ETALE EXTENSIONS OF λ-RINGS CHARLES REZK Abstract. Given a λ-ring A and a formally etale morphism f: A→B of commutative rings, one may ask: What are the possible λ-ring strutures on B such that f is a map of λ-rings? We give the answer: Such a lifted λ-ring structure on B is determined uniquely by a compatible choice of lifts of theAdamsoperationsψp fromAtoB forallprimespwhichsatisfyFrobeniuscongruences. Inother words, to extend a λ-ring structure along a formally etale morphism, we need not be concerned about the “non-linear” part of the λ-ring structures in question. Contents 1. Introduction 1 2. θp-rings 1 3. Lifting problems for θp-rings 4 4. λ-rings 7 5. Lifting problems for λ-rings 10 References 11 1. Introduction We assume the reader is familiar with λ rings. A ψ-ring B is a commutative ring equipped with “Adams operations”, i.e., ring endomorphisms ψp: B → B for each prime p which pairwise commute. A ψ-ring is said to satisfy the Frobenius condition if each ψp is a lift of Frobenius on B/pB. Every λ-ring has an underlying ψ-ring, which necessarily satisfies the Frobenius condition. Fix a λ-ring A and a map f: A → B of commutative rings which is formally etale. We will prove the following. • If B is a ψ-ring satisfying the Frobenius condition, and f: A → B is a map of ψ-rings, then there exists a unique λ-ring structure on B compatible with the given ψ-ring structure, making f a map of λ-rings (5.4). • If, in addition, h: B → C is a map of ψ-rings such that hf: A → C is a map of λ-rings, then h is a map of λ-rings (5.5). We will prove these by first dealing with the analogous result in the case of θp-rings, which are a p-local analogue of λ-rings. We will then prove the result for λ-rings using a characterization of λ-rings in terms of their underlying θp-ring structures (4.10). This characterization is of independent interest; I am unaware that it has appeared in the literature up to now.1 2. θp-rings 2.1. θp rings. Fix a prime p ∈ Z. A θp-ring is a commutative ring R equipped with a function θp: R → R satisfying the following identities: • θp(a+b) = θp(a)+θp(b)−C (a,b), where C (x,y) = (cid:80)p−1p−1(cid:0)p(cid:1)xiyp−i ∈ Z[x,y]; p p i=1 i Date: March 22, 2014. 1I have a vague sense that I have run across it before somewhere, but I cannot find it. 1 2 CHARLESREZK • θp(ab) = θp(a)bp+apθp(b)+pθp(a)θp(b); • θp(1) = 0. A morphism of θp-rings is a ring homomorphism which commutes with θp. We write θpRing for the category of θp-rings.2 Note that for n ∈ Z, the above identities for θp imply • θp(na) = nθp(a)−F (n)ap where F (n) = (np−n)/p ∈ Z. p p (This has the amusing consequence that for p odd we have θp(−a) = −θp(a), while on the other hand when p = 2 we have θ2(−a) = −θ2(a)−a2.) 2.2. θp-subrings. Given any collection {a } of elements of a θp-ring R, and any polynomial s s∈S f(x) ∈ Z[x |s ∈ S] with integer coefficients with variables indexed by S, it is straightforward to see s using the above identities that there exists a polynomial g(x,y) ∈ Z[x ,y |s ∈ S] such that s s θp(f(a )) = g(a ,θp(a )). s s s This implies the following. 2.3. Proposition. Let R be a θp-ring, and let X ⊆ R be a subset closed under θp (i.e., such that x ∈ X implies θp(x) ∈ X). Then the subring S ⊆ R generated as a ring by the set X is also a subset closed under θp, and thus is itself a θp-ring with θp-operation obtained by restriction from R. 2.4. θp-ideals. An ideal I ⊆ R in a θp-ring is a θp-ideal if for all x ∈ I and a ∈ R, we have that θp(a+x)−θp(a) ∈ I. It is straightforward to show that an ideal I ⊆ R is θp-ideal if and only if the quotient ring R/I inherits a (necessarily unique) θp-ring structure so that the quotient map R → R/I is a homomorphism of θp-rings. 