Finitistic dimension conjecture and radical-power extensions

Abstract

The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field should have finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this paper, we investigate those extensions of Artin algebras in which some radical-power of smaller algebras is a one-sided ideal in bigger algebras. Our results, however, are formulated more generally for an arbitrary ideal: Let B ⊆ A be an extension of Artin algebras and I an ideal of B such that the full subcategory of B/I-modules is B-syzygy-finite. Then: (1) If the extension is right-bounded (for example, pd (A B ) < ∞), I A rad(B) ⊆ B and fin.dim (A) < ∞, then fin.dim (B) < ∞.(2) If I rad(B) is a left ideal of A and A is torsionless-finite, then fin.dim (B) < ∞. Particularly, if I is specified to a power of the radical of B, then our results not only generalize some ones in the literature (see Corollaries 1.2 and 1.4), but also provide some completely new ways to detect algebras of finite finitistic dimensions.

Finitistic dimension conjecture and extensions of algebras

“…Here, we understand A 1 /I s−1 · · · I 1 = 0 if s = 1, which means that A 1 /I s−1 · · · I 1 being B-syzygy-finite always holds. Let us remark that Lemma 5.4 recovers [19, Theorem 3.1] if we take s = 1, and extends [15,Corollary 3.10] if we take s = 2.…”

“…In this direction, Wei gave in [16,Theorem 2.9] an affirmative answer under some homological conditions. In this section, we shall give a partial answer for the case s > 2 by imposing the condition concerning syzygy-finite algebras, which generalizes some results in [19,15].…”

Reduction techniques for the finitistic dimension

“…Clearly, this result is the predecessor of the vertex removal operation. More recently, Xi in a series of papers [23][24][25][26] introduced various methods for detecting finiteness of the finitistic dimension. It should be noted that Xi has connected the finitistic dimension of an algebra of the form eΛe, where e is an idempotent, with other homological dimensions, for instance, finiteness of the global dimension of Λ (less or equal to four), finiteness of the representation dimension of Λ/ΛeΛ (less or equal to three) and several other interesting relations.…”

“…At this point we would like to mention Xi's comment [23] on the available techniques that we have for the finitistic dimension. Very briefly, he writes that "not many practical methods are available so far to detect algebras of finite finitistic dimension and it is necessary to develop some methods even for some concrete examples".…”

Ring Constructions and Generation of the Unbounded Derived Module Category

“…One such approach is to consider a collection of related rings and ask whether, if some of the rings have a property, do the others as well. For example, this method has been applied to the finitistic dimension conjecture with recollements of derived categories [10,16], ring homomorphisms [35,[37][38][39], and operations defined on the quiver of a quiver algebra [8,13,14]. We apply the same philosophy to the property 'injectives generate'.…”