论文标题
deligne-illusie定理的一些改进
Some refinements of the Deligne-Illusie theorem
论文作者
论文摘要
我们将Deligne和Illusie的结果扩展到Liftings Modulo $ p^2 $和DE RHAM COMPLEX的分解。我们表明，对于一个平稳的方案$ x $，在一个完美的字段$ k $上的特征性$ p> 0 $，以$ \ max（p-1，2）$连续度的截断可以作为派生类别的对象重建，以其大多数程度（或等于confterion cluckition cluckity cloxpion classity of tentre cluckity oft contruction）的对象。因此，如果$ x $承认提升到$ w_2（k）$，则这些截断是可以分解的，在这种情况下，共轭光谱序列中的第一个非零差异似乎不到第$ \ max $ \ max（p，3）$（最近已经被Drinfeld，drinfeld，bhatt-lurrie和lirurie和limondal增强）。在不假设举重的情况下，我们描述了在共轭频谱序列的第二页上的两项截断和差异的分裂的gerbes，回答了Katz的问题。 $ p> 2 $中使用的主要技术结果纯粹属于同源代数。它涉及某些交换差分等级代数，其共同体学代数是外部代数，由我们“抽象的Koszul Complextes”称为外部代数，其中特征性$ p $的DE RHAM复合体就是一个例子。 在附录中，我们使用上述更强的分解结果证明Kodaira-Akizuki-Nakano消失和Hodge-de Rham变性都以$ f $ -split $（p+1）$ - 折叠为单位。
We extend the results of Deligne and Illusie on liftings modulo $p^2$ and decompositions of the de Rham complex in several ways. We show that for a smooth scheme $X$ over a perfect field $k$ of characteristic $p>0$, the truncations of the de Rham complex in $\max(p-1, 2)$ consecutive degrees can be reconstructed as objects of the derived category in terms of its truncation in degrees at most one (or, equivalently, in terms the obstruction class to lifting modulo $p^2$). Consequently, these truncations are decomposable if $X$ admits a lifting to $W_2(k)$, in which case the first nonzero differential in the conjugate spectral sequence appears no earlier than on page $\max(p,3)$ (these corollaries have been recently strengthened by Drinfeld, Bhatt-Lurie, and Li-Mondal). Without assuming the existence of a lifting, we describe the gerbes of splittings of two-term truncations and the differentials on the second page of the conjugate spectral sequence, answering a question of Katz. The main technical result used in the case $p>2$ belongs purely to homological algebra. It concerns certain commutative differential graded algebras whose cohomology algebra is the exterior algebra, dubbed by us "abstract Koszul complexes", of which the de Rham complex in characteristic $p$ is an example. In the appendix, we use the aforementioned stronger decomposition result to prove that Kodaira-Akizuki-Nakano vanishing and Hodge-de Rham degeneration both hold for $F$-split $(p+1)$-folds.
