nLab Joyal delta-ring

Idea

The notion of a $\lambda$-ring $R$ is closely tied to classical Adams operations $\psi^n \colon R \to R$, modeled by monoid homomorphisms from the multiplicative monoid $(\mathbb{N}_+, \cdot)$ to the monoid of ring homomorphisms on the underlying commutative ring $R$ and their compositions. As $\mathbb{N}_+$ is freely generated by primes $p$, the Adams operations $\psi^p$ play a distinguished role. Classically, they are “Frobenius lifts”, i.e., $\psi^p(x)$ agrees with $x^p$ modulo $p$: the Frobenius map $x \mapsto x^p$ which defines a ring endomorphism on $R/p R$ lifts to the homomorphism $\psi^p\colon R \to R$.

The notion of $\delta$-ring, originally due to André Joyal and developed independently by Alexandru Buium and others, axiomatizes the formal properties that hold for the “arithmetic derivative”

$$\delta(x) = \frac{\psi^p(x) - x^p}{p}.$$

Definition

Fix a prime number $p$ and a power $q = p^r$ ($r \geq 0$).

Definition

A $\delta$-ring is a commutative ring equipped with a unary operation $\delta: A \to A$ satisfying the following identities:

1. $\delta(x+y) = \delta(y) + \delta(y) - \sum_{i=1}^{q-1} \frac1{p}\binom{q}{i} x^i y^{q-i}$;

2. $\delta(x y) = x^q\delta(y) + \delta(x)y^q + p\delta(x)\delta(y)$, $\delta(1) = 0$.

We say that $A$ is a $\delta$-ring with respect to the pair $(p, q)$.

These conditions imply that the unary operation $f(x) = x^q + p\delta(x)$ is an endomorphism of the ring $A$:

$$f(x+y) = f(x) + f(y);$$

$$f(x y) = f(x)f(y)$$

$$f(1) = 1$$

We say that $f$ is the Frobenius endomorphism associated with $\delta$.

Conversely, let $f$ be an endomorphism of a commutative ring $A$ without $p$-torsion. Suppose that for all $x \in A$

$$f(x) \equiv x^q \mod p A.$$

We obtain a structure of $\delta$-ring on $A$ defined as follows:

$$\delta(x) = \frac1{p}(f(x) - x^q).$$

Examples

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