论文标题
Frobenius-Poincaré功能和Hilbert-Kunz多重性
Frobenius-Poincaré function and Hilbert-Kunz multiplicity
论文作者
论文摘要
我们概括了Hilbert-Kunz的概念，该级别的三重$（m，r，i）在特征$ p> 0 $中，通过证明任何复数数字$ y $，限制 $ \ inset {n \ to \ infty} {\ lim}（\ frac {1} {p^n}） （\ frac {m} {i^{[p^n]} m}）_ j \ right）e^{ - iyj/p^n} $$存在。我们证明，复杂变量$ y $中的限制函数是完整的，并将此功能命名为\ textit {frobenius-poincaré函数}。我们建立了Frobenius-Poincaré功能的各种属性，包括其与定义理想$ i $紧密关闭的关系；并将Frobenius-Poincaré函数与$ \ frac {r} {i^{[p^n]}} $的行为相关联。我们对Frobenius-Poincaré的描述在维度一和第二和其他示例中的功能提出了有关Frobenius-Poincaré通常功能的结构的问题。
We generalize the notion of Hilbert-Kunz multiplicity of a graded triple $(M,R,I)$ in characteristic $p>0$ by proving that for any complex number $y$, the limit $$\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(M)}\sum \limits_{j= -\infty}^{\infty}λ\left( (\frac{M}{I^{[p^n]}M})_j\right)e^{-iyj/p^n}$$ exists. We prove that the limiting function in the complex variable $y$ is entire and name this function the \textit{Frobenius-Poincaré function}. We establish various properties of Frobenius-Poincaré functions including its relation with the tight closure of the defining ideal $I$; and relate the study Frobenius-Poincaré functions to the behaviour of graded Betti numbers of $\frac{R}{I^{[p^n]}} $ as $n$ varies. Our description of Frobenius-Poincaré functions in dimension one and two and other examples raises questions on the structure of Frobenius-Poincaré functions in general.
