An approach to the finitistic dimension conjecture (original) (raw)

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

A model structure approach to the finitistic dimension conjectures

Mathematische Nachrichten, 2012

We explore the interlacing between model category structures attained to classes of modules of finite X -dimension, for certain classes of modules X . As an application we give a model structure approach to the Finitistic Dimension Conjectures and present a new conceptual framework in which these conjectures can be studied.

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 ...

Finitistic Dimension and a Homological Generalization of Semi-Primary Rings

Transactions of the American Mathematical Society, 1960

Introduction. If P is a ring and M a left P-module, then homological algebra attaches three dimensions to M, projective, weak, and injective(1)-By taking the supremum of one of these dimensions as M ranges over all left P-modules, one obtains one of the left "global" dimensions of R. Auslander and Buchsbaum

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.

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

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...

Applications of stratifying systems to the finitistic dimension

Journal of Pure and Applied Algebra, 2006

Given an Ext-injective stratifying system of-modules (, Y ,) satisfying that the projective dimension of Y is finite, we prove that the finitistic dimension of the algebra is equal to the finitistic dimension of the category I() = {X ∈ mod : Ext 1 (−, X)| F() = 0}. Moreover, using the theory of stratifying systems we obtain bounds for the finitistic dimension of. In particular, we get the optimal bound 2n − 2 for the finitistic dimension of a standardly stratified algebra with n simples.

The Φ-Dimension: A New Homological Measure

2016

In [15], K. Igusa and G. Todorov introduced two functions φ and ψ, 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 R-algebra A and the Igusa-Todorov function φ, we characterise the φ-dimension of A in terms either of the bi-functors Ext i A (−, −) or Tor's bi-functors Tor A i (−, −). Furthermore, by using the first characterisation of the φ-dimension, we show that the finiteness of the φdimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz's result [3, Corollary 1] as follows: For an artin algebra A, a tilting A-module T and the endomorphism algebra B = End A (T) op , we have that φ dim (A) − pd T ≤ φ dim (B) ≤ φ dim (A) + pd T.