AI 中文总结
本文针对平坦 $\bbZ_{(p)}$-概型，引入基于 Frobenius–Witt 余切复形的算术 Kodaira–Spencer 类，证明其与经典形变理论的 Frobenius 提升障碍等价，还定义了兼容基提升的相对障碍类。
AI 中文摘要
对于一个平坦的 $\boldsymbol{\bbZ_{(p)}}$-概型 $X$，我们引入了一个障碍类，用于判断其模 $p^2$ 的约化 $X_1$ 上是否存在 Frobenius 提升。该类的定义是在 Kodaira–Spencer 类的经典构造中，将余切复形替换为其算术类似物——Frobenius–Witt 余切复形。我们进一步证明，该类与经典形变理论定义的 Frobenius 提升障碍一致。我们还定义了它的相对版本，并证明该相对类是存在与基概型上给定 Frobenius 提升兼容的 Frobenius 提升的障碍类。
英文摘要
For a flat $\mathbb{Z}_{(p)}$-scheme $X$, we introduce an obstruction class to the existence of a Frobenius lift on its reduction $X_1$ modulo $p^2$. This class is defined by replacing the cotangent complex in the classical construction of the Kodaira--Spencer class with its arithmetic analogue, the Frobenius--Witt cotangent complex. We further prove that this class coincides with the obstruction to Frobenius lifting defined by classical deformation theory. We also define its relative version and prove that this is an obstruction class to the existence of a Frobenius lift compatible with the given one on the base scheme.
Comments13 pages. comments are very welcome
