There is also a notion of special lambda-ring. But in most cases by ‘’λ\lambda-ring’‘ is meant ‘’special λ\lambda-ring’’.



A λ\lambda-ring is a commutative ring which is in addition equipped with operations that behave as the operations of forming exterior powers (of vector spaces/representations) in a representation ring. The name derives from the common symbol Λ n\Lambda^n for the nnth exterior power.

Motivation from representation theory

Typically one can form direct sums of representations of some algebraic structure. The decategorification to isomorphism classes of such representations then inherits the structure of a commutative monoid. But nobody likes commutative monoids: we all have an urge to subtract. So, we throw in formal negatives and get an abelian group — the Grothendieck group.

In many situations, we can also take tensor products of representations. Then the Grothendieck group becomes something better than an abelian group. It becomes a ring: the representation ring. Moreover, in many situations we can also take exterior and symmetric powers of representations; indeed, we can often apply any Young diagram to a representation and get a new representation. Then the representation ring becomes something better than a ring: it becomes a λ\lambda-ring.

More generally, the Grothendieck group of a monoidal abelian category is always a ring, called a Grothendieck ring. If we start with a braided monoidal abelian category, this ring is commutative. But if we start with a symmetric monoidal abelian category, we get a λ\lambda-ring.

So, λ\lambda-rings are all about getting the most for your money when you decategorify a symmetric monoidal abelian category — for example the category of representations of a group, or the category of vector bundles on a topological space.

Unsurprisingly, the Grothendieck group of the free symmetric monoidal abelian category on one generator is the free λ\lambda-ring on one generator. This category is very important in representation theory. Objects in this category are called Schur functors, because for obvious reasons they act as functors on any symmetric monoidal abelian category. The irreducible objects in this category are called ‘Young diagrams’. Elements of the free λ\lambda-ring on one generator are called symmetric functions.

In terms of universal algebra

A λ\lambda-ring LL is a P-ring presented by the polynomial ring Symm=[h 1,h 1]Symm=\mathbb{Z}[h_1,h_1\dots] in countably many indeterminates over the integers or, equivalently, SymmSymm is the ring of symmetric functions in countably many variables. This means that (the underlying set valued functor of) LL is a copresheaf presented by SymmSymm such that

  1. L:CRingCRingL:\CRing \to CRing defines an endofunctor on the category of commutative rings.

  2. LL gives rise to a comonad on CRingC Ring.

A λ\lambda-ring is hence a commutative ring equipped with a co-action of this comonad. As always is the case with monads and comonads this definition can be formulated in terms of an adjunction.


The “orthodox” definition


A λ\lambda-structure on a commutative unital ring RR is defined to be a sequence of maps λ n\lambda^n for n0n\ge 0 satisfying

  1. λ 0(r)=1\lambda^0(r)=1 for all rRr\in R

  2. λ 1=id\lambda^1=id

  3. λ n(1)=0\lambda^n(1)=0, for n>1n\gt 1

  4. λ n(r+s)= k=0 nλ k(r)λ nk(s)\lambda^n(r+s)=\sum_{k=0}^n \lambda^k(r)\lambda^{n-k}(s), for all r,sRr,s\in R

  5. λ n(rs)=P n(λ 1(r),,λ n(r),λ 1(s),,λ n(s))\lambda^n(r s)=P_n(\lambda^1(r),\dots,\lambda^n(r),\lambda^1(s),\dots,\lambda^n(s)) for all r,sRr,s\in R

  6. λ m(λ n(r)):=P m,n(λ 1(r),,λ mn(r))\lambda^m(\lambda^n(r)):=P_{m,n}(\lambda^1(r),\dots,\lambda^{m n}(r)), for all rRr\in R

where P nP_nand P m,nP_{m,n} are certain (see the reference for their calculation) universal polynomials? with integer coefficients. RR is in this case called a λ\lambda-ring. Note that the λ n\lambda^n are not required to be morphisms of rings.

A homomorphism of λ\lambda-structures is defined to be a homomorphism of rings commuting with all λ n\lambda^n maps.


