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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeDec 24th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: \tableofcontents ## Definition (Definition 2.1 in Bhatt–Scholze.) Fix a prime $p$. A __δ-ring__ is a pair $(R,δ)$, where $R$ is a [[commutative ring]] and $\delta\colon R\to R$ is a map of underlying sets such that $\delta(0)=0$, $\delta(1)=0$, $$\delta(xy)=x^p \delta(y)+y^p \delta(x) + p\delta(x)\delta(y),$$ and $$\delta(x+y)=\delta(x)+\delta(y)+(x^p+y^p-(x+y)^p)/p.$$ ## Properties If $(R,\delta)$ is a δ-ring, then the map $\phi\colon R\to R$ given by $\phi(x)=x^p + p\delta(x)$ is a ring homomorphism that lifts the [[Frobenius endomorphism]] on $R/p$. For $p$-torsionfree rings, the above correspondence between δ-structures and lifts of the Frobenius endomorphism on $R/p$ to $R$ is bijective. This motivates the identities in the definition of a δ-structure. ## Related entries * [[perfectoid ring]] * [[perfectoid space]] * [[prism]] * [[prismatic cohomology]] ## References * [[Bhargav Bhatt]], [[Peter Scholze]], _Prisms and prismatic cohomology_, [arXiv](http://arxiv.org/abs/1905.08229v3). <a href="https://ncatlab.org/nlab/revision/%CE%B4-ring/1">v1</a>, <a href="https://ncatlab.org/nlab/show/%CE%B4-ring">current</a>

   Created:

   \tableofcontents

   Definition

   (Definition 2.1 in Bhatt–Scholze.)

   Fix a prime $p$. A δ-ring is a pair $(R,δ)$, where $R$ is a commutative ring and $\delta\colon R\to R$ is a map of underlying sets such that $\delta(0)=0$, $\delta(1)=0$,

   $$\delta(xy)=x^p \delta(y)+y^p \delta(x) + p\delta(x)\delta(y),$$

   and

   $$\delta(x+y)=\delta(x)+\delta(y)+(x^p+y^p-(x+y)^p)/p.$$

   Properties

   If $(R,\delta)$ is a δ-ring, then the map $\phi\colon R\to R$ given by $\phi(x)=x^p + p\delta(x)$ is a ring homomorphism that lifts the Frobenius endomorphism on $R/p$.

   For $p$-torsionfree rings, the above correspondence between δ-structures and lifts of the Frobenius endomorphism on $R/p$ to $R$ is bijective. This motivates the identities in the definition of a δ-structure.

   Related entries

   • perfectoid ring

   • perfectoid space

   • prism

   • prismatic cohomology

   References

   • Bhargav Bhatt, Peter Scholze, Prisms and prismatic cohomology, arXiv.

   v1, current