numbers – neverendingbooks

In 1956, Alexander Grothendieck (middle) introduced $λ$ -rings in an algebraic-geometric context to be commutative rings A equipped with a bunch of operations $λ^{i}$ (for all numbers $i \in N_{+}$ ) satisfying a list of rather obscure identities. From the easier ones, such as

$λ^{0} (x) = 1, λ^{1} (x) = x, λ^{n} (x + y) = \sum_{i} λ^{i} (x) λ^{n - i} (y)$

to those expressing $λ^{n} (x . y)$ and $λ^{m} (λ^{n} (x))$ via specific universal polynomials. An attempt to capture the essence of $λ$ -rings without formulas?

Lenstra’s elegant construction of the 1-power series rings $(Λ (A), \oplus, \otimes)$ requires only one identity to remember

$(1 - a t)^{- 1} \otimes (1 - b t)^{- 1} = (1 - a b t)^{- 1}$ .

Still, one can use it to show the existence of ringmorphisms $γ_{n} : Λ (A) \to A$ , for all numbers $n \in N_{+}$ . Consider the formal ‘logarithmic derivative’

$γ = \frac{t u (t)^{'}}{u (t)} = \sum_{i = 1}^{\infty} γ_{i} (u (t)) t^{i} : Λ (A) \to A [[t]]$

where $u (t)^{'}$ is the usual formal derivative of a power series. As this derivative satisfies the chain rule, we have

$γ (u (t) \oplus v (t)) = \frac{t (u (t) v (t))^{'}}{u (t) v (t)} = \frac{t (u (t)^{'} v (t) + u (t) v (t)^{'}}{u (t) v (t))} = \frac{t u (t)^{'}}{u (t)} + \frac{t v (t)^{'}}{v (t)} = γ (u (t)) + γ (v (t))$

and so all the maps $γ_{n} : Λ (A) \to A$ are additive. To show that they are also multiplicative, it suffices by functoriality to verify this on the special 1-series $(1 - a t)^{- 1}$ for all $a \in A$ . But,

$γ ((1 - a t)^{- 1}) = \frac{t \frac{a}{(1 - a t)^{2}}}{(1 - a t)} = \frac{a t}{(1 - a t)} = a t + a^{2} t^{2} + a^{3} t^{3} + \dots$

That is, $γ_{n} ((1 - a t)^{- 1}) = a^{n}$ and Lenstra’s identity implies that $γ_{n}$ is indeed multiplicative! A first attempt :

hassle-free definition 1 : a commutative ring $A$ is a $λ$ -ring if and only if there is a ringmorphism $s_{A} : A \to Λ (A)$ splitting $γ_{1}$ , that is, such that $γ_{1} \circ s_{A} = i d_{A}$ .

In particular, a $λ$ -ring comes equipped with a multiplicative set of ring-endomorphisms $s_{n} = γ_{n} \circ s_{A} : A \to A$ satisfying $s_{m} \circ s_{m} = s_{m n}$ . One can then define a $λ$ -ringmorphism to be a ringmorphism commuting with these endo-morphisms.

The motivation being that $λ$ -rings are known to form a subcategory of commutative rings for which the 1-power series functor is the right adjoint to the functor forgetting the $λ$ -structure. In particular, if $A$ is a $λ$ -ring, we have a ringmorphism $A \to Λ (A)$ corresponding to the identity morphism.

But then, what is the connection to the usual one involving all the operations $λ^{i}$ ? Well, one ought to recover those from $s_{A} (a) = (1 - λ^{1} (a) t + λ^{2} (a) t^{2} - λ^{3} (a) t^{3} + \dots)^{- 1}$ .

For $s_{A}$ to be a ringmorphism will require identities among the $λ^{i}$ . I hope an expert will correct me on this one, but I’d guess we won’t yet obtain all identities required. By the very definition of an adjoint we must have that $s_{A}$ is a morphism of $λ$ -rings, and, this would require defining a $λ$ -ring structure on $Λ (A)$ , that is a ringmorphism $s_{A H} : Λ (A) \to Λ (Λ (A))$ , the so called Artin-Hasse exponential, to which I’d like to return later.

For now, we can define a multiplicative set of ring-endomorphisms $f_{n} : Λ (A) \to Λ (A)$ from requiring that $f_{n} ((1 - a t)^{- 1}) = (1 - a^{n} t)^{- 1}$ for all $a \in A$ . Another try?

hassle-free definition 2 : $A$ is a $λ$ -ring if and only if there is splitting $s_{A}$ to $γ_{1}$ satisfying the compatibility relations $f_{n} \circ s_{A} = s_{A} \circ s_{n}$ .

But even then, checking that a map $s_{A} : A \to Λ (A)$ is a ringmorphism is as hard as verifying the lists of identities among the $λ^{i}$ . Fortunately, we get such a ringmorphism for free in the important case when A is of ‘characteristic zero’, that is, has no additive torsion. Then, a ringmorphism $A \to Λ (A)$ exists whenever we have a multiplicative set of ring endomorphisms $F_{n} : A \to A$ for all $n \in N_{+}$ such that for every prime number $p$ the morphism $F_{p}$ is a lift of the Frobenius, that is, $F_{p} (a) \in a^{p} + p A$ .

