Finitistic dimension and restricted flat dimension

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].