the possible quivers of selfinjective radical cube zero finite-dimensional algebras over an algebraically closed field having finite complexity. In the paper (Erdmann and Solberg, 2011) [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite-dimensional algebra to have such a theory of support varieties implies that the algebra has finite complexity. Hence this paper is a partial extension of [5].     

Ideals,radicals, and structure of additive categories

Simple and semisimple additive categories are studied. We prove, for example, that an artinian additive category is (semi)simple iff it is Morita equivalent to a division ring(oid). Semiprimitive additive categories (that is, those with zero radical) are those which admit anoether full, faithful functor into a category of modules over a division ringoid.     

Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes

Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpendicular category of P is equivalent to the full module category where and ?R is the Bongartz complement of P modulo its P-trace. Moreover, there is a ring epimorphism φ:R→S. We characterize the case when φ is a perfect localization. By [Riccardo Colpi, Alberto Tonolo, Jan Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997) 3225-3237], there exist mutually inverse isomorphisms μ^′ and ν^′ between the interval in the lattice of torsion classes in , and the lattice of all torsion classes in . We provide necessary and sufficient conditions for μ^′ and ν^′ to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when R is a Dedekind domain, and if P is finitely presented and R is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of μ^′ and ν^′ contains all injectives.     

On t-structures and torsion theories induced by compact objects

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.     

Locally finitely presented categories of sheaves

Let A be a locally finitely presented Grothendieck category. It is shown that a class of localizations of A in the sense of Bousfield is again locally finitely presented. The criterion is applied to torsion-free classes in A, sheaves and separated presheaves on a generalized ringed space, and representations of partially ordered sets.     

Flat covers in abelian and in non-abelian categories

A general existence theorem for flat covers in (e.g., quasi-abelian) locally finitely presented categories is obtained from an additive Ramsey type theorem. In the abelian case, it is shown that flat covers always exist. Applications to categories of separated presheaves or sheaves, localizations of Bousfield type, torsion-free classes of finite type, and categories of filtered objects or complexes, are given.     

A remark on rank varieties for a class of local algebras

We answer a question raised in [9] by showing the equality of two different definitions of rank variety for finitely generated modules over truncated polynomial algebras. We do this by establishing an isomorphism of algebras used in the two definitions of rank variety. Received: 23 April 2007     

The countable Telescope Conjecture for module categories

By the Telescope Conjecture for Module Categories, we mean the following claim: “Let R be any ring and (A,B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A,B) is of finite type.”We prove a modification of this conjecture with the word ‘finite’ replaced by ‘countable.’ We show that a hereditary cotorsion pair (A,B) of modules over an arbitrary ring R is generated by a set of strongly countably presented modules provided that B is closed under unions of well-ordered chains. We also characterize the modules in B and the countably presented modules in A in terms of morphisms between finitely presented modules, and show that (A,B) is cogenerated by a single pure-injective module provided that A is closed under direct limits. Then we move our attention to strong analogies between cotorsion pairs in module categories and localizing pairs in compactly generated triangulated categories.     

Grothendieck group and generalized mutation rule for 2-Calabi-Yau triangulated categories

We compute the Grothendieck group of certain 2-Calabi-Yau triangulated categories appearing naturally in the study of the link between quiver representations and Fomin-Zelevinsky cluster algebras. In this setup, we also prove a generalization of the Fomin-Zelevinsky mutation rule.     

On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.     

Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra.     

On the relative homology of cleft extensions of rings and abelian categories

We study the relative homological behaviour of the omnipresent class of cleft extensions of abelian categories. This class of extensions is a natural generalization of the trivial extensions studied in detail by Fossum, Griffith and Reiten and by Palmer and Roos. We apply our results to the relative homology of cleft extensions of rings.     

Cyclic homology of truncated quiver algebras

We determine the cyclic homology of truncated quiver algebras over a general commutative unital ring. Thus we improve and generalise the results in [8] and [5], where this problem has been studied for algebras over a field. We use the spectral sequence of a double complex that is obtained from minimal projective bimodule resolution and comparison morphisms.     

Morita equivalence of quasi-primal algebras and sheaves

In this paper, we show that two quasi-primal algebras are Morita equivalent if and only if their inverse semigroups of inner automorphisms are isomorphic, and if they have the same one-element subalgebras. The proof of this statement uses the representation theory of algebras by sections in sheaves.Presented by H. P. Gumm.     

Structure sheaves of definable additive categories

We prove that the 2-category of small abelian categories with exact functors is anti-equivalent to the 2-category of definable additive categories. We define and compare sheaves of localisations associated to the objects of these categories. We investigate the natural image of the free abelian category over a ring in the module category over that ring and use this to describe a basis for the Ziegler topology on injectives; the last can be viewed model-theoretically as an elimination of imaginaries result.     

Derived equivalences and Serre duality for Gorenstein algebras

We introduce a notion of Gorenstein algebras of codimension c and demonstrate that Serre duality theory plays an essential role in the theory of derived equivalences for Gorenstein algebras.     

Iyama’s finiteness theorem via strongly quasi-hereditary algebras

Let Λ be an artin algebra and X a finitely generated Λ-module. Iyama has shown that there exists a module Y such that the endomorphism ring Γ of X⊕Y is quasi-hereditary, with a heredity chain of length n, and that the global dimension of Γ is bounded by this n. In general, one only knows that a quasi-hereditary algebra with a heredity chain of length n must have global dimension at most 2n−2. We want to show that Iyama’s better bound is related to the fact that the ring Γ he constructs is not only quasi-hereditary, but even left strongly quasi-hereditary. By definition, the left strongly quasi-hereditary algebras are the quasi-hereditary algebras with all standard left modules of projective dimension at most 1.