On the finitistic dimension conjecture II: Related to finite global dimension
By Changchang Xi. 2006
74 citations recorded by the source corpus.
Abstract In this paper, we study the finitistic dimensions of artin algebras by establishing a relationship between the global dimensions of the given algebras, on the one hand, and the finitistic dimensions of their subalgebras, on the other hand. This is a continuation of the project in [J. Pure Appl. Algebra 193 (2004) 287–305]. For an artin algebra A we denote by gl . dim ( A ) , fin . dim ( A ) and rep . dim ( A ) the global dimension, finitistic dimension and representation dimension of A , respectively. The Jacobson radical of A is denoted by rad ( A ) . The main results in the paper are as follows: Let B be a subalgebra of an artin algebra A such that rad ( B ) is a left ideal in A . Then (1) if gl . dim ( A ) ⩽ 4 and rad ( A ) = rad ( B ) A , then fin . dim ( B ) ∞ . (2) If rep . dim ( A ) ⩽ 3 , then fin . dim ( B ) ∞ . The results are applied to pullbacks of algebras over semi-simple algebras. Moreover, we have also the following dual statement: (3) Let ϕ : B ⟶ A be a surjective homomorphism between two algebras B and A . Suppose that the kernel of ϕ is contained in the socle of the right B -module B B . If gl . dim ( A ) ⩽ 4 , or rep . dim ( A ) ⩽ 3 , then fin . dim ( B ) ∞ . Finally, we provide a class of algebras with representation dimension at most three: (4) If A is stably hereditary and rad ( B ) is an ideal in A , then rep . dim ( B ) ⩽ 3 .
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/a513c1c2295da3b8ce9d77025d29456b9bbdc0dd
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Citations in the corpus
- On the finitistic dimension conjecture, III: Related to the pair eAe⊆A (2008, background): “For further information about advances on the finitistic dimension conjecture as well as its relationship with other homological conjectures in the representation theory of Artin algebras, we may refer to [7, 29 ,30,33,34] and the references therein.”
- Constructions of stable equivalences of Morita type for finite‐dimensional algebras III (2007, unknown): “For further information on representation dimension we refer the reader to the original paper [1] or the recent papers [15, 16 ] and the references therein.”
- Finitistic dimension and Igusa-Todorov algebras (2009, methodology): “We refer the reader to the paper [19] for algebras satisfying the conditions in the proposition.”