nLab Joyal delta-ring



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

The notion of δ\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

δ(x)=ψ p(x)x pp.\delta(x) = \frac{\psi^p(x) - x^p}{p}.


Fix a prime number pp and a power q=p rq = p^r (r0r \geq 0).


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

  1. δ(x+y)=δ(y)+δ(y) i=1 q11p(qi)x iy qi\delta(x+y) = \delta(y) + \delta(y) - \sum_{i=1}^{q-1} \frac1{p}\binom{q}{i} x^i y^{q-i};

  2. δ(xy)=x qδ(y)+δ(x)y q+pδ(x)δ(y)\delta(x y) = x^q\delta(y) + \delta(x)y^q + p\delta(x)\delta(y), δ(1)=0\delta(1) = 0.

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

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

f(x+y)=f(x)+f(y);f(x+y) = f(x) + f(y);
f(xy)=f(x)f(y)f(x y) = f(x)f(y)
f(1)=1f(1) = 1

We say that ff is the Frobenius endomorphism associated with δ\delta.

Conversely, let ff be an endomorphism of a commutative ring AA without pp-torsion. Suppose that for all xAx \in A

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

We obtain a structure of δ\delta-ring on AA defined as follows:

δ(x)=1p(f(x)x q).\delta(x) = \frac1{p}(f(x) - x^q).



Last revised on July 25, 2023 at 07:45:39. See the history of this page for a list of all contributions to it.