There is also a notion of special lambda-ring. But in most cases by ‘’-ring’‘ is meant ‘’special -ring’’.
A -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 for the th exterior power. Hence -rings are one incarnation of the representation theory of the symmetric groups.
Equivalently, it turns out (Wilkerson 82) that a -ring is a commutative ring equipped with an endomorphism that lifts the Frobenius endomorphism after reduction mod at each prime number . As such -rings appear in Borger's absolute geometry and are related, in some way, to power operations (see there for more) in stable homotopy 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 -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 -ring.
So, -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 -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 -ring on one generator are called symmetric functions.
A -ring is a P-ring presented by the polynomial ring in countably many indeterminates over the integers or, equivalently, is the ring of symmetric functions in countably many variables. This means that (the underlying set valued functor of) is a copresheaf presented by such that
defines an endofunctor on the category of commutative rings.
gives rise to a comonad on .
A -structure on a commutative unital ring is defined to be a sequence of maps for satisfying
, for all
, for all
where and are certain (see the reference for their calculation) universal polynomials? with integer coefficients. is in this case called a -ring. Note that the are not required to be morphisms of rings.
There exists a -ring structure on the ring of power series with constant term where
a) addition on is defined to be multiplication of power series
b) multiplication is defined by
c) the -operations are defined by
Let denote the ring of symmetric functions, let be a -ring.
Then for every there is a unique homomorphism of -rings
Equivalently this result asserts that is the free -ring in the single variable .
This is due to (Hopkinson)
We define , then the assumption on to be a morphism of -rings yields .
(Hazewinkel 1.11, 16.1)
sending a commutative ring to the set of power series with constant term is representable by the polynomial ring in an infinity of indeterminates over the integers.
b) There is an adjunction where is the forgetful functor assigning to a -ring its underlying commutative ring.
The left inverse of the natural isomorphism is given by the ghost component .
(see also (Borger 08, section 1.8))
An instructive introduction to the “orthodox”- and preparation for the “heterodox” view (described below) on -rings is Hazewinkel’s survey article on Witt vectors, (Hazewinkel). There is also a reading guide to that article.
The theory of -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 of integers to produce -algebraic geometry. We show that -algebraic geometry is in a precise sense an algebraic geometry over a deeper base than 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:
is that given by taking each element to its th power
For a prime number, then a -typical -ring is
A big -ring is a commutative ring equipped with commuting endomorphisms, one for each prime number , such that each of them makes the ring -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 -rings in view of prop. 1.10 c) (see the adjunction) there.
the following originates from revision 19 but needs attention
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.
Let be an additively torsion-free commutative ring. Let be a commuting family of Frobenius lifts.
Then there is a unique -ring structure on whose Adams operations are the given Frobenius lifts .
A ring morphism between two -rings is a morphism of -rings (i.e. commuting with the -operations) iff commutes with the Adams operations.
There is an equivalence between the category of torsion-free -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.
Let be the inclusion. Let denote this comonad on . Then
a) is a comonad.
b) The category of coalgebras of is equivalent to the category of -rings.
c) is the big-Witt-vectors functor.
The “heterodox” generalizes to arbitrary Dedekind domains with finite residue field.
For instance over (instead of ), we would look at families of -operators indexed by the irreducible monic polynomials , and each would have to be congruent to the -th power map modulo , where is the size of .
This is originally due to Alexander Grothendieck.
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).
The relation to lifts of Frobenius homomorphisms is due to
Modern accounts include
John Baez, comment.
John R. Hopkinson, Universal polynomials in lambda-rings and the K-theory of the infinite loop space , thesis, pdf
Donald Knutson, -Rings and the Representation Theory of the Symmetric Group, Lecture Notes in Mathematics, Vol. 308, Springer, Berlin, 1973.
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
The -ring structure on equivariant elliptic cohomology is due to
Nora Ganter, Stringy power operations in Tate K-theory, Homology, Homotopy, Appl., 2013; arXiv:math/0701565
Nora Ganter, Power operations in orbifold Tate K-theory“; arXiv:1301.2754
Discussion in the context of Borger's absolute geometry is in