2.5. Proposition. Let R be a θp-ring, let X ⊆ R be a subset, and let I = (X) be the ideal in R generated by X. Then I is a θp-ideal if and only if θp(x) ∈ I for all x ∈ X. Proof. The “only if” part is clear, since θp(x) = θp(0+x)−θp(0). Suppose then that X ⊆ R satisfies the given condition; we want to show that I = (X) is a θp-ideal. LetJ = {x ∈ I | θp(x) ∈ I}. Notethat; (1)ifx,y ∈ J, thenθp(x+y) = θp(x)+θp(y)−C (x,y) ∈ p I and thus x+y ∈ J; (2) if x ∈ J and a ∈ R, then θp(ax) = θp(a)xp+apθp(x)+pθp(a)θp(x) ∈ I, and thus ax ∈ J. Thus J is a subideal of I containing X, whence J = I, and thus I is closed under the operation θp. Now note that if x ∈ I and a ∈ R, we have that θp(a+x)−θp(a) = θp(x)−C (a,x) ∈ I, p as desired. (cid:3) 2.6. Corollary. If R is a θp-ring and I,J ⊆ R are θp-ideals, then IJ ⊆ R is a θp-ideal. Proof. If x ∈ I and y ∈ J, then θp(xy) = θp(x)yp +xpθp(y)+pθp(x)θp(y) ∈ IJ, and thus IJ is a θp-ideal by (2.5). (cid:3) 2.7. ψp-rings. A ψp-ring is a pair (R,ψp) consisting of a commutative ring R equipped with a commutative ring map ψp: R → R. There is an evident category ψpRing of ψp-rings, whose morphisms are ring homomorphism which commute with ψp. (The “p” in the term “ψp-ring” and notation “ψp” and “ψpRing” is meant to stand for a prime p. In this paragraph it has served a purely decorative role, but it will matter in the following paragraph.) We say that a ψp-ring (R,ψp) satisfies the Frobenius condition if ψp is a lift of Frobenius, i.e., if for all a ∈ R we have that ψp(a) ≡ ap mod pR. We write ψpRing for the full subcategory of Fr ψpRing consisting of objects which satisfy the p-Frobenius condition. 2These are also known as δ-ring relative to p (Joyal [Joy85]) or a ring with p-derivation (Buium). We are following Bousfield’s [Bou96] terminology here. ETALE EXTENSIONS OF λ-RINGS 3 2.8. The Adams operation of a θp-ring. Given a θp-ring R, we define the Adams operation ψp = ψp: R → R by ψp(a) = ap + pθp(a). It is immediate that (R,ψp) is a ψp-ring, which furthermore satisfies the Frobenius condition. Thus, we have obtained a forgetful functor θpRing → ψpRing ⊂ ψpRing. Fr 2.9. The congruence criterion. Thefollowinggivesacompletecriterionforconstructingaθp-ring compatible with a given ψp-ring structure (R,ψp) the case that R has no p-torsion. 2.10. Proposition. Let R be a ψp-ring satsifying the Frobenius condition. If R is also p-torsion free, then there exists a unique θp-ring structure on R, compatible with the given ψp. Proof. Given such (R,ψp), define a function θp: R → R by θp(a) = (ψp(a)−ap)/p, and verify the identities for a θp-ring directly. (cid:3) ThisimpliesthattheforgetfulfunctorU: θpRing → ψpRing ⊆ ψpRingrestrictstoanequivalence Fr of categories θpRing −→∼ (ψpRing ) , tf Fr tf where these denote full subcategories of p-torsion free objects in θpRing and ψpRing respectively. Fr 2.11. θp-rings as coalgebras. Given a ring R, we define a ring V(R) = V (R) as follows. The p underlying set of V(R) is R×R, and addition and multiplication are defined by (x,y)+(x(cid:48),y(cid:48)) := (x+x(cid:48),y+y(cid:48)−C (x,x(cid:48))), p (x,y)·(x(cid:48),y(cid:48)) := (xx(cid:48),yx(cid:48)p+xpy(cid:48)+pyy(cid:48)). The map (cid:15) : V(R) → R defined by (cid:15) (x,y) = x is a ring homomorphism. V V A V-coalgebra is a pair (R,α) consisting of a ring R and a ring homomorphism α: R → V(R) such that (cid:15) α = id . A map of V-coalgebras is a ring homomorphism