Spahn lambda ring (changes)

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Λ\Lambda is the ring of symmetric function?s in countably many variables. A Λ\Lambda-ring is a P-ring? for P=ΛP = \Lambda.

Disambiguation

Λ\LambdaThere is also a notion of -rings are usually called special lambda ring?λ\lambda. But in most cases by ‘’-rings, and when people talk of λ\lambdaλ\lambda-ring’‘ is meant ‘’special -rings they usually mean what people λ\lambdaused-ring’’. to call special ∞-rings?.

John Baez?: There are vastly more approachable ways of defining the concept of λ\lambda-ring. Please, someone, type one in! Hazewinkel’s article below steers a nice middle ground between grunge and sophistication.

I believe we should also move most of the material at special lambda-ring? to this page.

Idea

A λ\lambda-ring LL is a P-ring? presented by the polynomial ring? Symm=ℤ[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:Cring→CRingL:\Cring \to CRing defines an endocunctor 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 definiton 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 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 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. 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 λ\lambda-ring on one generator are called symmetric functions?.

Connection to arithmetic

There is a second, ‘heterodox’ way to approach λ\lambda rings, which James Borger uses in his paper Λ\Lambda-rings and the field with one element. Quoting 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 is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.

Two points of view

On the n-Café, James Borger compared what he called the ‘orthodox’ and ‘heterodox’ approaches to λ\lambda-rings. He wrote:

A few words on the orthodox/heterodox approaches to Λ\Lambda-rings…

By the orthodox approach I mean everything involving symmetric functions, including symmetric groups, decategorifying linear tensor categories, K-theory, and so on. Everything mentioned above or pretty much anywhere takes this point of view. Indeed Grothendieck invented λ\lambda-rings to understand the extra structure on the Grothendieck group inherited from tensor operations (exterior powers etc) on tensor categories.

By the heterodox point of view, I mean everything involving Frobenius lifts. A Frobenius lift? at a prime pp on a commutative ring AA is an endomorphism which reduces to the pp-th power Frobenius map? x↦x px\mapsto x^p on the quotient ring A/pAA/pA. If the orthodox point of view is closely related to KK-theory, the heterodox point of view is closely related to cohomology? (e.g. crystalline cohomology?).

The connection between the two is as follows. Given any λ\lambda-ring structure in the orthodox sense, the pp-th Adams operation? ψ p\psi_p is a Frobenius lift, and the ψ p\psi_p for different primes pp all commute with each other. (You can see the split forming already: the Adams operations don’t exist on tensor categories until you decategorify.) Wilkerson’s theorem gives the converse, at least in the absence of torsion. It says this: Let AA be a commutative ring which is torsion free (in the additive sense), and let ψ p\psi_p be a family of commuting Frobenius lifts on AA indexed by the prime numbers pp. Then there is a unique λ\lambda-ring structure on AA whose Adams operations are the given Frobenius lifts ψ p\psi_p. The proof is an unenlightening (IMHO) exercise in composition of symmetric functions. Further, a ring map between two torsion-free λ\lambda-rings is a map of λ\lambda-rings if and only if it commutes with the Adams operations. Thus we have an explicit equivalence between the category of torsion-free λ\lambda-rings and the category of torsion-free commutative rings equipped with commuting Frobenius lifts.

[In fact, we can get around the torsion-free restriction with a dash of category theory. The category of -rings is both monadic? and comonadic? over the category of rings (commutative, which I will now stop writing). The point is that that the comonad (and hence the monad) is determined by what happens in the torsion-free setting, where it can be described in terms of Frobenius lifts. How does this work? The category of torsion-free -rings is also comonadic over the category of torsion-free rings. You can show this from either orthodox point of view or the heterodox point of view (where your definition of torsion-free -ring would be the one involving Frobenius lifts). Let be this comonad, on the category of torsion-free rings. Now let be the left Kan extension of this functor (or more precisely the composition of with the embedding of torsion-free rings into all rings) to the category of all rings. Then is a comonad on the category of all rings, and the category of its coalgebras is equivalent to the category of -rings, torsion free or not. By the way, this comonad is precisely the big Witt vector functor.]

The odd thing is that pretty much no one cares that the two approaches are equivalent, including me. Everything most people want to do is on the orthodox side, and everything I want to do is on the heterodox side. The deeper meaning of connection between the two is completely unclear to me. It could be that it should be viewed as an accident. Indeed the heterodox point of view generalizes to families of Frobenius lifts on other Dedekind domains with finite residue fields in a way that perhaps the orthodox point of view doesn’t. For instance over F p[x]F_p[x] (instead of Z\mathbf{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)). What is the analogue of such a structure in the world of symmetric functions? Is there a function-field version of symmetric functions? A number field version? It would be great if the answer to these were Yes. But the fact that such analogues are (apparently) unknown suggests that there is something to the split between the two points of view.

To understand what Borger is saying here, it’s helpful to read Hazewinkel’s survey article on Witt vectors, (Hazewinkel). To this end there is also a reading guide to that article.

