A Generalized Koszul Complex. I

Introduction. In [1], the Koszul complex was used to study the relationship between codimension and multiplicity. It also helped us investigate Macaulay modules and rings, and provided a context in which to prove the Cohen-Macaulay Theorem concerning the unmixedness of complete intersections. Now there is a generalization of the Cohen-Macaulay Theorem (known, we believe, as the generalized Cohen-Macaulay Theorem) which has to do with the unmixedness of an ideal generated by the minors of a matrix. Since the Koszul complex can be thought of as a complex associated with a map of R" -* R, where R is a commutative ring, it seemed likely that there should be a complex associated with a map of Rm-> R". In fact, as long as one is willing to go that far, why not look for some complex associated with an arbitrary map of modules over any commutative ring. In this paper, we define a complex associated with a map of modules. In §1, we discuss this complex in complete generality. In §2, we pretty much restrict our attention to maps of Rm -* R", and establish many of the formal properties of the complexes attached to this map which will be needed in the rest of this paper as well as in subsequent ones. §§3 and 4 are included here to give an indication of how the notion of E-sequence may be generalized, and to show that over local rings this generalized notion of £-sequence is (as in the usual case) independent of order. In subsequent papers we shall investigate these more general ideas, relating them to each other (as in [1]) and also applying them to the case of the singular variety of a variety which is defined by the ideal generated by the minors of the Jacobian matrix. It might also be mentioned that the usual Koszul complex plays a role in studying the invariant factors [2] of certain special modules, and we shall subsequently show the connection between the homology groups of the complex associated with a map/:Rm-> R", and the invariant factors of the cokernel of/ Another reason for looking at complexes associated with maps/:Rm-> R" is the following. If R is noetherian, and coker / has finite length, then coker Sp(f) has finite length for every p, where Sp(f) :SP(R'") -* Sp(R") denotes the induced map