On a Conjecture of R.P. Stanley; Part I—Monomial Ideals
On a Conjecture of R.P. Stanley; Part I—Monomial Ideals
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关于 R.P. Stanley 的猜想；第一部分——单项式理想
DOI：
10.1023/a:1021912724441
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发表时间：
2003
影响因子：
0.8
通讯作者：
J. Apel
中科院分区：
文献类型：
--
作者：
J. Apel
1982年，Richard P. Stanley证明了在一个n-生成的n-分次K-代数R上的任意n-生成的n-分次模M都可以分解为n个自由模νiSi的直和M = n i = 1 t νiSi，其中n个自由模νiSi必须满足一些附加条件。除了齐性条件的最重要的限制是，Si必须是子代数R的维数至少深度M.我们将研究这一猜想的特殊情况下，R是一个多项式环和M的理想R，我们遇到了一个强连接到广义对合基地。我们将得到一个标准，使我们能够从特定的对合基中提取深度M的上界。作为推论，我们得到任何单项式理想M，它拥有一个这种类型的对合基满足斯坦利猜想，在这种情况下，由基定义的对合分解也是M的斯坦利分解。此外，我们将表明，该标准适用于，例如，任何单项理想的深度最多为2，任何单项理想在最多3个变量，和任何单项理想，这是通用的关于一个变量。对合基理论为我们提供了在这些情况下计算斯坦利分解的算法部分。
In 1982 Richard P. Stanley conjectured that any finitely generated ℝn-graded module M over a finitely generated ℕn-graded K-algebra R can be decomposed in a direct sum M = ⊕i = 1t νiSi of finitely many free modules νiSi which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least depth M.We will 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 will 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 will 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 algorithmic part for the computation of Stanley decompositions in these situations.