1,495 research outputs found
Bass Numbers of Semigroup-Graded Local Cohomology
Given a module M over a ring R which has a grading by a semigroup Q, we
present a spectral sequence that computes the local cohomology of M at any
Q-graded ideal I in terms of Ext modules. This method is used to obtain
finiteness results for the local cohomology of graded modules over semigroup
rings; in particular we prove that for a semigroup Q whose saturation is
simplicial, the Bass numbers of such local cohomology modules are finite.
Conversely, if the saturation of Q is not simplicial, one can find a graded
ideal I and a graded R-module M whose local cohomology at I in some degree has
an infinite-dimensional socle. We introduce and exploit the combinatorially
defined essential set of a semigroup.Comment: 19 pages LaTeX, 1 figure (.eps) Definition 5.1 corrected;
transcription error in Theorem 7.1.3 fixe
Computing toric degenerations of flag varieties
We compute toric degenerations arising from the tropicalization of the full
flag varieties and embedded in a
product of Grassmannians. For and we
compare toric degenerations arising from string polytopes and the FFLV polytope
with those obtained from the tropicalization of the flag varieties. We also
present a general procedure to find toric degenerations in the cases where the
initial ideal arising from a cone of the tropicalization of a variety is not
prime.Comment: 35 pages, 6 figure
Algorithms for graded injective resolutions and local cohomology over semigroup rings
AbstractLet Q be an affine semigroup generating Zd, and fix a finitely generated Zd-graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Zd-graded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modules HIi(M) supported on any monomial (that is, Zd-graded) ideal I. Since these local cohomology modules are neither finitely generated nor finitely cogenerated, part of this task is defining a finite data structure to encode them
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