arXiv 1807.048592018-07math.AG

将向量丛提升为Witt向量丛

Lifting vector bundles to Witt vector bundles

Charles De Clercq, Mathieu Florence, Giancarlo Lucchini Arteche

备注：: Enriched version

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AI总结：

本文研究将概形上的向量丛提升为Witt向量丛的问题，给出线丛和射影丛情形下的正结果，推广形变理论至约化概形，并证明Grassmannian的平凡向量丛在特定条件下不可提升。

摘要：

设$X$为一个概形。设$r \geq 2$为一个整数。记$W_r(X)$为由$X$构造的长度为$r$的Witt向量概形。我们关注的问题是：将$X$上的向量丛扩展（即提升）为$W_r(X)$上的向量丛——促进Witt模和Witt向量丛的系统使用。首先，我们研究两个基本但重要的情形，其中该问题的答案是肯定的：线丛，以及仿射基上的射影丛的平凡向量丛。然后，我们对特征$p>0$的域$k$上光滑簇的形变理论中的经典结果给出一个简单的（重新）表述，并将其推广到约化$k$-概形。其中一些结果最近由Stefan Schröer以另一种形式重新获得。作为应用，我们证明：若$2 \leq m \leq n-2$，则Grassmannian $Gr_{\mathbb{F}_p}(m,n)$的平凡向量丛不能提升到$W_2(Gr_{\mathbb{F}_p}(m,n))$。最后，我们建立了与Zdanowicz关于某些射影丛不可提升性的工作的联系。

英文摘要：

Let $X$ be a scheme. Let $r \geq 2$ be an integer. Denote by $W_r(X)$ the scheme of Witt vectors of length $r$, built out of $X$. We are concerned with the question of extending (=lifting) vector bundles on $X$, to vector bundles on $W_r(X)$-promoting a systematic use of Witt modules and Witt vector bundles. To begin with, we investigate two elementary but significant cases, in which the answer to this question is positive: line bundles, and the tautological vector bundle of a projective bundle over an affine base. We then offer a simple (re)formulation of classical results in deformation theory of smooth varieties over a field $k$ of characteristic $p>0$, and extend them to reduced $k$-schemes. Some of these results were recently recovered, in another form, by Stefan Schröer. As an application, we prove that the tautological vector bundle of the Grassmannian $Gr_{\mathbb{F}_p}(m,n)$ does not extend to $W_2(Gr_{\mathbb{F}_p}(m,n))$, if $2 \leq m \leq n-2$. To conclude, we establish a connection to the work of Zdanowicz, on non-liftability of some projective bundles.

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