which commutes with α. V R 2.12. Proposition. There is an equivalence of categories θpRing −→∼ VCoalg, which sends the θp-ring (R,θp) to the V-coalgebra (R,α), where α(x) = (x,θp(x)). Proof. Straightforward. (cid:3) 2.13. Remark. One may also describe θp-rings as the coalgebras for a certain comonad W on Ring. The comonad W is in fact the cofree comonad on the augmented endofunctor (V,(cid:15) ). It is V well-known that the underlying functor W: Ring → Ring of this comonad is in fact the p-typical Witt functor.3 2.14. ψp-rings as coalgebras. Similarly as above, let G(R) = R × R as a ring, and define (cid:15) : G(R) → R by (cid:15) (x,y) = x. A G-coalgebra is a pair (R,β) with β: R → G(R) a ring G G homomorphism such that (cid:15) β = id . Let π: V(R) → G(R) denote the map π(x,y) = (x,xp+py). G R It is clear that π is a ring homomorphism natural in R, and that (cid:15) π = (cid:15) . G V 2.15. Proposition. There is an equivalence of categories ψpRing −→∼ GCoalg which sends (R,ψp) to the G-coalgebra (R,(id,ψp)). With respect to this equivalence and that of (2.12), the forgetful functor θpRing → ψpRing corresponds to the functor VCoalg → GCoalg which sends (R,α) to (R,πα). Proof. Immediate. (cid:3) 3I believe this is an observation of Joyal [Joy85]. 4 CHARLESREZK 2.16. Limits and colimits of θp-rings. 2.17. Proposition. The category θpRing of θp-rings has all small limits and colimits, and the forgetful functor θpRing → Ring which sends a θp-ring to its underlying commutative ring preserves limits and colimits. Proof. To prove the statement about limits, note that if A: C → θpRing is a functor from a small category, and U: θpRing → Ring denotes the underlying ring functor, we can define a operator θp on the set lim UA componentwise, and check that it satisfies the axioms for a θp-ring. It is then C straightforward to check that this realizes the limit of A: C → θpRing. To prove statement about colimits, let A: C → θpRing be a functor from a small category, and U: θpRing → Ring the underlying ring functor. Let S = colim UA, the colimit of the diagram in C Ring, with i(c): UA(c) → S denoting the tautological maps. Each θp-ring A(c) corresponds to a ring homomorphism α(c): UA(c) → V(UA(c)) which is a section π: V(UA(c)) → UA(c), and these maps fit together to give a ring homomorphism α: S = colim UA −c−o−lim−C−→α colim V(UA) −(−V−(i−(c−)→)) V(S), C C and that πα = id. Thus (S,α) defines a θp-ring structure on S, and it is straightforward to verify that S is the colimit of A. (cid:3) 3. Lifting problems for θp-rings 3.1. Lifting an ψp-ring structure to a θp-ring structure. Given a ψp-ring (R,ψp), a θp-ring structure on it is a θp-ring structure on R such that ψp(x) = xp+pθp(x). We now consider the following problem: given a ψp-ring (R,ψp), what are the possible θp-ring structures on it? In view of (2.12) and (2.15), we see that a θp-ring structure on (R,ψp) corresponds exactly to a homomorphism α fitting in the diagram V(R) (cid:60)(cid:60) α π (cid:15)(cid:15) (cid:47)(cid:47) R R×R (id,ψp) Thus, to understand this lifting problem we must examine the homomorphism π. Let V(R) = π(V(R)) ⊆ R×R denote the image of π. It is a subring of R×R, described as the subset {(x,y) | xp ≡ y mod p}. Let I(R) = Ker(π) = Ker(V(R) (cid:16) V(R)) ⊆ V(R). 