23 research outputs found
Artinian Gorenstein algebras with linear resolutions
Fix a pair of positive integers d and n. We create a ring R and a complex G
of R-modules with the following universal property. Let P be a polynomial ring
in d variables over a field and let I be a grade d Gorenstein ideal in P which
is generated by homogeneous forms of degree n. If the resolution of P/I by free
P-modules is linear, then there exists a ring homomorphism from R to P such
that P tensor G is a minimal homogeneous resolution of P/I by free P-modules.
Our construction is coordinate free
Bounds for the Multiplicity of Gorenstein algebras
We prove upper bounds for the Hilbert-Samuel multiplicity of standard graded
Gorenstein algebras. The main tool that we use is Boij-S\"oderberg theory to
obtain a decomposition of the Betti table of a Gorenstein algebra as the sum of
rational multiples of symmetrized pure tables. Our bound agrees with the one in
the quasi-pure case obtained by Srinivasan [J. Algebra, vol.~208, no.~2,
(1998)]
Gorenstein Hilbert Coefficients
We prove upper and lower bounds for all the coefficients in the Hilbert
Polynomial of a graded Gorenstein algebra with a quasi-pure resolution
over . The bounds are in terms of the minimal and the maximal shifts in the
resolution of . These bounds are analogous to the bounds for the
multiplicity found in \cite{S} and are stronger than the bounds for the Cohen
Macaulay algebras found in \cite{HZ}.Comment: 20 page
The Scarf complex and betti numbers of powers of extremal ideals
This paper is concerned with finding bounds on betti numbers and describing
combinatorially and topologically (minimal) free resolutions of powers of
ideals generated by a fixed number of square-free monomials. Among such
ideals, we focus on a specific ideal , which we call {\it
extremal}, and which has the property that for each the betti numbers
of are an upper bound for the betti numbers of for
any ideal generated by square-free monomials (in any number of
variables). We study the Scarf complex of the ideals and
use this simplicial complex to extract information on minimal free resolutions.
In particular, we show that has a minimal free resolution
supported on its Scarf complex when or when , and we
describe explicitly this complex. For any and , we also show that
is the smallest possible, or in other words equal
to the number of edges in the Scarf complex. These results lead to effective
bounds on the betti numbers of , with as above. For example, we obtain
that pd for all ideals generated by square-free monomials
and any