Amplitude inequalities for Differential Graded modules

Abstract

Differential Graded Algebras can be studied through their Differential Graded modules. Among these, the compact ones attract particular attention. This paper proves that over a suitable chain Differential Graded Algebra R, each compact Differential Graded module M satisfies amp M >= amp R, where amp denotes amplitude which is defined in a straightforward way in terms of the homology of a DG module. In other words, the homology of each compact DG module M is at least as long as the homology of R itself. Conversely, DG modules with shorter homology than R are not compact, and so in general, there exist DG modules with finitely generated homology which are not compact. Hence, in contrast to ring theory, it makes no sense to define finite global dimension of DGAs by the condition that each DG module with finitely generated homology must be compact.