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Finitistic dimension conjecture and radical-power extensions | | Chengxi Wang
; Changchang Xi
; | | Date: |
1 Sep 2015 | | 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 $Bsubseteq A$ 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, proj.dim$(A_B)$ is finite),
$I A, rad(B)subseteq B$ and fin.dim$(A)$ is finite, then fin.dim$(B)$ is
finite. (2) If $I, rad(B)$ is a left ideal of $A$ and $A$ is
torsionless-finite, then fin.dim$(B)$ is finite. 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 1.3 and 1.4), but also
provide some completely new ways to detect algebras of finite finitistic
dimensions. | | Source: | arXiv, 1509.0125 | | Services: | Forum | Review | PDF | Favorites | |
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