MAXIMAL Λ-ORDERS OVER Z -PRELIMINARY REPORT- JAMES BORGER, BART DE SMIT Abstract. We show that the category of Λ-rings that are finite ´ etale over Q as rings and also have a sub-Λ-ring which is an order is naturally equivalent to the category of finite discrete sets equipped with a continuous action of the monoid ˆ Z ◦ of pro-finite integers under multiplication. We also give an explicit description of the maximal such sub-Λ-ring. Introduction The most traditional definition of a Λ-action on a commutative ring R is a se- quence of set maps λ 1 ,λ 2 ,... from R to itself that satisfy certain complex implicitly- stated axioms. This notion was introduced by Grothendieck [3], under the name special λ-ring, to give an abstract context in which to discuss the structure on Grothendieck groups inherited from linear-algebraic operations such as tensor prod- ucts, exterior powers, and symmetric powers; and as far as we are aware, with only one exception [2], Λ-rings have been studied in the literature for this purpose only. However, it appears that the study of abstract Λ-rings—that is, those having no relation to K-theory—will have much to say about number theory, and the first author is in the midst of a long project exploring a general philosophy about this. For these applications, it is important to not restrict to Λ-rings that are finitely generated (as rings). But then, for exactly this reason, the question of how complex finitely generated Λ-rings can be becomes interesting. The purpose of this paper is to answer a question of a similar nature: what finite ´ etale Λ-rings over Q are of the form Q ⊗ A, where A is a Λ-ring that is finite flat over Z? We will consider Λ-actions only on rings R whose underlying abelian group is torsion-free, and for such rings, a Λ-action is the same as giving commuting ring endomorphisms ψ p of R, one for each prime p, lifting the Frobenius map modulo p— that is, such that ψ p (x) − x p ∈ pR for all x ∈ R. (The relation with the traditional definition is explained in Wilkerson [5].) An example that will be important for us is Z[µ r ]= Z[z]/(z r −1), where r is a positive integer and ψ p sends z to z p . A morphism f of torsion-free Λ-rings is the same as a ring map that satisfies f ◦ ψ p = ψ p ◦ f for all primes p. Note that if R is a Q-algebra, the congruence conditions in the definition above disappear. Also, Galois theory gives an equivalence between the category of finite ´ etale Q-algebras and the category of finite discrete sets equipped with a continuous action of the absolute Galois group G Q (with respect to a fixed algebraic closure ¯ Q). Combining these two remarks, we see that the category of Λ-rings that are finite ´ etale over Q is nothing more than the category of finite discrete sets equipped Date : December 13, 2004. 16:41. 1