WEEKLY ACTIVITIES

Time: 9h00, Wednesday, February 27, 2019
Location: Rom 612, Building A6, Institute of Mathematics

Abstract: Let $S=mathbb{K}[x_1,dots,x_n]$ be the polynomial ring in $n$ variables over a field $mathbb{K}$ and suppose that $M$ is a nonzero finitely generated $mathbb{Z}^n$-graded $S$-module. Let $uin M$ be a homogeneous element and $Zsubseteq {x_1,dots,x_n}$. The $mathbb {K}$-subspace $umathbb{K}[Z]$ generated by all elements $uv$ with $vin mathbb{K}[Z]$ is called a {it Stanley space} of dimension $|Z|$, if it is a free $mathbb{K}[Z]$-module. A decomposition $mathcal{D}$ of $M$ as a finite direct sum of Stanley spaces is called a {it Stanley decomposition} of $M$. The minimum dimension of a Stanley space in $mathcal{D}$ is called the {it Stanley depth} of $mathcal{D}$ and is denoted by ${rm sdepth} (mathcal {D})$.

The quantity $${rm sdepth}(M):=maxbig{{rm sdepth}(mathcal{D})mid mathcal{D} {rm is a Stanleydecomposition of} Mbig}$$ is called the {it Stanley depth} of $M$. We say that a $mathbb{Z}^n$-graded $S$-module $M$ satisfies the {it Stanley's inequality} if
$${rm depth}(M) leq {rm sdepth}(M).$$

In fact, in 1982 Stanley conjectured that every $mathbb{Z}^n$-graded $S$-module satisfies the Stanley's inequality.
This conjecture has been disproved by Duval, Goeckner, Klivans and Martin. However it is still interesting to find classes of modules which satisfy the Stanley's inequality.

It is a general philosophy that high powers of ideals have nice homological behavior. Thus, one would expect that the Stanley's inequality could be true for high powers of an ideal. In this talk we focus on this question and review the recent developments in this regard.