论文标题
关于特征仿射品种的De Rham同源性0
On the de Rham homology of affine varieties in characteristic 0
论文作者
论文摘要
Let $\mathbb K$ be a field of characteristic 0, let $S$ be a complete local ring with coefficient field $\mathbb K$, let $\mathbb K[[x_1,\dots,x_n]]$ be the ring of formal power series in variables $x_1,\dots, x_n$ with coefficients from $\mathbb K$, let $\mathbb k [[x_1，\ dots，x_n]] \ to s $为$ \ mathbb k $ -algebra surpocty，让$ e_ \ bullet^{\ bullet，\ bullet} $是相关的hodge-de hodge-de rham光谱序列，用于计算$ s $ $ s $的de rham同源物的计算。尼古拉斯·斯威塔拉（Nicholas Switala）证明，此频谱序列独立于$ e_2 $页面的陈述，而这些序列$ e^{p，q} _2 $都是$ \ mathbb k $的有限维度。 在本文中，我们将此结果扩展到仿射品种。 Namely, let $Y$ be an affine variety over $\mathbb K$, let $X$ be a non-singular affine variety over $\mathbb K$, let $Y\subset X$ be an embedding over $\mathbb K$ and let $E_\bullet^{\bullet,\bullet} $ be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of $ y $。然后，此频谱序列独立于以$ e_2 $页面开头的嵌入，而组$ e^{p，q} _2 $都是$ \ m athbb k $的有限维度。
Let $\mathbb K$ be a field of characteristic 0, let $S$ be a complete local ring with coefficient field $\mathbb K$, let $\mathbb K[[x_1,\dots,x_n]]$ be the ring of formal power series in variables $x_1,\dots, x_n$ with coefficients from $\mathbb K$, let $\mathbb K[[x_1,\dots,x_n]]\to S$ be a $\mathbb K$-algebra surjection and let $E_\bullet^{\bullet,\bullet} $ be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of $S$. Nicholas Switala proved that this spectral sequence is independent of the surjection beginning with the $E_2$ page, and the groups $E^{p,q}_2$ are all finite-dimensional over $\mathbb K$. In this paper we extend this result to affine varieties. Namely, let $Y$ be an affine variety over $\mathbb K$, let $X$ be a non-singular affine variety over $\mathbb K$, let $Y\subset X$ be an embedding over $\mathbb K$ and let $E_\bullet^{\bullet,\bullet} $ be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of $Y$. Then this spectral sequence is independent of the embedding beginning with the $E_2$ page, and the groups $E^{p,q}_2$ are all finite-dimensional over $\mathbb K$.
