2,809 research outputs found
Reduction numbers and initial ideals
The reduction number r(A) of a standard graded algebra A is the least integer
k such that there exists a minimal reduction J of the homogeneous maximal ideal
m of A such that Jm^k=m^{k+1}. Vasconcelos conjectured that the reduction
number of A=R/I can only increase by passing to the initial ideal, i.e
r(R/I)\leq r(R/in(I)). The goal of this note is to prove the conjecture.Comment: 6 page
Groebner bases for spaces of quadrics of codimension 3
Let be an Artinian standard graded -algebra
defined by quadrics. Assume that and that is algebraically
closed of characteristic . We show that is defined by a Gr\"obner
basis of quadrics with, essentially, one exception. The exception is given by
where is a complete intersection of 3 quadrics not containing
the square of a linear form.Comment: Minor changes, to appear in the J. Pure Applied Algebr
Koszul homology and extremal properties of Gin and Lex
In a polynomial ring with variables, for every homogeneous ideal
and for every we consider the Koszul homology with
respect to a sequence of of generic linear forms and define the
Koszul-Betti number of to be the dimension of the
degree part of . In characteristic 0, we show that the
Koszul-Betti numbers of any ideal are bounded above by those of any gin of
and also by those of the Lex-segment of . We also investigate the set
of all the gin of and show that the Koszul-Betti numbers of any
ideal in are bounded below by those of the gin-revlex of and
present examples showing that in general there is no is such that
the Koszul-Betti numbers of any ideal in are bounded above by those
of .Comment: 21 pages, preprint 200
KRS and determinantal ideals
The first sections contain a survey of the application of the
Knuth-Robinson-Schensted corerspondence to the computation of Groebner bases of
determinantal ideals. We also set up a conceptual framework for this
application in terms of so-called "KRS invariants". Then we show that the
initial ideal of a determinantal ideal "defined by shape" is given by its KRS
image. We furthermore characterize those among these ideals that even have a
Groebner basis of products of minors, and show that they can be characterized
in terms of Greene's KRS invariants. Furthermore it is shown that for the ideal
generated by all t-minors the formation of initial ideal and symbolic power
commutes. The last section contains a discussion of potential KRS invariants
related to so-called 1-cogenerated ideals.Comment: 22 pages, uses epic, eepi
The variety of exterior powers of linear maps
Let be a field and and be -vector spaces of dimension and
. Let be the canonical map from to . We investigate the Zariski closure of the image of
. In the case , is the cone over a Grassmannian,
but is larger than for . We analyze the
G=\GL(V)\times\GL(W)-orbits in via the corresponding -stable prime
ideals. It turns out that they are classified by two numerical invariants, one
of which is the rank and the other a related invariant that we call small rank.
Surprisingly, the orbits in arise from the images for
and simple algebraic operations. In the last section we determine the
singular locus of . Apart from well-understood exceptional cases, it is
formed by the elements of rank in .Comment: Few minor changes. Final version to appear in J. of Algebr
Castelnuovo-Mumford regularity of products of ideals
We discuss the behavior of the Castelnuovo-Mumford regularity under certain
operations on ideals and modules, like products or powers. In particular, we
show that reg(IM) can be larger than reg(M)+reg(I) even when I is an ideal of
linear forms and M is a module with a linear resolution. On the other hand, we
show that any product of ideals of linear forms has a linear resolution. We
also discuss the case of polymatroidal ideals and show that any product of
determinantal ideals of a generic Hankel matrix has a linear resolution.Comment: 14 page
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