arXiv 1012.32792010-12-15math.AG

曲线上的可构造nabla-模

Constructible nabla-modules on curves

Bernard Le Stum

备注：: 39 pages

摘要：

设$\mathcal V$是具有完全剩余域的混合特征的离散赋值环.设$X$是$\mathcal V$上的一条几何连通光滑真曲线.在对$X$的类纤维$X_{K}^{\mathrm{an}}$进行分析的基础上，引入了可构造收敛$\nabla$ -模的概念.一个可构造的模块是一个$\mathcal O_{X_{K}^{\mathrm{an}}}$ -模块，它不一定是相干的，但是在一个分层上通过$X$的特殊纤维$X_{k}$的局部闭合子集变得相干。连接、（过）收敛和弗罗贝纽斯结构的概念延续到这种情况。本文描述了从可构造收敛$\nabla$ -模范畴到$\mathcal D^\dagger_{\hat X \mathbf Q}$ -模范畴的特化函子.我们表明，如果$X$被赋予了一个提升的绝对Frobenius的$X$，然后专业化诱导构造$F$ - $\nabla$ -模块和反常完整$F$ - $\mathcal D^\dagger_{\hat X \mathbf Q}$ -模块之间的等价性。

英文摘要：

Let $\mathcal V$ be a discrete valuation ring of mixed characteristic with perfect residue field. Let $X$ be a geometrically connected smooth proper curve over $\mathcal V$. We introduce the notion of constructible convergent $\nabla$-module on the analytification $X_{K}^{\mathrm{an}}$ of the generic fibre of $X$. A constructible module is an $\mathcal O_{X_{K}^{\mathrm{an}}}$-module which is not necessarily coherent, but becomes coherent on a stratification by locally closed subsets of the special fiber $X_{k}$ of $X$. The notions of connection, of (over-) convergence and of Frobenius structure carry over to this situation. We describe a specialization functor from the category of constructible convergent $\nabla$-modules to the category of $\mathcal D^\dagger_{\hat X \mathbf Q}$-modules. We show that if $X$ is endowed with a lifting of the absolute Frobenius of $X$, then specialization induces an equivalence between constructible $F$-$\nabla$-modules and perverse holonomic $F$-$\mathcal D^\dagger_{\hat X \mathbf Q}$-modules.

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