Perhaps this captures the essence of $λ$ -rings best (without the risk of getting an headache) : in characteristic zero, they are the (commutative) rings having a multiplicative set of endomorphisms, generated by lifts of the Frobenius maps.

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Next time you visit your math-library, please have a look whether these books are still on the shelves : Michiel Hazewinkel‘s Formal groups and applications, William Fulton’s and Serge Lange’s Riemann-Roch algebra and Donald Knutson’s lambda-rings and the representation theory of the symmetric group.

I wouldn’t be surprised if one or more of these books are borrowed out, probably all of them to the same person. I’m afraid I’m that person in Antwerp…

Lately, there’s been a renewed interest in $λ$ -rings and the endo-functor W assigning to a commutative algebra its ring of big Witt vectors, following Borger’s new proposal for a geometry over the absolute point.

However, as Hendrik Lenstra writes in his 2002 course-notes on the subject Construction of the ring of Witt vectors : “The literature on the functor W is in a somewhat unsatisfactory state: nobody seems to have any interest in Witt vectors beyond applying them for a purpose, and they are often treated in appendices to papers devoting to something else; also, the construction usually depends on a set of implicit or unintelligible formulae. Apparently, anybody who wishes to understand Witt vectors needs to construct them personally. That is what is now happening to myself.”

Before doing a series on Borger’s paper, we’d better run through Lenstra’s elegant construction in a couple of posts. Let A be a commutative ring and consider the multiplicative group of all ‘one-power series’ over it $Λ (A) = 1 + t A [[t]]$ . Our aim is to define a commutative ring structure on $Λ (A)$ taking as its ADDITION the MULTIPLICATION of power series.

That is, if $u (t), v (t) \in Λ (A)$ , then we define our addition $u (t) \oplus v (t) = u (t) \times v (t)$ . This may be slightly confusing as the ZERO-element in $Λ (A), \oplus$ will then turn be the constant power series 1…

We are now going to define a multiplication $\otimes$ on $Λ (A)$ which is distributively with respect to $\oplus$ and turns $Λ (A)$ into a commutative ring with ONE-element the series $(1 - t)^{- 1} = 1 + t + t^{2} + t^{3} + \dots$ .

We will do this inductively, so consider $Λ_{n} (A)$ the (classes of) one-power series truncated at term n, that is, the kernel of the natural augmentation map between the multiplicative group-units $A [t] / (t^{n + 1})^{*} \to A^{*}$ .
Again, taking multiplication in $A [t] / (t^{n + 1})$ as a new addition rule $\oplus$ , we see that $(Λ_{n} (A), \oplus)$ is an Abelian group, whence a $Z$ -module.

For all elements $a \in A$ we have a scaling operator $ϕ_{a}$ (sending $t \to a t$ ) which is an A-ring endomorphism of $A [t] / (t^{n + 1})$ , in particular multiplicative wrt. $\times$ . But then, $ϕ_{a}$ is an additive endomorphism of $(Λ_{n} (A), \oplus)$ , so is an element of the endomorphism-RING $E n d_{Z} (Λ_{n} (A))$ . Because composition (being the multiplication in this endomorphism ring) of scaling operators is clearly commutative ( $ϕ_{a} \circ ϕ_{b} = ϕ_{a b}$ ) we can define a commutative RING $E$ being the subring of $E n d_{Z} (Λ_{n} (A))$ generated by the operators $ϕ_{a}$ .

The action turns $(Λ_{n} (A), \oplus)$ into an E-module and we define an E-module morphism $E \to Λ_{n} (A)$ by $ϕ_{a} \mapsto ϕ_{a} ((1 - t)^{- 1}) = (1 - a t)^{- a}$ .

All of this looks pretty harmless, but the upshot is that we have now equipped the image of this E-module morphism, say $L_{n} (A)$ (which is the additive subgroup of $(Λ_{n} (A), \oplus)$ generated by the elements $(1 - a t)^{- 1}$ ) with a commutative multiplication $\otimes$ induced by the rule $(1 - a t)^{- 1} \otimes (1 - b t)^{- 1} = (1 - a b t)^{- 1}$ .

Explicitly, $L_{n} (A)$ is the set of one-truncated polynomials $u (t)$ with coefficients in $A$ such that one can find elements $a_{1}, \dots, a_{k} \in A$ such that $u (t) \equiv (1 - a_{1} t)^{- 1} \times \dots \times (1 - a_{k})^{- 1} m o d t^{n + 1}$ . We multiply $u (t)$ with another such truncated one-polynomial $v (t)$ (taking elements $b_{1}, b_{2}, \dots, b_{l} \in A$ ) via

$u (t) \otimes v (t) = ((1 - a_{1} t)^{- 1} \oplus \dots \oplus (1 - a_{k})^{- 1}) \otimes ((1 - b_{1} t)^{- 1} \oplus \dots \oplus (1 - b_{l})^{- 1})$

and using distributivity and the multiplication rule this gives the element $\prod_{i, j} (1 - a_{i} b_{j} t)^{- 1} m o d t^{n + 1} \in L_{n} (A)$ .
Being a ring-qutient of $E$ we have that $(L_{n} (A), \oplus, \otimes)$ is a commutative ring, and, from the construction it is clear that $L_{n}$ behaves functorially.