There exists a λ\lambda-ring structure on the ring 1+R[[t]]1+ R[ [t] ] of power series with constant term 11 where

a) addition on 1+R[[t]]1+R[ [t] ] is defined to be multiplication of power series

b) multiplication is defined by

(1+ n=1 r nt n)(1+ n=1 s nt n):=1+ n=1 P n(r 1,,r n,s 1,,s n)t n(1+\sum_{n=1}^\infty r_n t^n)(1+\sum_{n=1}^\infty s_n t^n):=1+\sum_{n=1}^\infty P_n(r_1,\dots,r_n,s_1,\dots,s_n)t^n

c) the λ\lambda-operations are defined by

λ n(1+ m=1 r mt m)=1+ m=1 P m,n(r 1,,r mn)t m\lambda^n (1+\sum_{m=1}^\infty r_m t^m)=1+\sum_{m=1}^\infty P_{m,n}(r_1,\dots,r_{mn})t^m



Let Λ\Lambda denote the ring of symmetric functions, let RR be a λ\lambda-ring.

Then for every xRx\in R there is a unique homomorphism of λ\lambda-rings

Φ x:ΛR\Phi_x:\Lambda\to R

sending e 1xe_1\mapsto x, e nλ n(x)e_n\mapsto \lambda^n(x), p nψ n(x)p_n\mapsto\psi^n(x) where e ne_n denotes the nn-th elementary symmetric function and ψ n\psi^n denotes the nn-th Adams operation (explained in the reference).

Equivalently this result asserts that Λ\Lambda is the free λ\lambda-ring in the single variable e 1e_1.

This is due to (Hopkinson)


We define Φ x(e 1)=x\Phi_x(e_1)=x, then the assumption on Φ x\Phi_x to be a morphism of λ\lambda-rings yields Φ x(e n)=Φ x(λ n(e 1))=λ n(x)\Phi_x(e_n)=\Phi_x(\lambda^n(e_1))=\lambda^n(x).


(Hazewinkel 1.11, 16.1)

a) The endofunctor of the category of commutative rings

Λ:{CRingCRing A1+A[[t]]\Lambda:\begin{cases}C Ring \to C Ring\\A\mapsto 1 + A [ [t] ]\end{cases}

sending a commutative ring to the set of power series with constant term 11 is representable by the polynomial ring Symm[h 1,h 2,]Symm \coloneqq \mathbb{Z}[h_1, h_2,\dots] in an infinity of indeterminates over the integers.

b) There is an adjunction (forgetλΛ)(forget\, \lambda\dashv \Lambda) where forgetλ:λRingCRingforget\,\lambda: \lambda Ring\to CRing is the forgetful functor assigning to a λ\lambda-ring its underlying commutative ring.

The left inverse g S,A\g_{S,A} of the natural isomorphism q S,A:hom(forgetλ,A)hom(S,Λ(A))q_{S,A}:hom(forget\,\lambda,A)\to hom(S,\Lambda(A)) is given by the ghost component s 1s_1.

(see also (Borger 08, section 1.8))

An instructive introduction to the “orthodox”- and preparation for the “heterodox” view (described below) on λ\lambda-rings is Hazewinkel’s survey article on Witt vectors, (Hazewinkel). There is also a reading guide to that article.

The “heterodox” definition

There is a second, “heterodox” way to approach λ\lambda-rings with a strong connection to arithmetic discussed in detail in (Borger 08, section 1). An survey is in (Borger 09) where it says in the abstract:

The theory of Λ\Lambda-rings, in the sense of Grothendieck’s Riemann–Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z\mathbf{Z} of integers to produce Λ\Lambda-algebraic geometry. We show that Λ\Lambda-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z\mathbf{Z} and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this in a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.

First some standard notation:


For pp a prime number write 𝔽 p\mathbb{F}_p for the finite field whose underlying abelian group is the cyclic group /p\mathbb{Z}/p\mathbb{Z}.

For AA an 𝔽 p\mathbb{F}_p-algebra, then the Frobenius endomorphism

F p:AA F_p \colon A \longrightarrow A

is that given by taking each element to its ppth power

F p:xx p. F_p \colon x \mapsto x^p \,.

