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arXiv 1808.09035math.ACmath.AG
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关于分次D-模的完备化

On completion of graded D-modules

Nicholas Switala, Wenliang Zhang

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摘要：

设$R = k[x_1, \ldots, x_n]$是特征为零的域$k$上的多项式环，$\cR$是形式幂级数环$k[[x_1, \ldots, x_n]]$ .如果$M$是$R$上的$\D$ -模块，则$\cR \otimes_R M$自然是$\cR$上的$\D$ -模块。Hartshorne和Polini提出了一个问题：当$M$是分次的和完整的时，自然映射$H^i_{\dR}(M)\to H^i_{\dR}(\cR \otimes_R M)$（由$M\to \cR \otimes_R M$诱导）是否同构。我们对他们的问题给出了肯定的回答，作为下面更强结果的推论。设$M$是一个由$M\to \cR \otimes_R M$生成的分次$\D$ -模：对于每个整数$i$使得$\dim_kH^i_{\dR}(M)<\infty$，自然映射$H^i_{\dR}(M)\to H^i_{\dR}(\cR \otimes_R M)$（由$M\to \cR \otimes_R M$诱导）是一个同构.

英文摘要：

Let $R = k[x_1, \ldots, x_n]$ be a polynomial ring over a field $k$ of characteristic zero and $\cR$ be the formal power series ring $k[[x_1, \ldots, x_n]]$. If $M$ is a $\D$-module over $R$, then $\cR \otimes_R M$ is naturally a $\D$-module over $\cR$. Hartshorne and Polini asked whether the natural maps $H^i_{\dR}(M)\to H^i_{\dR}(\cR \otimes_R M)$ (induced by $M\to \cR \otimes_R M$) are isomorphisms whenever $M$ is graded and holonomic. We give a positive answer to their question, as a corollary of the following stronger result. Let $M$ be a finitely generated graded $\D$-module: for each integer $i$ such that $\dim_kH^i_{\dR}(M)<\infty$, the natural map $H^i_{\dR}(M)\to H^i_{\dR}(\cR \otimes_R M)$ (induced by $M\to \cR \otimes_R M$) is an isomorphism.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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