论文标题
在理想的弗罗贝尼乌斯力量的希尔伯特 - 塞缪尔系数上
On the Hilbert-Samuel coefficients of Frobenius powers of an ideal
论文作者
论文摘要
我们提供适当的条件，在该条件下，$ \ m athfrak {m} $的frobenius powers的渐近极限存在于Noetherian本地环$（R，\ Mathfrak {M Mathfrak {M}）中，具有主要特征性$ p> 0的主要特征。 We also prove that for a face ring $R$ of a simplicial complex and an ideal $J$ generated by pure powers of the variables, the generalized Hilbert-Kunz function $\ell(R/(J^{[q]})^k)$ is a polynomial for all $q,k$ and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of $J$ in terms of Hilbert-Samuel $ J的多样性。$我们通过对I. Smirnov提出的猜想进行反示例来结论，该猜想将理想的稳定性与理想的Frobenius功率的第一个希尔伯特系数的渐近极限联系起来。
We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an $\mathfrak{m}$-primary ideal exists in a Noetherian local ring $(R,\mathfrak{m})$ with prime characteristic $p>0.$ This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring $R$ of a simplicial complex and an ideal $J$ generated by pure powers of the variables, the generalized Hilbert-Kunz function $\ell(R/(J^{[q]})^k)$ is a polynomial for all $q,k$ and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of $J$ in terms of Hilbert-Samuel multiplicity of $J.$ We conclude by giving a counter-example to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal.
