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On a conjecture of R. P. Stanley. I: Monomial ideals. (English) Zbl 1031.13003

Summary: R. P. Stanley [Invent. Math. 68, 175-193 (1982; Zbl 0516.10009)] conjectured that any finitely generated \(\mathbb{Z}^n\)-graded module \(M\) over a finitely generated \(\mathbb{N}^n\)-graded \(\mathbb{K}\)-algebra \(R\) can be decomposed in a direct sum \(M=\oplus^t_{i=1}v_iS_i\) of finitely many free modules \(v_iS_i\) 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 study this conjecture for the special case that \(R\) is a polynomial ring and \(M\) an ideal of \(R\), where we encounter a strong connection to generalized involutive bases. We derive a criterion which allows us to extract an upper bound on depth \(M\) from particular involutive bases. As a corollary we obtain that any monomial ideal \(M\) which possesses an involutive basis of this type satisfies Stanley’s conjecture and in this case the involutive decomposition defined by the basis is also a Stanley decomposition of \(M\). Moreover, we show that the criterion applies, for instance, to any monomial ideal of depth at most 2, to any monomial ideal in at most 3 variables, and to any monomial ideal which is generic with respect to one variable. The theory of involutive bases provides us with the algorithm part for the computation of Stanley decompositions in these situations.
For part I of this paper see: J. Apel, J. Algebr. Comb. 17, No. 1, 57-74 (2003; see the following review Zbl 1031.13004).

MSC:

13C05 Structure, classification theorems for modules and ideals in commutative rings
13C10 Projective and free modules and ideals in commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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References:

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