419 research outputs found
Self-injective right artinian rings and Igusa Todorov functions
We show that a right artinian ring is right self-injective if and only if
(or equivalently ) for all finitely generated right
-modules , where are functions
defined by Igusa and Todorov. In particular, an artin algebra is
self-injective if and only if for all finitely generated right
-modules .Comment: 5 page
Igusa-Todorov functions for Artin algebras
In this paper we study the behaviour of the Igusa-Todorov functions for Artin
algebras A with finite injective dimension, and Gorenstein algebras as a
particular case. We show that the -dimension and -dimension are
finite in both cases. Also we prove that monomial, gentle and cluster tilted
algebras have finite -dimension and finite -dimension
The Igusa-Todorov function for comodules
We define the Igusa-Todorov function in the context of finite dimensional
comodules and prove that a coalgebra is left qcF if and only if it is left
semiperfect and its Igusa-Todorov function on each right finite dimensional
comodule is zero
Finitistic dimension through infinite projective dimension
We show that an artin algebra having at most three radical layers of infinite
projective dimension has finite finitistic dimension, generalizing the known
result for algebras with vanishing radical cube.Comment: 10 page
An approach to the finitistic dimension conjecture
Let be a finite dimensional -algebra over an algebraically closed
field and be the category of all finitely generated left
-modules. For a given full subcategory of we
denote by \pfd \mathcal{X} the projective finitistic dimension of
That is, \pfd \mathcal{X}:=\mathrm{sup} \{\pd X :
X\in\mathcal{X} \text{and} \pd X<\infty\}. \
It was conjectured by H. Bass in the 60's that the projective finitistic
dimension \pfd (R):=\pfd (\mathrm{mod} R) has to be finite. Since then, much
work has been done toward the proof of this conjecture. Recently, K. Igusa and
J. Todorov defined a function which turned
out to be useful to prove that \pfd (R) is finite for some classes of
algebras. In order to have a different approach to the finitistic dimension
conjecture, we propose to consider a class of full subcategories of
instead of a class of algebras, namely to take the class of
categories \F(\theta) of -filtered -modules for all stratifying
systems in $\mathrm{mod} R.
Split-by-nilpotent extensions algebras and stratifying systems
Let and be artin algebras such that is a
split-by-nilpotent extension of by a two sided ideal of
Consider the so-called change of rings functors
and In this paper, we find the necessary and
sufficient conditions under which a stratifying system in
\modu\Lambda can be lifted to a stratifying system in
\modu\,(\Gamma). Furthermore, by using the functors and we study the
relationship between their filtered categories of modules and some connections
with their corresponding standardly stratified algebras are stated. Finally, a
sufficient condition is given for stratifying systems in \modu\,(\Gamma) in
such a way that they can be restricted, through the functor to stratifying
systems in $\modu\,(\Lambda).
Filtering subcategories of modules of an artinian algebra
Let be an artinian algebra, and let be a subcategory of
mod that is closed under extensions. When is closed under
kernels of epimorphisms (or closed under cokernels of monomorphisms), we
describe the smallest class of modules that filters . As a
consequence, we obtain sufficient conditions for the finitistic dimension of an
algebra over a field to be finite. We also apply our results to the torsion
pairs. In particular, when a torsion pair is induced by a tilting module, we
show that the smallest classes of modules that filter the torsion and
torsion-free classes are completely compatible with the quasi-equivalences of
Brenner and Butler.Comment: 22 page
Pullback diagrams, syzygy finite classes and Igusa-Todorov algebras
For an abelian category , we define the category
PEx() of pullback diagrams of short exact sequences in
, as a subcategory of the functor category Fun() for a fixed diagram category . For any object in
we prove the existence of a short exact sequence of functors, where the objects are in
PEx() and for any . As an application, we prove that if is a triple of syzygy finite classes of objects in
satisfying some special conditions, then is
an Igusa-Todorov algebra. Finally, we study lower triangular matrix Artin
algebras and determine in terms of their components, under reasonable
hypothesis, when these algebras are syzygy finite or Igusa-Todorov
Igusa-Todorov functions for radical square zero algebras
We study the behaviour of the Igusa-Todorov functions for radical square zero
algebras. We show that the left and the right -dimensions coincide, in
this case. Some general results are given, but we concentrate more in the
radical square zero algebras. Our study is based on two notions of hearth and
member of a quiver . We give some bounds for the and the
-dimensions and we describe the algebras for which the bound of is
obtained. We also exhibit modules for which the -dimension is realised
The Phi-dimension: A new homological measure
K. Igusa and G. Todorov introduced two functions and which are
natural and important homological measures generalising the notion of the
projective dimension. These Igusa-Todorov functions have become into a powerful
tool to understand better the finitistic dimension conjecture.
In this paper, for an artin -algebra and the Igusa-Todorov function
we characterise the -dimension of in terms either of the
bi-functors or Tor's bi-functors
Furthermore, by using the first characterisation
of the -dimension, we show that the finiteness of the -dimension of
an artin algebra is invariant under derived equivalences. As an application of
this result, we generalise the classical Bongartz's result as follows: For an
artin algebra a tilting -module and the endomorphism algebra
we have that
$\mathrm{Fidim}\,(A)-\mathrm{pd}\,T\leq \mathrm{Fidim}\,(B)\leq
\mathrm{Fidim}\,(A)+\mathrm{pd}\,T.
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