arXiv subjects

Kanau Shimada

Publications and source records attributed to Kanau Shimada.

2 recordsLinked to original sources

arXiv2026

Arithmetic Kodaira--Spencer Class and Frobenius Liftings via Frobenius--Witt Cotangent Complex

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.

math.AG↗

arXiv2026

Frobenius--Witt cotangent complexes

We introduce the notion of the Frobenius--Witt cotangent complex, which can be considered as a derived variant of the module of Frobenius--Witt differentials defined by T. Saito. This new object also can be seen as an arithmetic variant of the notion of cotangent complex. We explain the suitability of these two viewpoints through a series of propositions. Furthermore, we establish a relationship between Frobenius--Witt cotangent complexes and the regularity of noetherian local rings, which can be considered as a derived variant of Saito's regularity criterion. This proof relies heavily on computations of Frobenius--Witt cotangent complexes in the case of perfectoid rings. We also study the deformation theory of delta structures using Frobenius--Witt cotangent complexes.

math.AG↗