A generalized Koszul complex. I

Abstract

DAVID A. BUCHSBAUM (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