Published in Portugaliae Mathematica, Volume 43, Issue 3, January 1, 1985, pages 307-316. Copyright © 1985 European Mathematical Society (EMS).
ABSTRACT. For a cochain complex one can have the cohomology functor. In this paper we introduce the notion of precohomology for a cochain that is not a complex, i. e., dq+1 o dq may not be zero. Such a cochain, with objects and morphisms of an abelian category A, is called a cochain precomplex whose category is denoted by Pco (A). If a cochain precomplex is actually a cochain complex, then the notion of precohomology coincides with that of cohomology, i. e., precohomology is a gene¬ralization of cohomology. For a left exact functor F from an abelian category A to an abelian category B, the hyperprecohomology of F is defined, and some properties are given. In the last section, a generalization of an inverse limit, called a prein¬verse limit, is introduced. We discuss some of the links between precohomology and preinverse limit.