多项式环上多阶模$M$的Stanley分解是近年来研究的热点.有一个自然的深度概念，与斯坦利分解，称为斯坦利深度。Stanley解释说，模块$M$的Stanley深度总是至少为$M$的（经典）深度。在本文中，我们引入了一种较弱的分解类型，我们称之为希尔伯特分解，因为它只依赖于$M$的希尔伯特函数，以及类似的深度概念，称为希尔伯特深度。由于斯坦利分解是希尔伯特分解，后者为斯坦利分解的存在性设置了上界。希尔伯特分解的优点是它们更容易找到。我们测试我们的新概念上的syzygy模块的剩余类域的$K[X_1,...,X_n]$（通常确定与$K$）。对$k$次合冲模写出$M(n,k)$，证明了M（n，1）的Hilbert深度为$\lfloor(n+1)/2\rfloor$ .进一步证明了对于$n > k \ge \lfloor n/2\rfloor$，$M(n,k)$的Hilbert深度等于$n-1$ .我们推测斯坦利深度也是如此。对于范围$n/2 > k > 1$，似乎不可能得出希尔伯特深度的简洁公式。相反，我们提供了非常精确的渐近结果，因为$n$变得很大。
英文摘要：
Stanley decompositions of multigraded modules $M$ over polynomials rings have been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition, called the Stanley depth. Stanley conjectured that the Stanley depth of a module $M$ is always at least the (classical) depth of $M$. In this paper we introduce a weaker type of decomposition, which we call Hilbert decomposition, since it only depends on the Hilbert function of $M$, and an analogous notion of depth, called Hilbert depth. Since Stanley decompositions are Hilbert decompositions, the latter set upper bounds to the existence of Stanley decompositions. The advantage of Hilbert decompositions is that they are easier to find. We test our new notion on the syzygy modules of the residue class field of $K[X_1,...,X_n]$ (as usual identified with $K$). Writing $M(n,k)$ for the $k$-th syzygy module, we show that theHilbert depth of M(n,1) is $\lfloor(n+1)/2\rfloor$. Furthermore, we show that, for $n > k \ge \lfloor n/2\rfloor$, the Hilbert depth of $M(n,k)$ is equal to $n-1$. We conjecture that the same holds for the Stanley depth. For the range $n/2 > k > 1$, it seems impossible to come up with a compact formula for the Hilbert depth. Instead, we provide very precise asymptotic results as $n$ becomes large.
We consider simplicial complexes admitting a free action by an abelian group. Specifically, we establish a refinement of the classic result of Hochster describing the local cohomology modules of the associated Stanley--Reisner ring, demonstrating that the topological structure of the free action extends to the algebraic setting. If the complex in question is also Buchsbaum, this new description allows for a specialization of Schenzel's calculation of the Hilbert series of some of the ring's Artinian reductions. In further application, we generalize to the Buchsbaum case the results of Stanley …
Let $R$ be a positively graded algebra over a field. We say that $R$ is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all of its roots on the unit circle. Such rings arise naturally in commutative algebra, numerical semigroup theory and Ehrhart theory. If $R$ is standard graded, we prove that, under the additional hypothesis that $R$ is Koszul or has an irreducible $h$-polynomial, Hilbert-cyclotomic algebras coincide with complete intersections. In the Koszul case, this is a consequence of some classical results about the vanishing of deviations of a graded algebra.
We begin with proving a formula relating the Hilbert series of a graded algebra $A$ and the Poincaré series for $A$ in two variables. This gives the Fröberg formula in the case where the bigraded $Tor^A(k,k)$ is concentrated on the diagonal, which we take as definition of $A$ being "Koszul". We look at a resolution in the commutative case obtained from the Koszul complex in the "trivially Golod" case. The algebra structure of $Ext_A(k,k)$ is introduced in different ways. Its subalgebra generated by the one-dimensional elements is by definition the "Koszul" dual of $A$. We define the"generalize…
Given an infinite field $\mathbb{k}$ and a simplicial complex $Δ$, a common theme in studying the $f$- and $h$-vectors of $Δ$ has been the consideration of the Hilbert series of the Stanley--Reisner ring $\mathbb{k}[Δ]$ modulo a generic linear system of parameters $Θ$. Historically, these computations have been restricted to special classes of complexes (most typically triangulations of spheres or manifolds). We provide a compact topological expression of $h_{d-1}^\mathfrak{a}(Δ)$, the dimension over $\mathbb{k}$ in degree $d-1$ of $\mathbb{k}[Δ]/(Θ)$, for any complex $Δ$ of dimension $d-1$. I…
