696 research outputs found
Computing invariants of algebraic group actions in arbitrary characteristic
Let G be an affine algebraic group acting on an affine variety
X. We present an algorithm for computing generators of the invariant ring
K[X]^G in the case where G is reductive. Furthermore, we address the case where
G is connected and unipotent, so the invariant ring need not be finitely
generated. For this case, we develop an algorithm which computes K[X]^G in
terms of a so-called colon-operation. From this, generators of K[X]^G can be
obtained in finite time if it is finitely generated. Under the additional
hypothesis that K[X] is factorial, we present an algorithm that finds a
quasi-affine variety whose coordinate ring is K[X]^G. Along the way, we develop
some techniques for dealing with non-finitely generated algebras. In
particular, we introduce the finite generation locus ideal.Comment: 43 page
Koszul algebras and regularity
This is a survey paper on commutative Koszul algebras and Castelnuovo-Mumford
regularity. We describe several techniques to establish the Koszulness of
algebras. We discuss variants of the Koszul property such as strongly Koszul,
absolutely Koszul and universally Koszul. We present several open problems
related with these notions and their local variants
Generalized multiplicities of edge ideals
We explore connections between the generalized multiplicities of square-free
monomial ideals and the combinatorial structure of the underlying hypergraphs
using methods of commutative algebra and polyhedral geometry. For instance, we
show the -multiplicity is multiplicative over the connected components of a
hypergraph, and we explicitly relate the -multiplicity of the edge ideal of
a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of
its special fiber ring. In addition, we provide general bounds for the
generalized multiplicities of the edge ideals and compute these invariants for
classes of uniform hypergraphs.Comment: 24 pages, 6 figures. The results of Theorem 4.6 and Theorem 9.2 are
now more general. To appear in Journal of Algebraic Combinatoric
Hilbert series of modules over Lie algebroids
We consider modules over Lie algebroids which are of
finite type over a local noetherian ring . Using ideals such
that and the length we can define in a natural way the Hilbert series of
with respect to the defining ideal . This notion is in particular studied
for modules over the Lie algebroid of -linear derivations that preserve an ideal , for example when
, the ring of convergent power series. Hilbert series over
Stanley-Reisner rings are also considered.Comment: 42 pages. This is a substantial revision of the previous versio
Moment-angle complexes, monomial ideals, and Massey products
Associated to every finite simplicial complex K there is a "moment-angle"
finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth,
compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study
the cohomology ring, the homotopy groups, and the triple Massey products of a
moment-angle complex, relating these topological invariants to the algebraic
combinatorics of the underlying simplicial complex. Applications to the study
of non-formal manifolds and subspace arrangements are given.Comment: 30 pages. Published versio
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