A finiteness theorem in homological algebra

Abstract

In (1), (2), (3) and (4) it is shown that homological algebra (5) can be applied to stable homotopy-theory. In this application, we deal with A-modules, where A is the mod p Steenrod algebra. In the present paper, we shall prove a finiteness theorem for the cohomology of the Steenrod algebra. This theorem is stated as Corollary 2 below. It is purely algebraic, but it is not claimed that it has any algebraic interest; it is inspired solely by the application mentioned above. Here it has the following uses.