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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 7th 2017
   • (edited Jul 7th 2017)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexBhatt's [lecture notes](http://www-personal.umich.edu/~bhattb/teaching/mat679w17/lectures.pdf) on perfectoid spaces has >a characteristic $p$ ring $R$ is _perfect_ if the Frobenius $\phi: R \to R$ is an isomorphism; if instead $\phi$ is merely assumed to be surjective, we say that $R$ is _semiperfect_. We have other 'perfect' entries, such as [[perfect field]]. Is there a connection?

   Bhatt’s lecture notes on perfectoid spaces has

   a characteristic $p$ ring $R$ is perfect if the Frobenius $\phi: R \to R$ is an isomorphism; if instead $\phi$ is merely assumed to be surjective, we say that $R$ is semiperfect.

   We have other ’perfect’ entries, such as perfect field. Is there a connection?

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 7th 2017
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   Author: DavidRoberts
   Format: MarkdownItexFor a characteristic p field, the Frobenius is injective. I was reading Scholze's ICM article and he says a field is perfect if it is surjective.

   For a characteristic p field, the Frobenius is injective. I was reading Scholze’s ICM article and he says a field is perfect if it is surjective.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 7th 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThe Wikipedia [page](https://en.wikipedia.org/wiki/Perfect_field) links them: >More generally, a ring of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism.[1] (This is equivalent to the above condition "every element of k is a pth power" for integral domains.)

   The Wikipedia page links them:

   More generally, a ring of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism.[1] (This is equivalent to the above condition “every element of k is a pth power” for integral domains.)

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 7th 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSo I've made a start at [[perfect ring]].

   So I’ve made a start at perfect ring.