Resolutions by mapping cones

By Jürgen Herzog, Yukihide Takayama. 2001

209 citations recorded by the source corpus.

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

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/861f815feb6fed2077fbb3a3792b3a1636d4c5c4

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Citations in the corpus

Integral closure of ideals, rings, and modules (2006, background): “(See [122] for more general resolutions by mapping cone constructions.”
Simplicial trees are sequentially Cohen–Macaulay ☆ (2003, background): “1 (Herzog and Takayama [10], linear quotients).”
Monomial ideals whose powers have a linear resolution (2003, background): “It is known [ 7 ] that polymatroidal ideals have linear resolutions and that powers of polymatroidal ideals are again polymatroidal (see [2] and [8]).”
How to compute the Stanley depth of a monomial ideal (2007, background): “It is shown in [13] that projdimI = max{m1, .”
Castelnuovo-Mumford regularity of products of ideals (2002, methodology): “Examples of such ideals are stable, and squarefree stable ideals in the sense of Eliahou-Kervaire [10] and Aramova-Herzog-Hibi [1], as well as polymatroidal ideals, as noted in [ 13 ].”
Stanley decompositions and partitionable simplicial complexes (2006, background): “In our case it follows that IΔ∨ is minimally generated by the monomial of the form xi1 ... xim where xi j ∈ supp(v j) for j = 1,..., m. Thus we see that IΔ∨ is the matriodal ideal of the transversal matroid attached to the sets supp(v1),..., supp(vm), see [CoHe, Section 5]. In [ HeTa , Lemma 1.3] an”
Symbolic powers of cover ideal of very well-covered and bipartite graphs (2016, unknown): “It is also known [18] that polymatroidal ideals have linear resolution and that powers of polymatroidal ideals are again polymatroidal (see [16]).”