On Artin algebras arising from Morita contexts

We study Morita rings \Lambda_{(\phi,\psi)}=\bigl({smallmatrix} A &_AN_B_BM_A & B {smallmatrix}\bigr) in the context of Artin algebras from various perspectives. First we study covariant finite, contravariant finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms $\phi$ and $\psi$ are zero. Further we give bounds for the global dimension of a Morita ring $\Lambda_{(0,0)}$, regarded as an Artin algebra, in terms of the global dimensions of $A$ and $B$ in the case when both $\phi$ and $\psi$ are zero. We illustrate our bounds with some examples. Finally we investigate when a Morita ring is a Gorenstein Artin algebra and then we determine all the Gorenstein-projective modules over the Morita ring with $A=N=M=B=\Lambda$, where $\Lambda$ is an Artin algebra.Comment: 29 pages, revised versio

Recollements of Module Categories

We establish a correspondence between recollements of abelian categories up to equivalence and certain TTF-triples. For a module category we show, moreover, a correspondence with idempotent ideals, recovering a theorem of Jans. Furthermore, we show that a recollement whose terms are module categories is equivalent to one induced by an idempotent element, thus answering a question by Kuhn.Comment: Comments are welcom

Derived equivalences between generalized matrix algebras

summary:We construct derived equivalences between generalized matrix algebras. We record several corollaries. In particular, we show that the $n$-replicated algebras of two derived equivalent, finite-dimensional algebras are also derived equivalent

Corner replacement for Morita contexts

We consider how Morita equivalences are compatible with the notion of a corner subring. Namely, we outline a canonical way to replace a corner subring of a given ring with one which is Morita equivalent, and look at how such an equivalence ascends. We use the language of Morita contexts, and then specify these more general results. We work in the setting of rings with local units and unital bimodules, to extend the reach of potential applications, say to categories of functors.Comment: 16 pages, comments welcom

Generalised Lat-Igusa-Todorov Algebras and Morita Contexts

In this paper we define (special) GLIT classes and (special) GLIT algebras. We prove that GLIT algebras, which generalise Lat-Igusa-Todorov algebras, satisfy the finitistic dimension conjecture and give several properties and examples. In addition we show that special GLIT algebras are exactly those that have finite finitistic dimension. Lastly we study Morita algebras arising form a Morita context and give conditions for them to be (special) GLIT in terms of the algebras and bimodules used in their definition. As a consequence we obtain simple conditions for a triangular matrix algebra to be (special) GLIT and also prove that the tensor product of a GLIT K-algebra with a path algebra of a finite quiver without oriented cycles is GLIT.Comment: 20 page

On singular equivalences of Morita type with level and Gorenstein algebras

Rickard proved that for certain self-injective algebras, a stable equivalence induced from an exact functor is a stable equivalence of Morita type, in the sense of Brou\'{e}. In this paper we study singular equivalences of finite dimensional algebras induced from tensor product functors. We prove that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a suitable complex of bimodules, induces a singular equivalence of Morita type with level, in the sense of Wang. This recovers Rickard's theorem in the self-injective case.Comment: Final version. The proof of the main Corollary is now fixed as well as some typos in 3.6, 3.7. To appear in the Bulletin of the London Mathematical Societ