Abstract:

In 1982, Richard Stanley conjectured that any finitely generated $\Z^n$-graded module $M$ over a finitely generated $\N^n$-graded $\k$-algebra $R$ can be decomposed in a direct sum $M = \bigoplus_{i=1}^t \nu_i S_i$ of finitely many free modules $\nu_i S_i$ such that all $S_i$ are subalgebras of $R$ of dimension at least $\depth(M)$ satisfying some additional conditions. It is still not known in general whether this is true or not. I will give an overview over existing results focusing on the case that $R$ is a polynomial ring and $M = R/I$ or $M = I$ for some monomial ideal $I$. Interestingly enough, an affirmative answer to the conjecture would imply that every Cohen-Macaulay simplicial complex is partitionable.