Scheme of characteristic p ($\mathrm{char}X=p$)

A scheme $X$ has characteristic $p$ when $p\cdot 1=0$ in its structure sheaf of a scheme. Equivalently, its structure morphism $X\to \mathrm{Spec}\mathbb{Z}$ factors through $\mathrm{Spec}{\mathbb{F}}_p$.

The absolute Frobenius morphism of a scheme of characteristic p is the identity on the underlying topological space and raises every local function to its $p$th power. On an affine chart $\mathrm{Spec}A$ it is induced contravariantly by the Frobenius endomorphism $A\to A$, $a\mapsto a^p$. It need not be an isomorphism of schemes, even when it acts invertibly on global sections.