nLab perfect ring

Definition

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

Perfections

For a ring $R$ of characteristic $p$, let $R_{perf} = \underset{\rightarrow}{lim}_{\phi} R$ and $R^{perf} = \underset{\leftarrow}{lim}_{\phi} R$

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

Related entries

• perfect field
• perfectoid space

References

• Bhargav Bhatt, lecture notes.