nLab
Lambda-ring

Disambiguation

There is also a notion of special lambda-ring. But in most cases by ”$\lambda$-ring” is meant ”special $\lambda$-ring”.

Idea

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

1. $L:CRing\to \mathrm{CRing}$ defines an endofunctor on the category of commutative rings.

2. $L$ gives rise to a comonad on $C\mathrm{Ring}$.

A $\lambda$-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.

Motivation from algebra

In many situations, we can take direct sums of representations of some algebraic gadget. So, decategorifying, the set of isomorphism classes of representations becomes 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 so-called Grothendieck group.

In many situations, we can also take tensor products of representations. Then our Grothendieck group becomes something better than an abelian group. It becomes a ring: the representation ring. But, we’re not done! 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 our representation ring becomes something better than a ring. It becomes a $\lambda$-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 $\lambda$-ring!

So, $\lambda$-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 space.

Unsurprisingly, the Grothendieck group of the free symmetric monoidal abelian category on one generator is the free $\lambda$-ring on one generator. This category is very important in representation theory. Object 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 $\lambda$-ring on one generator are called symmetric functions.

Definition

The “orthodox” view on $\lambda$-rings

Definition

A $\lambda$-structure on a commutative unital ring $R$ is defined to be a sequence of maps ${\lambda }^n$ for $n\ge 0$ satisfying

1. ${\lambda }^0(r)=1$ for all $r\in R$

2. ${\lambda }^1=\mathrm{id}$

3. ${\lambda }^n(1)=0$, for $n>1$

4. ${\lambda }^n(r+s)={\sum }_{k=0}^n{\lambda }^k(r){\lambda }^{n-k}(s)$, for all $r,s\in R$

5. ${\lambda }^n(rs)=P_n({\lambda }^1(r),\dots ,{\lambda }^n(r),{\lambda }^1(s),\dots ,{\lambda }^n(s))$ for all $r,s\in R$

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

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

A morphism of $\lambda$-structures is defined to be a morphism of rings commuting with all ${\lambda }^n$ maps.

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

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

b) multiplication is defined by

$$(1+\sum _{n=1}^{\mathrm{\infty }}{r}_n{t}^n)(1+\sum _{n=1}^{\mathrm{\infty }}{s}_n{t}^n):=1+\sum _{n=1}^{\mathrm{\infty }}{P}_n(r_1,\dots ,r_n,s_1,\dots ,s_n)t^n$$(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 $\lambda$-operations are defined by

$${\lambda }^n(1+\sum _{m=1}^{\mathrm{\infty }}{r}_m{t}^m)=1+\sum _{m=1}^{\mathrm{\infty }}{P}_{m,n}(r_1,\dots ,r_{\mathrm{mn}})t^m$$\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

(Hopkins)

Remark

Let $\Lambda$ denote the ring of symmetric functions, let $R$ be a $\lambda$-ring.

Then for every $x\in R$ there is a unique morphism of $\lambda$-rings

$${\Phi }_x:\Lambda \to R$$\Phi_x:\Lambda\to R

sending $e_1\mapsto x$, $e_n\mapsto {\lambda }^n(x)$, $p_n\mapsto {\psi }^n(x)$ where $e_n$ denotes the $n$-th elementary symmetric function? and ${\psi }^n$ denotes the $n$-th Adams operation? (explained in the reference).

Equivalently this result asserts that $\Lambda$ is the free $\lambda$-ring in the single variable $e_1$.

(Hopkins)

Proof

We define ${\Phi }_x(e_1)=x$, then the assumption on ${\Phi }_x$ to be a morphism of $\lambda$-rings yields ${\Phi }_x(e_n)={\Phi }_x({\lambda }^n(e_1))={\lambda }^n(x)$.

