Article ID Journal Published Year Pages File Type
4587925 Journal of Algebra 2008 12 Pages PDF
Abstract

We investigate the relations between finitistic dimensions and restricted flat dimensions (introduced by Foxby [L.W. Christensen, H.-B. Foxby, A. Frankild, Restricted homological dimensions and Cohen–Macaulayness, J. Algebra 251 (1) (2002) 479–502]). In particular, we show the following result. (1) If T is a selforthogonal left module over a left noetherian ring R with the endomorphism ring A, then . (2) If is classical partial tilting, then . (3) If A=A0⊆A1⊆⋯⊆Am=R are Artin algebras with the same identity such that, for each 0⩽i⩽m−1, radAi is a right ideal in Ai+1 and rfd(Ai+1Ai)<∞ (e.g., Ai+1Ai is of finite projective dimension, or finite Gorenstein projective dimension, or finite Tor-bound dimension), then implies . As applications, we disprove Foxby's conjecture [H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004) 167–193] on restricted flat dimensions by providing a counterexample and give a partial answer to a question posed by Mazorchuk [V. Mazorchuk, On finitistic dimension of stratified algebras, arXiv:math.RT/0603179, 6.4].

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory