big Witt vectors for everyone (1/2)

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 -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 . Our aim is to define a commutative ring structure on taking as its ADDITION the MULTIPLICATION of power series.

That is, if , then we define our addition . This may be slightly confusing as the ZERO-element in will then turn be the constant power series 1…

We are now going to define a multiplication on which is distributively with respect to and turns into a commutative ring with ONE-element the series .

We will do this inductively, so consider the (classes of) one-power series truncated at term n, that is, the kernel of the natural augmentation map between the multiplicative group-units . Again, taking multiplication in as a new addition rule , we see that is an Abelian group, whence a -module.

For all elements we have a scaling operator (sending ) which is an A-ring endomorphism of , in particular multiplicative wrt. . But then, is an additive endomorphism of , so is an element of the endomorphism-RING . Because composition (being the multiplication in this endomorphism ring) of scaling operators is clearly commutative () we can define a commutative RING being the subring of generated by the operators .

The action turns into an E-module and we define an E-module morphism by .

All of this looks pretty harmless, but the upshot is that we have now equipped the image of this E-module morphism, say (which is the additive subgroup of generated by the elements ) with a commutative multiplication induced by the rule .

Explicitly, is the set of one-truncated polynomials with coefficients in such that one can find elements such that . We multiply with another such truncated one-polynomial (taking elements ) via

and using distributivity and the multiplication rule this gives the element . Being a ring-qutient of we have that is a commutative ring, and, from the construction it is clear that behaves functorially.

For rings such that we are done, but in general may be strictly smaller. The idea is to use functoriality and do the relevant calculations in a larger ring where we can multiply the two truncated one-polynomials and observe that the resulting truncated polynomial still has all its coefficients in .

Here’s how we would do this over : 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 and . Then, over the field we have that and hence we can compute their product as before to be . But then, all coefficients of this truncated K-polynomial are invariant under all permutations of the roots and the roots and so is invariant under all elements of the Galois group. But then, these coefficients are algebraic numbers in whence integers. That is, . It should already be clear from this that the rings contain a lot of arithmetic information!

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

The idea being that every one-polynomial now has one root, namely in . Further, is a free A-module with basis elements all with .

Good! We now have at least one root, but we can continue this process. At level 2, will be the set of all non-constant one-polynomials over and we use them to construct the free overring (which now has the property that every has at least two roots in ). And, again, we repeat this process and obtain in succession the rings . Finally, we define having the property that every one-polynomial over A splits entirely in linear factors over .

But then, for all we can compute . Remains to show that the resulting truncated one-polynomial has all its entries in A. The ring contains two copies of namely and and the intersection of these two rings in exactly (here we use the freeness property and the additional fact that 1 is one of the base elements). But then, by functoriality of , the element lies in the intersection . Done!

Hence, we have endo-functors in the category of all commutative rings, for every number n. Reviewing the construction of one observes that there are natural transformations and therefore also natural transformations . Taking the inverse limits we therefore have the ‘one-power series’ endo-functor 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 can be used to define the category of -rings.