Definition

Definition

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

  1. λ 0(r)=1\lambda^0(r)=1 for all r∈Rr\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 n−k\lambda^n(r+s)=\sum_{k=0}^n \lambda^k(r)\lambda^{n-k}, for all r,s∈Rr,s\in R

  5. λ n(rs)=P n(λ 1(r),…,λ n(r),λ 1(s),…,λ n(s))\lambda^n(rs)=P_n(\lambda^1(r),\dots,\lambda^n(r),\lambda^1(s),\dots,\lambda^n(s)) for all r,s∈Rr,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^{mn}(r)), for all r∈Rr\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 morphism of λ\lambda-structures is defined to be a morphism 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

(Hopkins)

The ‘’orthodox’‘ view on λ\lambda-rings
Remark Definition

Let A Λ λ \Lambda \lambda -structure denote on the a commutative unital ring ofsymmetric functions?RR , let is defined to be a sequence of mapsRλ n R \lambda^n be for a λ n≥0 \lambda n\ge 0 -ring. satisfying

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

  1. λ 0(r)=1\lambda^0(r)=1 for all r∈Rr\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 n−k\lambda^n(r+s)=\sum_{k=0}^n \lambda^k(r)\lambda^{n-k}, for all r,s∈Rr,s\in R

  5. λ n(rs)=P n(λ 1(r),…,λ n(r),λ 1(s),…,λ n(s))\lambda^n(rs)=P_n(\lambda^1(r),\dots,\lambda^n(r),\lambda^1(s),\dots,\lambda^n(s)) for all r,s∈Rr,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^{mn}(r)), for all r∈Rr\in R

Φ x:Λ→R\Phi_x:\Lambda\to 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.

sending A morphism ofe 1λ↦x e_1\mapsto \lambda x , -structures is defined to be a morphism of rings commuting with alle n↦λ n(x) e_n\mapsto \lambda^n \lambda^n(x) , maps.p n↦ψ 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 There this exists result a asserts that Λ λ \Lambda \lambda -ring is structure on the free ringλ1+R[[t]] \lambda 1+ R[ [t] ] -ring in of the power single series variable with constant terme 11 e_1 1 . 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

(Hopkins)

Proof Remark

We Let defineΦ xΛ(e 1)=x \Phi_x(e_1)=x \Lambda , then denote the assumption ring on ofΦ x\Phi_xsymmetric functions? , to let be a morphism of λ R \lambda R -rings yields be aΦ xλ(e n)=Φ x(λ n(e 1))=λ n(x) \Phi_x(e_n)=\Phi_x(\lambda^n(e_1))=\lambda^n(x) \lambda . -ring.

Then for every x∈Rx\in R there is a unique morphism of λ\lambda-rings
Φ x:Λ→R\Phi_x:\Lambda\to R

sending e 1↦xe_1\mapsto x, e n↦λ n(x)e_n\mapsto \lambda^n(x), p n↦ψ 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.

(Hopkins)
Remark Proof and Theorem

( We defineHazewinkelΦ x(e 1)=x\Phi_x(e_1)=x , 1.11, then 16.1) a) The endofunctor of the category assumption of on commutative ringsΦ 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).

Λ:{CRing→CRing A↦1+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:=ℤ[h 1,h 2,…]Symm:=\mathbb{Z}[h_1, h_2,\dots] in an infinity of indeterminates over the integers.

b) There is an adjunction (forgetλ⊣Λ)(forget\, \lambda\dashv \Lambda) where forgetλ:λRing→CRingforget\,\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.
Remark and Theorem

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

Λ:{CRing→CRing A↦1+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:=ℤ[h 1,h 2,…]Symm:=\mathbb{Z}[h_1, h_2,\dots] in an infinity of indeterminates over the integers.

b) There is an adjunction (forgetλ⊣Λ)(forget\, \lambda\dashv \Lambda) where forgetλ:λRing→CRingforget\,\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.

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’‘ view on λ\lambda-rings

There is a second, ‘’heterodox’‘ way to approach λ\lambda-rings with a strong connection to arithmetic? used by James Borger in his paper Λ\Lambda-rings and the field with one element. Quoting 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 is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.

Let pp be a prime number. Recall that for any commutative ring RR the Frobenius morphism? is defined by F R:x↦x pF_R:x\mapsto x^p.
Defiition

Let AA be a commutative ring or a lambda ring. A morphism Fl:A→AFl:A\to A is called a Frobenis lift if the restriction Fl| A/pAFl \,|_{A/pA} of FlFl to the quotient ring is the Frobenius morphism F A/pAF_{A/pA}.
Example

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.
Theorem

(Wilkerson’s theorem) Let AA be an additively torsion-free commutative ring. Let {psi 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 {psi p}\{psi_p\}.
Theorem

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 operation?.
Corollary

There is an equivalence between the category of torsion-free λ\lambda-rings and the category of torsion-free commutative rings.

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

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.
Proposition

Let i:CRing ¬tor↪CRing\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) W:=Lan ii∘W ′:CRing→CRingW:=Lan_i i\circ W^\prime: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?.
Remark

The ‘’heterodox’‘ generalizes to arbitrary Dedekind domains? with finite residue field.

For instance over F p[x]F_p[x] (instead of ℤ\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)).

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, λ\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

Last revised on May 31, 2012 at 15:32:32. See the history of this page for a list of all contributions to it.

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