3.2. Proposition. Let R be a commutative ring, and let V(R) and I(R) be defined as above. (1) As an abelian group, the quotient group (R×R)/V(R) is isomorphic to R/pR. (2) As an abelian group, I(R) is isomorphic to R[p] = Ker[p: R → R], the additive group of p-torsion elements in R. (3) The ideal I(R) ⊆ V(R) is square-zero, and thus 0 → I(R) → V(R) (cid:16) V(R) ⊆ R×R presents V(R) as a square-zero extension of V(R) by I(R). (4) Let φ: V(R) → R/pR be the ring homomorphism defined by φ(x,y) = xp = y. As a V(R)- module, I(R) is isomorphic to φ∗(R[p]), the module obtained by restricting scalars along φ from the evident R/pR-module structure on R[p]. Proof. The ring V(R) is isomorphic to the limit of the diagram x(cid:55)→xp y←y R −−−→ R/pR ←−−− R, (cid:91) and (1) follows immediately. ETALE EXTENSIONS OF λ-RINGS 5 Note that as a set, I(R) = {(0,z) | pz = 0} ⊆ V(R). Statements (2) and (3) are immediate from the description of the ring structure on V(R). For statement (4), let (x,y) ∈ V(R), and choose any lift (x,u) ∈ V(R), so that pu = y −xp. Then for (0,z) ∈ I(R), we have (x,u)·(0,z) = (0,xpz+puz) = (0,xpz) = (0,yz). This verifies the claim about the module structure on I(R). (cid:3) Thus,toconstructaθp-ringstructureon(R,ψp),wemust(1)showthattheimageof(id,ψp): R → R×R lies in V(R), and (2) lift the resulting map R → V(R) to a homomorphism α: R → V(R). Step (1) exactly says that ψp must satisfy the Frobenious condition. 3.3. Remark. If R has no p-torsion, then I(R) = 0, and thus V(R) → V(R) is an isomorphism. Thus we recover the congruence criterion (2.10) for torsion free θp-rings. 3.4. θp-ring structures and p-localization. Fix a ψp-ring (R,ψp). Tensoring with Z gives rise (p) to a ψp-ring (R ,ψp). (p) 3.5. Proposition. There is a one-to-one correspondence (cid:8)θp-ring structures on (R,ψp)(cid:9) ←→ (cid:8)θp-ring structures on (R ,ψp)(cid:9) (p) Proof. From (3.2), we have natural exact sequences of abelian groups π 0 → R[p] → V(R) −→ R×R → R/pR → 0. ∼ ∼ When we plug in the homomorphism j: R → R , and observe that R[p] −→ R [p] and R/pR −→ (p) (p) R /pR , we obtain a pullback square of rings of the form (p) (p) V(j) (cid:47)(cid:47) V(R) V(R ) (p) π π (cid:15)(cid:15) (cid:15)(cid:15) (cid:47)(cid:47) R×R R ×R (p) (p) j×j Also,V(R )isaZ -module,asitfitsinafiniteexactsequencewhoseothertermsareZ -modules. (p) (p) (p) To define the correspondence asserted by the proposition, we observe that there is a bijection {α: R → V(R) | πα = (id,ψp)} ←→ {α(cid:48): R → V(R ) | πα(cid:48) = (id,ψp)}, (p) (p) sending α to the unique homomorphism α(cid:48) such that α(cid:48)j = V(j)α. (cid:3) Thus, the problem of lifting a ψp-structure to a θp-structure is (unsurprisingly) a purely p-local problem. 3.6. The relative lifting problem for θp-rings. Nowweconsiderthefollowingproblem. Suppose we are given a θp-ring (A,θp) and a ψp-ring (B,ψp) which satisfies the Frobenius condition, together with homomophism f: A → B of ψp-rings (using the underlying ψp-structure of the (A,θp)). What are the possible θp-ring structures on (B,ψp) making f a map of θp-rings? In view of the previous sections, we see that providing such a structure amounts to producing a dotted arrow α in B A V(f)αA (cid:47)(cid:47) V(B) (cid:58)(cid:58) f αB π (cid:15)(cid:15) (cid:15)(cid:15) (cid:47)(cid:47) B V(B) ⊆ B×B (id,ψp) making the diagram commute. 6 CHARLESREZK Recall that f: A → B is formally etale if for every ring R and square-zero ideal I ⊆ R, and every commutative diagram of ring homomorphisms of the form (cid:47)(cid:47) A (cid:61)(cid:61)R f (cid:15)(cid:15) (cid:15)(cid:15) (cid:47)(cid:47) B R/I there exists a unique dotted arrow making the diagram commute. 