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perfect ring

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Contents

Definition

A ring RR of characteristic pp is said to be perfect if the Frobenius map ϕ:RR\phi: R \to R is an isomorphism. If instead ϕ\phi is merely assumed to be surjective, RR is said to be semiperfect.

Perfections

For a ring RR of characteristic pp, let R perf=lim ϕRR_{perf} = \underset{\rightarrow}{lim}_{\phi} R and R perf=lim ϕRR^{perf} = \underset{\leftarrow}{lim}_{\phi} R

Both R perfR_{perf} and R perfR^{perf} are perfect. The canonical map RR perfR\to R_{perf} (respectively, R perfRR^{perf} \to R) is universal for maps into (respectively, from) perfect rings. Moreover, the projection R perfRR^{perf} \to R is surjective exactly when RR is semiperfect.

References

Created on July 7, 2017 at 10:56:15. See the history of this page for a list of all contributions to it.