On the Notion of Precohomology

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+l 0 d q 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 limzt, called a prein­ verse limit, is introduced, We discuss some of the links between precohomology and preinverse limit. Introduction Let Z be the ring of integers and let A be an abelian category. Suppose a sequence of objects and morphisms in A is given dq-1 dq dq+l -~ Cq-l -~-~ Cq -~ Cq+l --~ which may not satisfy dq 0 dq-l = 0 for certain q E Z. Then one may not be able to take the cohomology at Cq. We will introduce a functor Received September 4, 1984, Revised November 28, 1984, * AMS Subject Classification (1980): Primary 18G40, 18G35; Secondary 18E25. Key vVords and Phrases: Precohomology, Hyperprecohomology of a left exact functor, Preinverse limit.