3.7. Proposition. If (A,θp) is a θp-ring, (B,ψp) a ψp-ring satisfying the Frobenius condition, and f: A → B is a map of ψp-rings which is formally etale as a map of commutative rings, then there exists a unique θp-ring structure on (B,ψp) making f a map of θp-rings. Proof. Immediate in view of the above remarks and (3.2)(3). (cid:3) The lift of the previous proposition is natural. 3.8. Proposition. Consider a commutative diagram g (cid:47)(cid:47) A (cid:62)(cid:62)C f (cid:15)(cid:15) h B of ring maps, such that (i) A, B, and C are θp-rings, (ii) f and g are maps of θp-rings, (iii) h is a map of ψp-rings, and (iv) f is formally etale. Then h is a map of θp-rings. Proof. Consider the diagrams A A V(g)αA (cid:47)(cid:47) V(C) A V(f)αA (cid:47)(cid:47) V(B) V(h) (cid:47)(cid:47) V(C) (cid:55)(cid:55) (cid:55)(cid:55) f g αC π f αB π π (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) B C C ×C B B×B C ×C h (id,ψp) (id,ψp) h×h C B The solid arrow diagrams commute, using that f and g are maps of θp-rings. The two outer rectangles are actually identical, since V(g)α = V(h)V(f)α and (id,ψp)h = (h,ψph) = (h,hψp) = (h×h)(id,ψp). A A C C B B Thus the homomorphisms α h,V(h)α : B → V(C) must coincide, since f is formally etale and C B they are solutions to the same lifting problem, whence h is a map of θp-rings as desired. (cid:3) Given a θp-ring A, let θpRing(A) denote the category of θp-rings under A, and ψpRing (A) the Fr category of ψp-rings under A which satisfy the Frobenius congruence. Let θpRing(A) and f.etale ψpRing (A) denote the respective full subcategories consisting of objects f: A → B such that Fr f.etale f is formally etale. 3.9. Proposition. The evident forgetful functor θpRing(A) → ψpRing (A) is an equiva- f.etale Fr f.etale lence of categories. Proof. Clear from (3.7) and (3.8). (cid:3) The statement of the previous proposition remains true if we replace “formally etale” with any subclass of maps, such as “etale” or “weakly etale”. 3.10. Remark. In fact, we can generalize the above a little bit. Say that f: A → B is p-formally etale if f ⊗Z : A → B . In view of (3.5), the propositions (3.7), (3.8), and (3.9) apply with (p) (p) (p) “formally etale” replaced by “p-formally etale”. ETALE EXTENSIONS OF λ-RINGS 7 4. λ-rings A λ-ring is a commutative ring R equipped with functions λn: R → R for n ≥ 0, satisfying identities of the form • λ0(x) = 1 and λ1(x) = x, • λn(x+y) = (cid:80) λi(x)λj(y), i+j=n • λn(xy) = P (λ1(x),...,λn(x);λ1(y),...,λn(y)), n • λmλn(x) = P (λ1(x),...,λmn(x)), m,n where P and P are certain polynomials with integer coefficients. We refer the reader elsewhere n m,n for a complete definition, for instance [Yau10]. We write λRing for the category of λ-rings. The purpose of this section is to give a characterization of λ-rings in terms of θp-rings. That is, we will show that a λ-ring is nothing more than a commutative ring R equipped with θp-ring structures for each prime p, which are compatible in the sense that ψpθq = θqψp for all distinct primes p,q. 4.1. Remark. It is known to those who know these things that data of a λ-ring structure on R is equivalent to: a θp-structure on R for each prime, together with for each pair of distnict primes p,q a somewhat non-trivial relation relating θpθq(x) and θqθp(x) up to terms which do not involve compositions of θ-operations. (Probably this is in [Joy85]; see also [Bor11, §1.19].) Our characterization allows us to avoid explicit mention of this relation. 