scholarly journals Idempotent reduction for the finitistic dimension conjecture

2020 ◽  
Vol 148 (5) ◽  
pp. 1891-1900
Author(s):  
Diego Bravo ◽  
Charles Paquette
Keyword(s):  
Finitistic Dimension ◽  
Dimension Conjecture

scholarly journals On the finitistic dimension conjecture of Artin algebras

2008 ◽  
Vol 320 (1) ◽  
pp. 253-258 ◽  
Author(s):  
Aiping Zhang ◽  
Shunhua Zhang
Keyword(s):  
Finitistic Dimension ◽  
Artin Algebras ◽  
Dimension Conjecture

A geometric approach to the finitistic dimension conjecture

1996 ◽  
Vol 67 (6) ◽  
pp. 448-456 ◽  
Author(s):  
F. H. Membrillo-Hern�ndez ◽  
L. Salmer�n
Keyword(s):  
Geometric Approach ◽  
Finitistic Dimension ◽  
Dimension Conjecture

THE FINITISTIC DIMENSION CONJECTURE AND RELATIVELY PROJECTIVE MODULES

2013 ◽  
Vol 15 (02) ◽  
pp. 1350004 ◽  
Author(s):  
CHANGCHANG XI ◽  
DENGMING XU
Keyword(s):  
Left Ideal ◽  
General Setting ◽  
Projective Dimension ◽  
Homological Dimension ◽  
Projective Modules ◽  
Finitistic Dimension ◽  
Artin Algebra ◽  
Finite Dimensional ◽  
New Formulation ◽  
Dimension Conjecture

The famous finitistic dimension conjecture says that every finite-dimensional 𝕂-algebra over a field 𝕂 should have finite finitistic dimension. This conjecture is equivalent to the following statement: If B is a subalgebra of a finite-dimensional 𝕂-algebra A such that the radical of B is a left ideal in A, and if A has finite finitistic dimension, then B has finite finitistic dimension. In the paper, we shall work with a more general setting of Artin algebras. Let B be a subalgebra of an Artin algebra A such that the radical of B is a left ideal in A. (1) If the category of all finitely generated (A, B)-projective A-modules is closed under taking A-syzygies, then fin.dim (B) ≤ fin.dim (A) + fin.dim (BA) + 3, where fin.dim (A) denotes the finitistic dimension of A, and where fin.dim (BA) stands for the supremum of the projective dimensions of those direct summands of BA that have finite projective dimension. (2) If the extension B ⊆ A is n-hereditary for a non-negative integer n, then gl.dim (A) ≤ gl.dim (B) + n. Moreover, we show that the finitistic dimension of the trivially twisted extension of two algebras of finite finitistic dimension is again finite. Also, a new formulation of the finitistic dimension conjecture in terms of relative homological dimension is given. Our approach in this paper is completely different from the one in our earlier papers.

Cut cotorsion pairs

2021 ◽  
pp. 1-38
Author(s):  
Mindy Huerta ◽  
Octavio Mendoza ◽  
Marco A. Pérez
Keyword(s):  
General Framework ◽  
Abelian Category ◽  
Triangulated Categories ◽  
Finitistic Dimension ◽  
Coherent Sheaves ◽  
Chain Complexes ◽  
Cotorsion Pairs ◽  
Gorenstein Homological Algebra ◽  
Serre Subcategories ◽  
Dimension Conjecture

Abstract We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

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