Lambda-rings for formula-phobics

In 1956, Alexander Grothendieck (middle) introduced -rings in an algebraic-geometric context to be commutative rings A equipped with a bunch of operations (for all numbers ) satisfying a list of rather obscure identities. From the easier ones, such as

to those expressing and via specific universal polynomials. An attempt to capture the essence of -rings without formulas?

Lenstra’s elegant construction of the 1-power series rings requires only one identity to remember

.

Still, one can use it to show the existence of ringmorphisms , for all numbers . Consider the formal ‘logarithmic derivative’

where is the usual formal derivative of a power series. As this derivative satisfies the chain rule, we have

and so all the maps are additive. To show that they are also multiplicative, it suffices by functoriality to verify this on the special 1-series for all . But,

That is, and Lenstra’s identity implies that is indeed multiplicative! A first attempt :

hassle-free definition 1 : a commutative ring is a -ring if and only if there is a ringmorphism splitting , that is, such that .

In particular, a -ring comes equipped with a multiplicative set of ring-endomorphisms satisfying . One can then define a -ringmorphism to be a ringmorphism commuting with these endo-morphisms.

The motivation being that -rings are known to form a subcategory of commutative rings for which the 1-power series functor is the right adjoint to the functor forgetting the -structure. In particular, if is a -ring, we have a ringmorphism corresponding to the identity morphism.

But then, what is the connection to the usual one involving all the operations ? Well, one ought to recover those from .

For to be a ringmorphism will require identities among the . I hope an expert will correct me on this one, but I’d guess we won’t yet obtain all identities required. By the very definition of an adjoint we must have that is a morphism of -rings, and, this would require defining a -ring structure on , that is a ringmorphism , the so called Artin-Hasse exponential, to which I’d like to return later.

For now, we can define a multiplicative set of ring-endomorphisms from requiring that for all . Another try?

hassle-free definition 2 : is a -ring if and only if there is splitting to satisfying the compatibility relations .

But even then, checking that a map is a ringmorphism is as hard as verifying the lists of identities among the . Fortunately, we get such a ringmorphism for free in the important case when A is of ‘characteristic zero’, that is, has no additive torsion. Then, a ringmorphism exists whenever we have a multiplicative set of ring endomorphisms for all such that for every prime number the morphism is a lift of the Frobenius, that is, .

Perhaps this captures the essence of -rings best (without the risk of getting an headache) : in characteristic zero, they are the (commutative) rings having a multiplicative set of endomorphisms, generated by lifts of the Frobenius maps.