4.2. Facts about lambda rings. We note the following facts about λ-rings. (1) λ-rings are a variety of universal algebra, and thus λRing is a locally presentable category. In particular, any functor U: λRing → C which preserves small colimits admits a right adjoint. (2) LimitsandcolimitsinλRing exist, andtheevidentforgetfulfunctorλRing → Ring preserves limits and colimits. (3) The free λ-ring on one generator F has the form F ≈ Z[λn(x)|n ≥ 1], where x = λ1(x) is the generator. (4) Any λ-ring R has natural Adams operations ψn: R → R for n ≥ 1, which are ring homomorphisms; furthermore, ψmψn = ψmn and ψ1 = id. (5) For p prime, ψp(x) ≡ xp mod pR in any λ-ring R. (6) For every prime p there exist natural functions θp: R → R on any λ-ring such that ψp(x) = xp+pθp(x), and (R,θp) is in fact a θp-ring, as can be shown by checking the appropriate formulas in the free λ-ring on one generator F, which is torsion free. (7) For distinct primes p and q, we have that ψpθq = θqψp as functions on any λ-ring, as can be shown by verifying that ψpθq(x) = θqψp(x) in F. (8) Let δ: F → Z be the λ-ring homomorphism from the free λ-ring on one generator sending the generator x to 0, and let J = Kerδ. Then J/J2 is a free abelian group on {λn(x)} . n≥1 (9) We have that ψn(x) ≡ (−1)n−1nλn(x) mod J2. Thus, any sequence {u } of elements n n≥0 in J such that nu ≡ ψn(x) mod J2 is a basis for J/J2. n 4.3. Θ-rings. A Θ-ring is the data (R,{θp}) consisting of a commutative ring R and a choice for each prime p ∈ Z of a θp-structure on R, such that for all distinct primes p and q, we have that ψpθq = θqψp, where ψp(x) = xp+pθp(x) is the Adams operation associated to θp. We note that it is also the case that ψpθp = θpψp, as this is true in any θp-ring. A morphism A → B of Θ-rings is a map which commutes with all the structure, i.e., a ring homomorphism f: A → B such that fθp = θpf for all p. We write ΘRing for the category of Θ-rings. 8 CHARLESREZK An ideal I ⊆ R of a Θ-ring is a Θ-ideal if it is a θp-ideal for all p. It is clear that if I is a Θ-ideal, then R/I admits a unique Θ-ring structure as a quotient of the structure on R. 4.4. Facts about subrings of Θ-rings. We collect some facts for use in the proof in the next section. Let R be a Θ-ring, and consider an ordinary subring S ⊆ R. Write ΘS ⊆ R for the ordinary subring generated by the set S ∪(cid:83) θp(S). It is straightforward to show (see §2.2) that if S is p generated as a subring by a subset X ⊆ R, then ΘS is generated as a subring by the subset X ∪{θp(x) | x ∈ S, p prime}. It is clear from (2.3) that if S ⊆ R is a subring, then (cid:83) ΘkS is the Θ-subring in R generated by k S (i.e., the smallest subring of R containing S and closed under the θp operations). Although a subring is not generally an ideal, it is a subgroup, and so it makes sense to talk about congruence modulo a subring: we say x ≡ y ∈ S if x−y ∈ S, when S ⊆ R is a subring and x,y ∈ R. 4.5. Proposition. Let R be a Θ-ring, S ⊆ R an ordinary subring, and x ∈ S. Then for all primes p and q, we have that θpθq(x) ≡ θqθp(x) mod ΘS. Proof. If p = q this is obvious. For distinct primes p and q we have ψpθq(x) = θq(x)p+pθpθq(x) ≡ pθpθq(x) mod ΘS, and θqψp(x) = θq(xp+pθp(x)) = θq(xp)+(cid:0)pθqθp(x)−F (p)θp(x)q(cid:1)−C (xp,pθp(x)) q p ≡ pθqθp(x) mod ΘS. Therefore p(θpθq(x) − θqθp(x)) ∈ ΘS. By symmetry we also have q(θpθq(x) − θqθp(x)) ∈ ΘS, and since p and q are relatively prime, a suitable integer combination of these congruences gives θpθq(x)−θqθp(x) ∈ ΘS, as desired. (cid:3) 4.6. Remark. The argument of (4.5) actually shows that θpθq(x)−θqθp(x) = f(x,θp(x),θq(x)) where f is some polynomial with integer coefficients. It is not hard to describe this polynomial explicitly, e.g., [BS09, Def. 2.2] or [Bor11, (1.19.4)]. Let P denote the set of finite sequences P = (p ,...,p ) of primes p ∈ Z, including the empty 1 k i sequence, and write |P| = k for the length of the sequence P. For x ∈ R and P ∈ P write θP(x) = θp1···θpk(x). 