论文标题
地位希尔伯特系列和渐近特性的合理性
Rationality of Equivariant Hilbert Series and Asymptotic Properties
论文作者
论文摘要
An FI- or an OI-module $\mathbf{M}$ over a corresponding noetherian polynomial algebra $\mathbf{P}$ may be thought of as a sequence of compatible modules $\mathbf{M}_n$ over a polynomial ring $\mathbf{P}_n$ whose number of variables depends linearly on $n$.为了研究模块的不变式$ \ mathbf {m} _n $的依赖性$ n $，如果分级$ \ mathbf {m} $，则会引入一个模棱两可的希尔伯特系列。如果$ \ mathbf {m} $也是有限生成的，则表明该系列是一个合理的函数。此外，如果以简化的形式编写此功能，则获得了有关分母的不可还原因素的精确信息。这是应用程序的关键。因此，模块的krull尺寸$ \ mathbf {m} _n $最终以$ n $线性形成线性，而$ \ mathbf {m} _n $的多重性最终在$ n $中成倍成倍增长。此外，对于任何固定度$ j $，$ \ mathbf {m} _n $的$ j $组件的矢量空间尺寸最终以$ n $多样化生长。结果，在固定同源度中，任何分级的betti数量的$ \ mathbf {m} _n $和固定的内部学位最终在$ n $中生长。此外，还获得了证据来支持一个猜想，即Castelnuovo-Mumford的规律性和$ \ Mathbf {M} _n $的投影维度最终在$ n $中生长。还表明，模块$ \ mathbf {m} $的宽度$ n $ components $ \ mathbf {m} _n $最终以它们的eprivariant Hilbert系列可以表征Artinian。提出了使用普通语言和有限自动机，提出了用于计算epariant Hilbert系列的算法。
An FI- or an OI-module $\mathbf{M}$ over a corresponding noetherian polynomial algebra $\mathbf{P}$ may be thought of as a sequence of compatible modules $\mathbf{M}_n$ over a polynomial ring $\mathbf{P}_n$ whose number of variables depends linearly on $n$. In order to study invariants of the modules $\mathbf{M}_n$ in dependence of $n$, an equivariant Hilbert series is introduced if $\mathbf{M}$ is graded. If $\mathbf{M}$ is also finitely generated, it is shown that this series is a rational function. Moreover, if this function is written in reduced form rather precise information about the irreducible factors of the denominator is obtained. This is key for applications. It follows that the Krull dimension of the modules $\mathbf{M}_n$ grows eventually linearly in $n$, whereas the multiplicity of $\mathbf{M}_n$ grows eventually exponentially in $n$. Moreover, for any fixed degree $j$, the vector space dimensions of the degree $j$ components of $\mathbf{M}_n$ grow eventually polynomially in $n$. As a consequence, any graded Betti number of $\mathbf{M}_n$ in a fixed homological degree and a fixed internal degree grows eventually polynomially in $n$. Furthermore, evidence is obtained to support a conjecture that the Castelnuovo-Mumford regularity and the projective dimension of $\mathbf{M}_n$ both grow eventually linearly in $n$. It is also shown that modules $\mathbf{M}$ whose width $n$ components $\mathbf{M}_n$ are eventually Artinian can be characterized by their equivariant Hilbert series. Using regular languages and finite automata, an algorithm for computing equivariant Hilbert series is presented.
