Restricted Homological Dimensions of Complexes

Abstract

We define and study the notions of restricted Tor-dimension and Ext-dimension for unbounded complexes of left modules over associative rings. We show that, for a right (respectively, left) homologically bounded complex, our definition agrees with the small restricted flat (respectively, injective) dimension defined by Christensen et al. Furthermore, we show that the restricted Tor-dimension defined in this paper is a refinement of the Gorenstein flat dimension of an unbounded complex in some sense. In addition, we give some results concerning restricted homological dimensions under a base change over commutative Noetherian rings.