4.7. Proposition. Let R be a Θ-ring and S ⊆ R ordinary subring. If x,y ∈ ΘS are such that x ≡ y mod S, then θP(x) ≡ θP(y) mod Θ|P|S for all P ∈ P. Proof. If |P| = 0, there is nothing to prove. If P = (p), then if x = y+a with a ∈ S, we have that θp(x)−θp(y) = θp(a)−C (y,a) ∈ ΘS. p The case of |P| > 1 is handled by induction on the length: given θP(x) ≡ θP(y) mod Θ|P|S with θP(x),θP(y) ∈ Θ|P|+1S, it follows using the length-one case already proved that θqθP(x) ≡ θqθP(y) mod Θ|P|+1S and θqθP(x),θqθP(x) ∈ Θ|P|+2S. (cid:3) 4.8. Proposition. Let R be a Θ-ring, S ⊆ R an ordinary subring, and x ∈ S. Let P,Q ∈ P be two sequences of the same length, where Q is obtained from P by reordering its elements. Then θP(x) ≡ θQ(x) mod Θ|P|−1S. ETALE EXTENSIONS OF λ-RINGS 9 Proof. It suffices to consider the case of pairs of sequences obtained by reordering an adjacent pair of elements. Thus, let p,q be primes, U,V ∈ P and let P = (U,p,q,V) and Q = (U,q,p,V). Then θV(x) ∈ Θ|V|S, whence θpθqθV(x) ≡ θqθpθV(x) mod Θ|V|+1S by (4.5). As θpθqθV(x),θqθpθV(x) ∈ Θ|V|+2S, it follows that ΘUθpθqΘV(x) ≡ θUθqθpθV(x) mod Θ|U|+1+|V|S = Θ|P|−1S by (4.7). (cid:3) 4.9. Equivalence of Θ-rings and λ-rings. There is an evident “forgetful” functor U: λRing → ΘRing, by §4.2(6), (7). 4.10. Proposition. The forgetful functor U: λRing → ΘRing is an equivalence of categories. Proof. Let R be a Θ-ring. By (2.3), it is clear that for any x ∈ R, the subring S ⊆ R generated as a ring by the set {θP(x)} is a sub-Θ-ring. We may thus construct the free Θ-ring on one generator E as the P∈P quotient of the polynomial ring Z[θP(x) | P ∈ P] by the ideal consisting of relations which hold in every Θ-ring. Let P ⊆ P denote the subset consisting of sequences P = (p ,...,p ) such that p ≤ ··· ≤ p . ≤ 1 k 1 k By inductive application of (4.8), we see that E is in fact generated as a commutative ring by the set {θP(x)} . That is, if we write S for the subring of E generated by {θP(x) | P ∈ P , |P| ≤ k}, P∈P≤ k ≤ we show by induction on k that ΘkS ⊆ S , using (4.8). Thus E = (cid:83) ΘkS = (cid:83) S . 0 k k 0 k k Let δ: E → Z be the Θ-homomorphism sending x (cid:55)→ 0, and let I = Kerδ. Clearly I is a Θ-ideal, and thus I2 is a Θ-ideal by (2.6). Furthermore, as an ideal I is generated by the set {θP(x)} , P∈P ≤ since these elements generate E as a ring and are contained in I. Therefore I/I2 is generated as an abelian group by the θP(x) with P ∈ P . Note also that pθp(x) ≡ ψp(x) mod I2, and thus that ≤ nθP(x) ≡ ψn(x) mod I2 where P = (p ,...,p ) ∈ P is such that n = (cid:81)p . 1 k i NowletF denotethefreeλ-ringononegenerator,withaugmentationidealJ,andletφ: E → U(F) be the Θ-ring map sending generator to generator. By §4.2(3), (9), the induced map I/I2 → J/J2 ≈ Z{“n−1ψn(x)”} n≥1 the generators θP(x) of I/I2 with P ∈ P bijectively to a basis of J/J2, whence I/I2 → J/J2 is a ≤ bijection, whence φ is an isomorphism. The forgetful functor U: λRing → ΘRing preserves small colimits (since in both λRing and ΘRing colimits are computed as in commutative rings, by §4.2(2) and (2.17) respectively), and thus U admits a right adjoint G: ΘRing → λRing by §4.2(1). Now let η: Id → GU and (cid:15): UG → Id be unit and counit of the U (cid:97) G adjunction. For a λ-ring R, the composite U φ∗ R ≈ λRing(F,R) −→ ΘRing(UF,UR) −→ ΘRing(E,UR) ≈ R is the identity function on the set R, and thus U: λRing(F,R) → ΘRing(UF,UR) is a bijection