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Register a new userFinite group characters on free resolutions
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Authors
Federico Galetto
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Under reasonable assumptions, a group action on a module extends to the minimal free resolutions of the module. Explicit descriptions of these actions can lead to a better understanding of free resolutions by providing, for example, convenient expressions for their differentials or alternative characterizations of their Betti numbers. This article introduces an algorithm for computing characters of finite groups acting on minimal free resolutions of finitely generated graded modules over polynomial rings.
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We present some partial results regarding subadditivity of maximal shifts in finite graded free resolutions.
Let $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$ are the binomials in $I$. We show that for an appropriate choice of bases every $A$-homogeneous minimal free resolution of $R/I$ is simple. We introduce the gcd-complex $D_{gcd}(bf b)$ for a degree $mathbf{b}in A$. We show that the homology of $D_{gcd}(bf b)$ determines the $i$-Betti numbers of degree $bf b$. We discuss the notion of an indispensable complex of $R/I$. We show that the Koszul complex of a complete intersection lattice ideal $I$ is the indispensable resolution of $R/I$ when the $A$-degrees of the elements of the generating $R$-sequence are incomparable.
We show that the property of a standard graded algebra R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module corresponding to any degree sequence of length at most depth(R). We also give a relation in terms of graded Betti numbers, called the Herzog-Kuhl equations, for a pure R-module M to satisfy the condition dim(R) - depth(R) = dim(M) - depth(M). When R is Cohen-Macaulay, we prove an analogous result characterizing all graded Cohen-Macaulay R-modules.
Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d=2), and strictly contains the class of `strongly stable hypergraphs corresponding to pure shifted simplicial complexes. The polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms, and are realized as subcomplexes of mixed subdivisions of the Minkowski sums of simplices. Resolutions of more general hypergraphs are obtained by considering decompositions into cointerval hypergraphs.
An explicit combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic $0$ is defined canonically, without any choices, using higher-dimensional generalizations of combined spanning trees for cycles and cocycles (hedges) in the upper Koszul simplicial complexes of $I$ at lattice points in $mathbb{Z}^n$. The differentials in these sylvan resolutions are expressed as matrices whose entries are sums over lattice paths of weights determined combinatorially by sequences of hedges (hedgerows) along each lattice path. This combinatorics enters via an explicit matroidal expression for the Moore-Penrose pseudoinverses of the differentials in any CW complex as weighted averages of splittings defined by hedges. This Hedge Formula also yields a projection formula from CW chains to boundaries. The translation from Moore-Penrose combinatorics to free resolutions relies on Wall complexes, which construct minimal free resolutions of graded ideals from vertical splittings of Koszul bicomplexes. The algebra of Wall complexes applied to individual hedgerows yields explicit but noncanonical combinatorial minimal free resolutions of arbitrary monomial ideals in any characteristic.
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