For rings $A$ such that $L_{n} (A) = Λ_{n} (A)$ we are done, but in general $L_{n} (A)$ may be strictly smaller. The idea is to use functoriality and do the relevant calculations in a larger ring $A \subset B$ where we can multiply the two truncated one-polynomials and observe that the resulting truncated polynomial still has all its coefficients in $A$ .

Here’s how we would do this over $Z$ : take two irreducible one-polynomials u(t) and v(t) of degrees r resp. s smaller or equal to n. Then over the complex numbers we have
$u (t) = (1 - α_{1} t) \dots (1 - α_{r} t)$ and $v (t) = (1 - β_{1}) \dots (1 - β_{s} t)$ . Then, over the field $K = Q (α_{1}, \dots, α_{r}, β_{1}, \dots, β_{s})$ we have that $u (t), v (t) \in L_{n} (K)$ and hence we can compute their product $u (t) \otimes v (t)$ as before to be $\prod_{i, j} (1 - α_{i} β_{j} t)^{- 1} m o d t^{n + 1}$ . But then, all coefficients of this truncated K-polynomial are invariant under all permutations of the roots $α_{i}$ and the roots $β_{j}$ and so is invariant under all elements of the Galois group. But then, these coefficients are algebraic numbers in $Q$ whence integers. That is, $u (t) \otimes v (t) \in Λ_{n} (Z)$ . It should already be clear from this that the rings $Λ_{n} (Z)$ contain a lot of arithmetic information!

For a general commutative ring $A$ we will copy this argument by considering a free overring $A^{(\infty)}$ (with 1 as one of the base elements) by formally adjoining roots. At level 1, consider $M_{0}$ to be the set of all non-constant one-polynomials over $A$ and consider the ring

$A^{(1)} = ⨂_{f \in M_{0}} A [X] / (f) = A [X_{f}, f \in M_{0}] / (f (X_{f}), f \in M_{0})$

The idea being that every one-polynomial $f \in M_{0}$ now has one root, namely $α_{f} = \overset{―}{X_{f}}$ in $A^{(1)}$ . Further, $A^{(1)}$ is a free A-module with basis elements all $α_{f}^{i}$ with $0 \leq i < d e g (f)$ .

Good! We now have at least one root, but we can continue this process. At level 2, $M_{1}$ will be the set of all non-constant one-polynomials over $A^{(1)}$ and we use them to construct the free overring $A^{(2)}$ (which now has the property that every $f \in M_{0}$ has at least two roots in $A^{(2)}$ ). And, again, we repeat this process and obtain in succession the rings $A^{(3)}, A^{(4)}, \dots$ . Finally, we define $A^{(\infty)} = \underset{\to}{l i m} A^{(i)}$ having the property that every one-polynomial over A splits entirely in linear factors over $A^{(\infty)}$ .

But then, for all $u (t), v (t) \in Λ_{n} (A)$ we can compute $u (t) \otimes v (t) \in Λ_{n} (A^{(\infty)})$ . Remains to show that the resulting truncated one-polynomial has all its entries in A. The ring $A^{(\infty)} \otimes_{A} A^{(\infty)}$ contains two copies of $A^{(\infty)}$ namely $A^{(\infty)} \otimes 1$ and $1 \otimes A^{(\infty)}$ and the intersection of these two rings in exactly $A$ (here we use the freeness property and the additional fact that 1 is one of the base elements). But then, by functoriality of $L_{n}$ , the element
$u (t) \otimes v (t) \in L_{n} (A^{(\infty)} \otimes_{A} A^{(\infty)})$ lies in the intersection $Λ_{n} (A^{(\infty)} \otimes 1) \cap Λ_{n} (1 \otimes A^{(\infty)}) = Λ_{n} (A)$ . Done!

Hence, we have endo-functors $Λ_{n}$ in the category of all commutative rings, for every number n. Reviewing the construction of $L_{n}$ one observes that there are natural transformations $L_{n + 1} \to L_{n}$ and therefore also natural transformations $Λ_{n + 1} \to Λ_{n}$ . Taking the inverse limits $Λ (A) = \underset{\leftarrow}{l i m} Λ_{n} (A)$ we therefore have the ‘one-power series’ endo-functor
$Λ : comm \to comm$
which is ‘almost’ the functor W of big Witt vectors. Next time we’ll take you through the identification using ‘ghost variables’ and how the functor $Λ$ can be used to define the category of $λ$ -rings.

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Tag: numbers

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