nLab δ-ring

Definition

(Definition 2.1 in Bhatt–Scholze.)

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

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

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

The original notion is due to André Joyal:

• André Joyal, δ-anneaux et vecteurs de Witt, C. R. Math. Rep. Acad. Sci. Canada 7 (1985), no. 3, 177–182.

Recent developments can be found in

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