Conference abstracts

Session S10 - Homological Methods

July 28, 16:00 ~ 16:30

## On Unimodality of Hilbert Function Graded Algebras

### University of Missouri, USA   -   srinivasanh@missouri.edu

We will survey the problem of determining Unimodality of Hilbert Functions especially for graded algebras of small codimensions. Let $R= \oplus R_n$ be a standard graded algebra over a field $k$ an $I$ be a homogeneous ideal so that $S = R/I = \oplus S_n$ is a graded algebra of dimension zero. Then the Hilbert function of $R/I$, denoted by $h_{I}(n) = h_S(n) = dim_k S_n$ is a function such that $h_S(0) = 1, h_S(1) = e$, the embedding dimension of $S$ and $h_S(n) = 0$, for $n>s$, where $s$ is the socle degree of $S$. Hilbert function is called unimodal if $h_0\le h_1\le \cdots \le h_{t-1} \le h_t\ge h_{t+1} \ge cdots \ge h_s \ge h_{s+1} = 0$ for some $t$. Hilbert functions of Gorenstein algebras are also symmetric. So, if they are unimodal, $t = s/2$ or $(s+1)/2$. It is known that Hilbert function of Goresntein algebras are unimodal in codimensio three and it is as yet open in codimension 4. There are examples of non unimodal Cohen Macaulay algebras codimension 3 and Gorenstein algebras in codimension 5 and higher. We will discuss the problem and some recent resutls in Codimension 3 level algebras and Gorenstein algebras of codimension 4.