4,213 research outputs found
Gr\"obner bases of syzygies and Stanley depth
Let F. be a any free resolution of a Z^n-graded submodule of a free module
over the polynomial ring K[x_1, ..., x_n]. We show that for a suitable term
order on F., the initial module of the p'th syzygy module Z_p is generated by
terms m_ie_i where the m_i are monomials in K[x_{p+1}, ..., x_n]. Also for a
large class of free resolutions F., encompassing Eliahou-Kervaire resolutions,
we show that a Gr\"obner basis for Z_p is given by the boundaries of generators
of F_p. We apply the above to give lower bounds for the Stanley depth of the
syzygy modules Z_p, in particular showing it is at least p+1. We also show that
if I is any squarefree ideal in K[x_1, ..., x_n], the Stanley depth of I is at
least of order the square root of 2n.Comment: 13 page
On the Stanley Depth of Squarefree Veronese Ideals
Let be a field and . In 1982, Stanley defined what is
now called the Stanley depth of an -module , denoted \sdepth(M), and
conjectured that \depth(M) \le \sdepth(M) for all finitely generated
-modules . This conjecture remains open for most cases. However, Herzog,
Vladoiu and Zheng recently proposed a method of attack in the case when with being monomial -ideals. Specifically, their method
associates with a partially ordered set. In this paper we take advantage of
this association by using combinatorial tools to analyze squarefree Veronese
ideals in . In particular, if is the squarefree Veronese ideal
generated by all squarefree monomials of degree , we show that if , then \sdepth(I_{n,d})= \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d,
and if and , then d+3\le \sdepth(I_{n,d}) \le
\floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d.Comment: 10 page
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