20 research outputs found
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 and are zero. Further we give
bounds for the global dimension of a Morita ring , regarded as
an Artin algebra, in terms of the global dimensions of and in the case
when both and 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 , where is an Artin algebra.Comment: 29 pages, revised versio
Change of rings and singularity categories
We investigate the behavior of singularity categories and stable categories
of Gorenstein projective modules along a morphism of rings. The natural context
to approach the problem is via change of rings, that is, the classical adjoint
triple between the module categories. In particular, we identify conditions on
the change of rings to induce functors between the two singularity categories
or the two stable categories of Gorenstein projective modules. Moreover, we
study this problem at the level of `big singularity categories' in the sense of
Krause. Along the way we establish an explicit construction of a right adjoint
functor between certain homotopy categories. This is achieved by introducing
the notion of 0-cocompact objects in triangulated categories and proving a dual
version of Bousfield's localization lemma. We provide applications and examples
illustrating our main results.Comment: v2: 40 pages, minor changes in Section 6, including a shift in the
definition of a 0-cocompact objec
A functorial approach to monomorphism categories II: Indecomposables
We investigate the (separated) monomorphism category
of a quiver over an Artin algebra .
We show that there exists a representation equivalence in the sense of
Auslander from to
, where
is the category of finitely generated modules and
and
denote the respective injectively
stable categories. Furthermore, if has at least one arrow, then we show
that this is an equivalence if and only if is hereditary. In general,
the representation equivalence induces a bijection between indecomposable
objects in and
non-injective indecomposable objects in , and
we show that the generalized Mimo-construction, an explicit minimal right
approximation into , gives an inverse to this
bijection. We apply these results to describe the indecomposables in the
monomorphism category of a radical-square-zero Nakayama algebra, and to give a
bijection between the indecomposables in the monomorphism category of two
artinian uniserial rings of Loewy length with the same residue field.
The main tool to prove these results is the language of a free monad of an
exact endofunctor on an abelian category. This allows us to avoid the technical
combinatorics arising from quiver representations. The setup also specializes
to yield more general results, in particular in the case of representations of
(generalised) speciesComment: 41 pages. Comments welcome
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