12 Finitistic Dimension Through Infinite Projective Dimension (original) (raw)

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

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 ...

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.

Idempotent reduction for the finitistic dimension conjecture

Proceedings of the American Mathematical Society, 2020

In this note, we prove that if Λ \Lambda is an Artin algebra with a simple module S S of finite projective dimension, then the finiteness of the finitistic dimension of Λ \Lambda implies that of ( 1 − e ) Λ ( 1 − e ) (1-e)\Lambda (1-e) where e e is the primitive idempotent supporting S S . We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if Λ \Lambda is the quotient of a path algebra by an admissible ideal I I whose defining relations do not involve a certain arrow α \alpha , then the finitistic dimension of Λ \Lambda is finite if and only if the finitistic dimension of Λ / Λ α Λ \Lambda /\Lambda \alpha \Lambda is finite.

On the Projective Dimension of Artinian Modules

2021

Let (R,mathfrakm)(R, mathfrak{m})(R,mathfrakm) be a Noetherian local ring and MMM, NNN be two finitely generated RRR-modules. In this paper it is shown that RRR is a Cohen-Macaulay ring if and only if RRR admits a non-zero Artinian RRR-module AAA of finite projective dimension; in addition, for all such Artinian RRR-modules AAA, it is shown that mathrmpdR,A=dimRmathrm{pd}_R, A=dim RmathrmpdR,A=dimR. Furthermore, as an application of these results it is shown that$$pdd H^i_{{frak p}R_{frak p}}(M_{frak p}, N_{frak p})leq pd H^{i+dim R/{frak p}}_{frak m}(M,N)$$for each frakpinmathrmSpecR{frak p}in mathrm{Spec} RfrakpinmathrmSpecR and each integer igeq0igeq 0igeq0. This result answers affirmatively a question raised by the present authors in [13].

Global Dimensions of Some Artinian Algebras

In this article we obtain lower and upper bounds for global dimensions of a class of artinian algebras in terms of global dimensions of a finite subset of their artinian subalgebras. Finding these bounds for the global dimension of an artinian algebra AAA is realized via an explicit algorithm we develop. This algorithm is based on a directed graph (not the Auslander-Reiten quiver) we construct, and it allows us to decide whether an artinian algebra has finite global dimension in good number of cases.

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.

Projective dimensions and Nakayama algebras

"Representations of algebras and related topics"

Institutt for matematiske fag, NTNU NO-7491 Trondheim Norway dagma@math.ntnu.no Dedicated to Professor Vlastimil Dlab on the occation of his sixtieth birthday.

Approximations of injective modules and finitistic dimension

Communications in Algebra

Let Λ be an artin algebra and let P <∞ Λ the category of finitely generated right Λ-modules of finite projective dimension. We show that P <∞ Λ is contravariantly finite in mod Λ if and only if the direct sum E of the indecomposable Ext-injective modules in P <∞ Λ form a tilting module in mod Λ. Moreover, we show that in this case E coincides with the direct sum of the minimal right P <∞ Λ-approximations of the indecomposable Λ-injective modules and that the projective dimension of E equal to the finitistic dimension of Λ.