= Frobenius map on an algebra cocenter
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{title2=$F([a])=[a^p]$}
In <characteristic> $p>0$, the displayed map on the <cocenter of a finite-dimensional algebra> is additive and Frobenius-semilinear. Expand a $p$th power into words: nonconstant cyclic-word orbits have length $p$ and cancel modulo commutators. Also $(ab)^p$ and $(ba)^p$ have the same cocenter class, so the map kills additive commutators and is well defined.
Over a perfect <field> with split semisimple quotient, the stable image of this map has dimension equal to the number of <simple modules> of the <algebra>. Indeed, the kernel of $C(A)\to C(A/J(A))$ has representatives in the nilpotent <Jacobson radical>, so an iterate kills it. On the <semisimple algebra> quotient, each matrix-block coordinate is a field trace, and $\operatorname{tr}(a^p)=\operatorname{tr}(a)^p$. For a split semisimple quotient, Frobenius is bijective on every coordinate. In a <group algebra> over an <algebraically closed field>, the stable image has the <p-regular element> conjugacy classes as its basis, proving the simple-module counting theorem.
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