Liftability of singularities and their Frobenius morphism modulo $p^2$
Liftability of singularities and their Frobenius morphism modulo $p^2$
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奇点的可提升性及其 Frobenius 态射模 $p^2$
DOI：
10.1093/imrn/rnw297
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Maciej Zdanowicz
中科院分区：
文献类型：
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作者：
Maciej Zdanowicz
我们研究了奇异方案的 $W_2(k)$ 可提升性。我们证明了平面族 $X \to S$ 中 $W_2(k)$ 可提升方案轨迹的可构造性。此外，我们在完美域 $k$ 上构建了 Frobenius 分割方案 $X$ 的显式 $W_2(k)$ 提升，驳斥了 Bhatt 的存在性结果。此外，我们研究了 Frobenius 态射提升的存在性。特别是，我们证明在维度 $n \geq 4$ 中，普通双点不允许与 Frobenius 兼容的 $W_2(k)$ 提升，并且规范表面奇点是 Frobenius 可提升的。结合 Bhatt 的结果，后一个结果意味着超过 $k$ 的具有正则奇点的表面的晶体上同调群不是有限维的。作为我们结果的推论，我们对 $W_2(k)$-可提升性、Frobenius 可提升性和经典 $F$-奇点类型的概念进行了彻底的比较。
We investigate the $W_2(k)$-liftability of singular schemes. We prove constructibility of the locus of $W_2(k)$-liftable schemes in a flat family $X \to S$. Moreover, we construct an explicit $W_2(k)$-lifting of a Frobenius split scheme $X$ over a perfect field $k$, reproving Bhatt's existential result. Furthermore, we study existence of liftings of the Frobenius morphism. In particular, we prove that in dimension $n \geq 4$ ordinary double points do not admit a $W_2(k)$-lifting compatible with Frobenius, and that canonical surface singularities are Frobenius liftable. Combined with Bhatt's results, the latter result implies that the crystalline cohomology groups over $k$ of surfaces with canonical singularities are not finite dimensional. As a corollary of our results, we provide a thorough comparison between the notions of $W_2(k)$-liftability, Frobenius liftability and classical $F$-singularity types.