Remark and Theorem

(Hazewinkel 1.11, 16.1) a) The endofunctor of the category of commutative rings

$$\Lambda :\{\begin{array}{l}C\mathrm{Ring}\to C\mathrm{Ring}\\ A\mapsto 1+A[[t]]\end{array}$$\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 $1$ is representable by the polynomial ring $\mathrm{Symm}≔\mathbb{Z}[h_1,h_2,\dots ]$ in an infinity of indeterminates over the integers.

b) There is an adjunction $(\mathrm{forget}\lambda ⊣\Lambda )$ where $\mathrm{forget}\lambda :\lambda \mathrm{Ring}\to \mathrm{CRing}$ is the forgetful functor assigning to a $\lambda$-ring its underlying commutative ring.

The left inverse $g_{S,A}$ of the natural isomorphism $q_{S,A}:\mathrm{hom}(\mathrm{forget}\lambda ,A)\to \mathrm{hom}(S,\Lambda (A))$ is given by the ghost component? $s_1$.

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

The “heterodox” view on $\lambda$-rings

There is a second, “heterodox” way to approach $\lambda$-rings with a strong connection to arithmetic used by James Borger in his paper $\Lambda$-rings and the field with one element. Quoting the abstract:

The theory of $\Lambda$-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 $Z$ of integers to produce $\Lambda$-algebraic geometry. We show that $\Lambda$-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than $Z$ and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.

Let $p$ be a prime number. Recall that for any commutative ring $R$ the Frobenius morphism is defined by $F_R:x\mapsto x^p$.

Defiition

Let $A$ be a commutative ring or a lambda ring. A morphism $\mathrm{Fl}:A\to A$ is called a Frobenis lift if the restriction $\mathrm{Fl}{\mid }_{A/\mathrm{pA}}$ of $\mathrm{Fl}$ to the quotient ring is the Frobenius morphism $F_{A/\mathrm{pA}}$.

Example

The $p$-th Adams operation? ${\psi }_p$ 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.

Theorem

(Wilkerson’s theorem) Let $A$ be an additively torsion-free commutative ring. Let $\{{\psi }_p\}$ be a commuting family of Frobenius lifts.

Then there is a unique $\lambda$-ring structure on $A$ whose Adams operations are the given Frobenius lifts $\{{\psi }_p\}$.

Theorem

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

Corollary

There is an equivalence between the category of torsion-free $\lambda$-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.

Remark

a) The category $\lambda \mathrm{Ring}$ of $\lambda$-rings is monadic and comonadic? over the category of $C\mathrm{Ring}$ of commutative rings.

b) The category $\lambda {\mathrm{Ring}}_{\neg \mathrm{tor}}$ of $\lambda$-rings is monadic and comonadic? over the category of $C{\mathrm{Ring}}_{\neg \mathrm{tor}}$ of commutative rings.

Proposition

Let $i:C{\mathrm{Ring}}_{\neg \mathrm{tor}}\hookrightarrow C\mathrm{Ring}$ be the inclusion. Let $W^{\prime }$ denote this comonad on $C{\mathrm{Ring}}_{\neg \mathrm{tor}}$. Then

a) $W≔{\mathrm{Lan}}_{i}i\circ W^{\prime }:C\mathrm{Ring}\to \mathrm{CRing}$ is a comonad.

b) The category of coalgebras of $W$ is equivalent to the category of $\lambda$-rings.

c) $W$ is the big-Witt-vectors functor?.

Remark

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

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

Related concepts

• Grothendieck group

• p-divisible groups.

• field with one element

• blueprints, blue scheme

References

• John Baez, comment.

• Hazewinkel, formal groups and applications

• Hazewinkel, Witt vectors, pdf.

• John R. Hopkins, universal polynomials in lambda rings and the K-theory of the infinite loop space tmf, thesis, pdf

• Donald Knutson, $\lambda$-Rings and the Representation Theory of the Symmetric Group, Lecture Notes in Mathematics, Vol. 308, Springer, Berlin, 1973.

• concretenonsense blog

• Donald Yau, LAMBDA-RINGS, World Scientific, 2010.

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

• Pierre Cartier: Lambda-rings and Witt vectors

• Lars Hesselholt: The de Rham-Witt complex

• Alexandru Buium: Arithmetic differential equations

• James Borger: Lambda-algebraic geometry

conference site

participants