We construct a long exact sequence involving the homology of an FI-module. Using the long exact sequence, we give two methods to bound the Castelnuovo–Mumford regularity of an FI-module which is generated and related in finite degree. We also prove that for an FImodule which is generated and related in finite degree, if it has a nonzero higher homology, then its homological degrees are strictly...

Recall that the (Mumford-Castelnuovo) regularity of M is the least integer ρ such that for each i all free generators of Fi lie in degree ≤ i + ρ, that is βi,j = 0, for j > i + ρ. In terms of Macaulay [Mac] regularity is the number of rows in the diagram produced by the “betti” command. A Betti number βi,j 6= 0 will be called extremal if βl,r = 0 for all l ≥ i and r ≥ j + 1, that is if βi,j is ...

We characterize all graphs whose binomial edge ideals have a linear resolution. Indeed, we show that complete graphs are the only graphs with this property. We also compute some graded components of the first Betti number of the binomial edge ideal of a graph with respect to the graphical terms. Finally, we give an upper bound for the Castelnuovo-Mumford regularity of the binomial edge ideal of...

We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity Ext and Tor, with respect to homological degree, over complete intersection rings. derive from a theorem Gulliksen linearity result for modules in high degrees. show similar under additional hypothesis that enough Tor are supported dimension at most one; we then provide examples showing could be pretty hectic when latter ...

We call a vertex x of a graph G = (V,E) a codominated vertex if NG[y] ⊆ NG[x] for some vertex y ∈ V \{x}, and a graph G is called codismantlable if either it is an edgeless graph or it contains a codominated vertex x such that G − x is codismantlable. We show that (C4, C5)-free vertex-decomposable graphs are codismantlable, and prove that if G is a (C4, C5, C7)-free well-covered graph, then ver...

Let A be a noetherian AS regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and grA the category of finitely generated graded left A-modules. Following Jørgensen, we define the CastelnuovoMumford regularity reg(M•) of a complex M• ∈ D(grA) in terms of the local cohomologies or the minimal projective resolution of M•. Let A be the quadratic dual ring of A. ...

In this article we study the structure of residual intersections via constructing a finite complex which is acyclic under some sliding depth conditions on the cycles of the Koszul complex. This complex provides information on an ideal which coincides with the residual intersection in the case of geometric residual intersection; and is closely related to it in general. A new success obtained thr...

Castelnuovo-Mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules. A recent result of Doering, Gunston, Vasconcelos shows that both can be compared in case of a graded algebra. We extend this result to modules and analyze when the estimate is in fact an equality. A complete classification is obtained i...