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.