For pp a prime number, then a pp-typical Λ\Lambda-ring is

such that under tensor product with 𝔽 p\mathbb{F}_p it becomes the Frobenius morphism, def.2:

𝔽 p F A=F p:𝔽 p A𝔽 p A. \mathbb{F}_p \otimes_{\mathbb{Z}} F_A = F_p \; \colon \; \mathbb{F}_p \otimes_{\mathbb{Z}} A \longrightarrow \mathbb{F}_p \otimes_{\mathbb{Z}} A \,.

A big Λ\Lambda-ring is a commutative ring equipped with commuting endomorphisms, one for each prime number pp, such that each of them makes the ring pp-typical, respectively, as above.

This is def. 1.7 in (Borger 08), formulated for the special case of example 1.15 there (which is stated in terms of Witt vectors) and translated to Λ\Lambda-rings in view of prop. 1.10 c) (see the adjunction) there.

the following originates from revision 19 but needs attention


The pp-th Adams operation ψ p\psi_p is a Frobenius lift. Moreover given any two prime numbers then their Adams operations commute with each other.

The following two theorems are crucial for the “heterodox” point of view. We will see later that in fact we do not need the torsion-freeness assumption.


(Wilkerson’s theorem) Let AA be an additively torsion-free commutative ring. Let {ψ p}\{\psi_p\} be a commuting family of Frobenius lifts.

Then there is a unique λ\lambda-ring structure on AA whose Adams operations are the given Frobenius lifts {ψ p}\{\psi_p\}.


A ring morphism ff between two λ\lambda-rings is a morphism of λ\lambda-rings (i.e. commuting with the λ\lambda-operations) iff ff commutes with the Adams operations.


There is an equivalence between the category of torsion-free λ\lambda-rings and the category of torsion-free commutative rings equipped with commuting Frobenius lifts..

Now we will argue that these statements hold for arbitrary commutative rings.


a) The category λRing\lambda Ring of λ\lambda-rings is monadic and comonadic over the category of CRingC Ring of commutative rings.

b) The category λRing ¬tor\lambda Ring_{\neg tor} of λ\lambda-rings is monadic and comonadic over the category of CRing ¬torC Ring_{\neg tor} of commutative rings.


Let i:CRing ¬torCRing\i:C Ring_{\neg tor}\hookrightarrow C Ring be the inclusion. Let W W^\prime denote this comonad on CRing ¬torC Ring_{\neg tor}. Then

a) WLan iiW :CRingCRingW \coloneqq Lan_i i\circ W^\prime \colon C Ring\to CRing is a comonad.

b) The category of coalgebras of WW is equivalent to the category of λ\lambda-rings.

c) WW is the big-Witt-vectors functor.


The “heterodox” generalizes to arbitrary Dedekind domains with finite residue field.

For instance over F p[x]F_p[x] (instead of \mathbb{Z}), we would look at families of ψ\psi-operators indexed by the irreducible monic polynomials f(x)f(x), and each ψ f(x)\psi_{f(x)} would have to be congruent to the qq-th power map modulo f(x)f(x), where qq is the size of F p[x]/(f(x))F_p[x]/(f(x)).


Free and co-free Λ\Lambda-rings – Symmetric function and Witt vectors


The forgetful functor U:ΛRingCRingU \;\colon\; \Lambda Ring \longrightarrow CRing from Λ\Lambda-rings to commutative rings has

(SymmUW):ΛRingWUSymmCRing. (Symm \dashv U \dashv W) \;\colon\; \Lambda Ring \stackrel{\overset{Symm}{\leftarrow}}{\stackrel{\overset{U}{\longrightarrow}}{\underset{W}{\leftarrow}}} CRing \,.


This statement appears in (Hazewinkel 08, p. 87, p. 97, 98). The right adjoint in a more general context is in (Borger 08, prop. 1.10 (c)).


On the level of toposes (etale toposes) over these sites of rings, this statement reappears as an essential geometric morphism from the etale topos of Spec(Z) to that over “F1” in Borger's absolute geometry (Borger 08, exposition in Borger 09).


school/conference in Leiden: Frobenius lifts and lambda rings 5-10. October 2009 featuring

conference site


Discussion in the context of Borger's absolute geometry is in

Revised on July 23, 2014 08:09:18 by Urs Schreiber (