Spahn lambda ring (Rev #5, changes)

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

$\Lambda$ -rings are usually called $\lambda$ -rings, and when people talk of $\lambda$ -rings they usually mean what people used to call special ∞-rings?.

John Baez?: There are vastly more approachable ways of defining the concept of $\lambda$ -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

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?.

Connection to arithmetic

There is a second, ‘heterodox’ way to approach $\lambda$ rings, which James Borger uses 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 $\mathbf{Z}$ 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 $\mathbf{Z}$ 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 $\lambda$ -rings. He wrote:

A few words on the orthodox/heterodox approaches to $\Lambda$ -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 $\lambda$ -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 $p$ on a commutative ring $A$ is an endomorphism which reduces to the $p$ -th power Frobenius map? $x\mapsto x^p$ on the quotient ring $A/pA$ . If the orthodox point of view is closely related to $K$ -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 $\lambda$ -ring structure in the orthodox sense, the $p$ -th Adams operation? $\psi_p$ is a Frobenius lift, and the $\psi_p$ for different primes $p$ 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 $A$ be a commutative ring which is torsion free (in the additive sense), and let $\psi_p$ be a family of commuting Frobenius lifts on $A$ indexed by the prime numbers $p$ . Then there is a unique $\lambda$ -ring structure on $A$ whose Adams operations are the given Frobenius lifts $\psi_p$ . The proof is an unenlightening (IMHO) exercise in composition of symmetric functions. Further, a ring map between two torsion-free $\lambda$ -rings is a map of $\lambda$ -rings if and only if it commutes with the Adams operations. Thus we have an explicit equivalence between the category of torsion-free $\lambda$ -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 $F_p[x]$ (instead of $\mathbf{Z}$ ), 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 $F_p[x]/(f(x))$ . 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. Here’s a little guide to that:

Hazewinkel talks about both the ‘orthodox’ and ‘heterodox’ approaches to $\lambda$ -rings. Indeed he starts out with a lot of material on Witt vectors and their relation to $p$ -adics. The $\lambda$ -rings only make their debut on page 87, where the operation of ‘taking the Witt vectors’ of a commutative ring is revealed to be the right adjoint to the forgetful functor from $\lambda$ -rings to commutative rings.
He then goes ahead and defines $\lambda$ -rings on page 88. At first his definition looks a bit frustrating, because Hazewinkel defines ? $\lambda$ -ring? using the concept of ?morphism of $\lambda$ -rings?! But it’s not actually circular; it’s really just a trick to spare us certain ugly equations that appear in the usual definition.
Just for fun, note the unusual remark in footnote 62 on page 88: warning the reader to ‘steer clear’ of a certain book by two famous authors.
On page 92, Hazewinkel proves the Wilkerson theorem getting $\lambda$ -rings from rings equipped with Adams operations $\psi_p$ . And then, at the bottom of page 94, he goes heterodox and defines ? $\psi$ -rings? to be commutative rings equipped with Adams operations — and notes that over a field of characteristic zero, $\lambda$ -rings are the same as $\psi$ -rings.
On page 97, he describes ‘taking the Witt vectors’ as a comonad on the category of commutative rings. This is just another way of talking about the right adjointness property he mentioned on page 87; now he’s proving it.
On page 98 he goes orthodox again, and shows that Symm, the ring of symmetric functions in countably many variables, is the free $\lambda$ -ring on one generator.
On page 102 starts explaining ‘plethysm’.

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

$\lambda^0(r)=1$ for all $r\in R$
$\lambda^1=id$
$\lambda^n(1)=0$ , for $n\gt 1$
$\lambda^n(r+s)=\sum_{k=0}^n \lambda^k(r)\lambda^{n-k}$ , for all $r,s\in R$
$\lambda^n(rs)=P_n(\lambda^1(r),\dots,\lambda^n(r),\lambda^1(s),\dots,\lambda^n(s))$ for all $r,s\in R$
$\lambda^m(\lambda^n(r)):=P_{m,n}(\lambda^1(r),\dots,\lambda^{mn}(r))$ , 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}^\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}^\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

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{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 $Symm:=\mathbb{Z}[h_1, h_2,\dots]$ in an infinity of indeterminates over the integers.

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

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

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.
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

Revision on May 30, 2012 at 01:16:09 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.