The Φ-Dimension: A New Homological Measure (original) (raw)
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The Phi-dimension: A new homological measure
arXiv: Representation Theory, 2013
K. Igusa and G. Todorov introduced two functions phi\phiphi and psi,\psi,psi, 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 RRR-algebra AAA and the Igusa-Todorov function phi,\phi,phi, we characterise the phi\phiphi-dimension of AAA in terms either of the bi-functors mathrmExtiA(−,−)\mathrm{Ext}^{i}_{A}(-, -)mathrmExtiA(−,−) or Tor's bi-functors mathrmTorAi(−,−).\mathrm{Tor}^{A}_{i}(-,-).mathrmTorAi(−,−). Furthermore, by using the first characterisation of the phi\phiphi-dimension, we show that the finiteness of the phi\phiphi-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,A,A, a tilting AAA-module TTT and the endomorphism algebra B=mathrmEndA(T)op,B=\mathrm{End}_A(T)^{op},B=mathrmEndA(T)op, we have that $\mathrm{Fidim}\,(A)-\mathrm{pd}\,T\leq \mathrm{Fidim...
On the Finitistic Global Dimension Conjecture for Artin Algebras
2005
We find a simple condition which implies finiteness of fini- tistic global dimension for artin algebras. As a consequence we obtain a short proof of the finitistic global dimension conjecture for radical cubed zero algebras. The same condition also holds for algebras of representa- tion dimension less then or equal to three. Hence the finitistic dimension conjecture holds in that
12 Finitistic Dimension Through Infinite Projective Dimension
2012
Abstract. We show that an artin algebra Λ having at most three radical layers of infinite projective dimension has finite finitistic dimension, generalizing the known result for algebras with vanishing radical cube. We also give an equivalence between the finiteness of fin.dim.Λ and the finiteness of a given class of Λ-modules of infinite projective dimension. 1. Introduction. Let Λ be an artin algebra, and consider mod Λ the class of finitely generated left Λ-modules. The finitistic dimension of Λ is then defined to be fin.dim. Λ = sup{pdM: M ∈ mod Λ and pdM < ∞}, where pd M denotes the projective dimension of M. It was conjectured by Bass in
Tilting Theory and the Finitistic Dimension Conjectures
2002
Let R be a right noetherian ring and let P<1 be the class of all nitely presented modules of nite projective dimension. We prove that ndimR = n< 1 i there is an (innitely generated) tilting module T such that pdT = n and T? =( P<1)? .I fR is an artin algebra, then T can be taken to be
Global Dimension of Some Artinian Algebras
arXiv preprint arXiv:1206.3726, 2012
The structure of arbitrary associative commutative unital artinian algebras is well-known: they are finite products of associative commutative unital local algebras pg.351, Cor. 23.12]. In the semi-simple case, we have the Artin-Wedderburn Theorem which states that any semi-simple artinian algebra (which is assumed to be associative and unital but not necessarily commutative) is a direct product of matrix algebras over division rings pg.35, Par. 3.5]. Along these lines, we observe a simple classification of artinian algebras and their representations in Proposition 1.3.2 (hereby referred as the Classification Lemma) in terms of a category in which each object has a local artinian endomorphism algebra. This category is constructed using a fixed set of primitive (not necessarily central) idempotents in the underlying algebra. The Classification Lemma is a version of Freyd's Representation Theorem [4, Sect. 5.3]: from an artinian algebra A we create a category C A on finitely many objects, and then the category of A-modules can be realized as a category of functors which admit C A as their domain. This construction can also be thought as a higher dimensional analogue of the semi-trivial extensions of [10] for artinian algebras.
2008
Let Λ be a hereditary Artin algebra and M a Λ-module that is both a generator and a cogenerator. We are going to describe the possibilities for the global dimension of End(M) in terms of the cardinalities of the Auslander-Reiten orbits of indecomposable Λ-modules. Résumé. Soit Λ une algèbre d'Artin héréditaire et M un Λ-module qui est un générateur-cogénérateur. Nous allons décrire toutes les possibilités pour la dimension globale de End(M)à l'aide des cardinalités des orbites d'Auslander-Reiten des Λ-modules indécomposables.
Finitistic Dimension Conjectures via Gorenstein Projective Dimension
2020
It is a well-known result of Auslander and Reiten that contravariant finiteness of the class P^fin_∞ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, in this work we examine the Gorenstein counterpart of Auslander–Reiten condition, namely contravariant finiteness of the class GP^fin_∞ (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class GP^fin_∞ implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander–Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in ...
An approach to the finitistic dimension conjecture
2007
It was conjectured by H. Bass in the 60's that the projective finitistic dimension pfd(R):=pfd(mathrmmodR)\pfd (R):=\pfd (\mathrm{mod} R)pfd(R):=pfd(mathrmmodR) 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 Psi:mathrmmodRtoBbbN,\Psi:\mathrm{mod} R\to \Bbb{N},Psi:mathrmmodRtoBbbN, which turned out to be useful to prove that pfd(R)\pfd (R)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 mathrmmodR\mathrm{mod} RmathrmmodR instead of a class of algebras, namely to take the class of categories F(theta)\F(\theta)F(theta) of theta\thetatheta-filtered RRR-modules for all stratifying systems (theta,leq)(\theta,\leq)(theta,leq) in mathrmmodR.\mathrm{mod} R.mathrmmodR.