for every λ-ring R. The composite map U (4.11) R ≈ λRing(F,R) −→ ΘRing(UF,UR) ≈ λRing(F,GUR) ≈ GUR is therefore a bijection, whence η: Id → GU is an isomorphism. For a Θ-ring S, the composite map (φ∗)−1 φ∗◦U (4.12) S ≈ ΘRing(E,S) −−−−→ ΘRing(UF,S) ≈ λRing(F,GS) −−−→ ΘRing(E,UGS) ≈ UGS ∼ is a bijection of sets (using that (4.11) is always an isomorphism), and it is straightforward to check that the composite of (4.12) is an inverse to (cid:15): UGS → S. Thus we have shown that U (cid:97) G is an equivalence of categories. (cid:3) 10 CHARLESREZK 5. Lifting problems for λ-rings In view of (4.10), we will treat λ-rings as synonymous with Θ-rings. 5.1. Ψ-rings. A Ψ-ring is the data (R,{ψp}) consisting of a commutative ring R and a choice for each prime p ∈ Z of a ψp-ring structure on R, such that for all distinct primes p and q we have that ψpψq = ψqψp. We write ΨRing for the category of Ψ-rings. We say that a Ψ-ring satisfies the Frobenius condition if each ψp does, i.e., if ψp(x) ≡ xp mod p for all p. We write ΨRing ⊆ ΨRing for the full subcategory of Ψ-rings satisfying the Fr Frobenius condition. There is an evident forgetful functor λRing → ΨRing ⊆ ΨRing. Fr We observe that if F: Ring → Ring is any functor and R is a Ψ-ring, then F(R) inherits a natural Ψ-ring structure, with operations ψp = F(ψp). For instance, this applies to the functors F(R) R V : Ring → Ring of §2.11. p 5.2. Lifting a Ψ-ring structure to a λ-ring structure. We now consider the following problem: given a Ψ-ring R, what are the possible λ-ring structures on R compatible with the given Ψ-ring structure? In view of (4.10), (2.12), and (2.15), such a λ-ring structure on R corresponds exactly to a choice for each prime p of Ψ-ring homomorphism α fitting in the commutative diagram p V (R) (cid:60)(cid:60) p αp π (cid:15)(cid:15) (cid:47)(cid:47) R R×R (id,ψp) That is, a Ψ-ring map α : R → V (R) such that πα = (id,ψp) corresponds exactly to a θp-ring p p p structure on (R,ψp) such that ψqθp = θpψq for all primes q (cid:54)= p, by the discussion in §3.1. (Recall that for any θp-ring structure on (R,ψp) we automatically have that ψpθp = θpψp.) A choice of such α for all primes p thus amounts to a λ-ring structure on R by (4.10). p Clearly, a necessary condition to lift Ψ-ring structure on R to a λ-ring structure is that the Ψ-ring structure should satisfy the Frobenius condition. As a corollary of (3.2)(2) we obtain Wilkerson’s criterion: if R is torsion free as an abelian group, then any Ψ-ring structure on R satisfying the Frobenius condition lifts uniquely to a λ-ring structure. 5.3. The relative lifting problem for λ-rings. Now consider the following problem. Suppose we are given a λ-ring A , a Ψ-ring B which satisfies the Frobenius condition, and a homomorphism f: A → B of Ψ-rings. What are the possible λ-ring structures on B making f a map of λ-rings? 5.4. Proposition. If A is a λ-ring, B a Ψ-ring satisfying the Frobenius condition, and f: A → B is a map of Ψ-rings which is formally etale as a map of commutative rings, then there exists a unique λ-ring structure on B making f a map of λ-rings. Proof. In view of (3.7), there are unique θp-ring structures on B making f a map of θp-rings for each p. In view of (3.8), these θp-structures on B must commute with the given ψq-operations on B for q (cid:54)= p. That is, apply (3.8) for θp-rings to the commutative triangle of ring maps A fψAq (cid:47)(cid:47)(cid:58)(cid:58) B f (cid:15)(cid:15) ψq B B where f and thus fψq are θp-ring maps, to show that ψq : B → B is a map of θp-rings. (cid:3) A B