= Perfect ring of characteristic p
{title2=$a\mapsto a^p\text{ is bijective}$}
A <commutative ring> of prime characteristic $p$ is perfect in the Frobenius sense when its <Frobenius endomorphism> is an automorphism. It is necessarily reduced: if $a$ is nilpotent, some $p$-power iterate is zero, and injectivity gives $a=0$. This use of perfect is distinct from the homological notion of a perfect ring. The <Absolute Frobenius morphism> of a <scheme> is an <isomorphism> exactly when all its <local rings> are perfect in this sense, because it is the identity on points and is an isomorphism on every stalk.
Back to article page
