19 research outputs found
Lower bounds on Betti numbers
We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these
On generators of bounded ratios of minors for totally positive matrices
We provide a method for factoring all bounded ratios of the form where is a
totally positive matrix, into a product of more elementary ratios each of which
is bounded by 1, thus giving a new proof of Skandera's result. The approach we
use generalizes the one employed by Fallat et al. in their work on principal
minors. We also obtain a new necessary condition for a ratio to be bounded for
the case of non-principal minors.Comment: 27 pages, 2 figure
LOWER BOUNDS ON BETTI NUMBERS
We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these
On the growth of deviations
The deviations of a graded algebra are a sequence of integers that determine
the Poincare series of its residue field and arise as the number of generators
of certain DG algebras. In a sense, deviations measure how far a ring is from
being a complete intersection. In this paper we study extremal deviations among
those of algebras with a fixed Hilbert series. In this setting, we prove that,
like the Betti numbers, deviations do not decrease when passing to an initial
ideal and are maximized by the Lex-segment ideal. We also prove that deviations
grow exponentially for Golod rings and for certain quadratic monomial algebras.Comment: Corrected some minor typos in the version published in PAM
Edge ideals and DG algebra resolutions
Let where and is a homogeneous ideal in
. The acyclic closure of over is a DG algebra
resolution obtained by means of Tate's process of adjoining variables to kill
cycles. In a similar way one can obtain the minimal model , a DG algebra
resolution of over . By a theorem of Avramov there is a tight connection
between these two resolutions. In this paper we study these two resolutions
when is the edge ideal of a path or a cycle. We determine the behavior of
the deviations , which are the number of variables in
in homological degree . We apply our results to the
study of the -algebra structure of the Koszul homology of