ON A CONJECTURE OF R.P.STANLEY, II: QUOTIENTS MODULO MONOMIAL IDEALS

JOACHIM APEL

Abstract. In 1982 Richard P. 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 = ⨁_{i=1}^{t} ν_{i}S_{i} of finitely many free modules ν_{i}S_{i} which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the S_{i} have to be subalgebras of R of dimension at least depth M.

We will study this conjecture for modules M = R/I, where R is a polynomial ring and I a monomial ideal. We will present a necessary and sufficient condition for squarefree monomial Cohen-Macaulay rings to fulfill Stanley’s Conjecture. Moreover, we prove that Stanley’s Conjecture holds for the quotient modulo any generic monomial ideal, the quotient modulo any monomial ideal in at most three variables, and for any cogeneric Cohen-Macaulay ring. The presented theorems are constructive and allow to extract explicit algorithms for the computation of Stanley decompositions. Further, we will discuss some relationships between our algebraic results and combinatorial properties of related complexes. Finally, we will give an outlook to Stanley decompositions of arbitrary graded polynomial modules. In particular, we obtain a more general result in the 3-variate case.