Resolutions by mapping cones

Many well-known free resolutions arise as iterated mapping cones. Prominent examples are the Eliahou-Kervaire resolution of stable monomial ideals (as noted by Evans and Charalambous [10]), and the Taylor resolution. The idea of the iterated mapping cone construction is the following: Let I ⊂ R be an ideal generated by f1, . . . , fn, and set Ij = (f1, . . . , fj). Then for j = 1, . . . , n there are exact sequences 0 −→ R/(Ij−1 : fj) −→ R/Ij−1 −→ R/Ij −→ 0. Assuming that the free R-resolution F of R/Ij−1 is already known, and a free Rresolution G of R/(Ij−1 : fj) is also known, one obtains the resolution of R/Ij as a mapping cone of a complex homomorphism ψ : G → F which is a lifting of R/(Ij−1 : fj) → R/Ij−1. Of course one cannot expect that such a resolution will be minimal in general. However this construction yields an inductive procedure to compute a resolution of R/I provided for each j, a resolution of R/(Ij−1 : fj) is known as well as the comparison map ψ. So it is natural to consider classes of ideals for which the colon ideals Ij−1 : fj are generated by regular sequences. But even in this nice case it still hard to construct the comparison maps. In the first section of this paper we therefore restrict ourselves to the case that I is a monomial ideal in a polynomial ring, and that the colon ideals in question are generated by subsets of the variables. In this case we say that I has linear quotients. At a first glance these hypotheses seem to be very restrictive. On the other hand, there are many interesting examples of such ideals. All stable and squarefree stable ideals belong to this class, as well as all matroidal ideals (see Section 1 for the definitions). It is easy to see that I has linear quotients, if and only if the first syzygy module of I has a quadratic Gröbner basis, which, in case of a Stanley ideal I∆ attached to the simplicial complex ∆, is equivalent to saying that the Alexander dual ∆ of ∆ is nonpure shellable. This fact was communicated to us by Sköldberg. It is also easy to see that I has linear quotients if and only if I satisfies condition (4.1) of Batzies and Welker [3], a condition which the authors call shellable. It is clear that our approach only requires to describe the comparison maps in order to compute explicit free resolutions of ideals with linear quotients as iterated mapping cones. Our description of the comparison maps is modeled after Eliahou and Kervaire and is based on decomposition functions. A function g which assigns to each monomial in I (in a natural way) a monomial generator of I, see 1.7, is called a decomposition function. If it satisfies a certain additional condition which is described in Definition 1.9, then we call it regular. Stable and squarefree stable