论文标题
投影单曲线的坐标环的衍生模块和Hilbert-Kunz多重性
Derivation module and the Hilbert-Kunz multiplicity of the co-ordinate ring of a projective monomial curve
论文作者
论文摘要
令$ n_0，n_1，\ ldots，n_p $为一个正整数的顺序，以便$ n_0 <n_1 <\ cdots <n_p $和$ \ mathrm {gcd}（n_0，n_1，n_1，\ ldots，n_p）= 1 $。令$ s = \ langle（0，n_p），（n_0，n_p-n_0），\ ldots，（n_ {p-1}，n_p-n_ {p-1}），（n_p，0）\ rangle $ be affine sefine be affine sefine be affine sefine sefine sefine sefine sepine sepine sepine sepine semogep in $ \ mathbb {n} n}^2 $。 Semigroup Ring $ K [S] $是投影空间中投影单元曲线的坐标环$ \ Mathbb {p} _K^{p+1} $，该{p+1} $由\ begin {center} $ x_0 = v^{ u^{n_0} v^{n_p-n_0}，\ quad \ ldots，\ quad x_p = u^{n_ {p-1}} v^{n_p-n_ {p-1}}}，\ quad x____________________ \ end {center}在本文中，我们考虑了$ n_0，n_1，\ ldots，n_p $形成算术序列的情况，并为派生模块$ \ mathrm {der} _K（k [k [k [s]）提供明显的最小发电机集。此外，我们给出了一个明确的公式，用于射影单曲线的坐标环的Hilbert-Kunz多重性。
Let $n_0, n_1, \ldots, n_p$ be a sequence of positive integers such that $n_0 < n_1 < \cdots < n_p$ and $\mathrm{gcd}(n_0,n_1, \ldots,n_p) = 1$. Let $S = \langle (0,n_p), (n_0,n_p-n_0),\ldots,(n_{p-1},n_p-n_{p-1}), (n_p,0) \rangle$ be an affine semigroup in $\mathbb{N}^2$. The semigroup ring $k[S]$ is the co-ordinate ring of the projective monomial curve in the projective space $\mathbb{P}_k^{p+1}$, which is defined parametrically by \begin{center} $x_0 = v^{n_p}, \quad x_1 = u^{n_0}v^{n_p-n_0},\quad \ldots , \quad x_p= u^{n_{p-1}}v^{n_p-n_{p-1}}, \quad x_{p+1} = u^{n_p}$. \end{center} In this article, we consider the case when $n_0, n_1, \ldots, n_p$ forms an arithmetic sequence, and give an explicit set of minimal generators for the derivation module $\mathrm{Der}_k(k[S])$. Further, we give an explicit formula for the Hilbert-Kunz multiplicity of the co-ordinate ring of a projective monomial curve.
