perfectoid spaces-6

In this post, we will say something on Witt vectors, which will be used in Scholze’s paper. All the materials in this post can be found in Serre’s book, ‘Corps Locaux’.

We shall not discuss the original motivation for Witt to introduce the Witt vector. One interesting motivation is the following simple result:

Suppose that A is a complete discrete valuation ring, with fraction field K and residue field \kappa. Suppose that S is a set of representatives of \kappa in A, and \pi is a uniformizer of A. Then each element a in A is a ‘power series’ of \pi with coefficients in S, that is, a can be written uniquely as a=\sum_{n=0}^{\infty}s_n\pi^n with s_n\in S.

Here a uniformizer of A is just a generator for the maximal ideal of A. Since A is a discrete valuation ring, this means that \pi always exists. The essential part in the proof to this proposition is that, the ideals m=(\pi)\supset m^2\supset m^3\supset... form a system of open neighborhood of 0. And of course, elements in the fraction field K can also be written as Laurent power series as above.

Now note that A is a ring, which has addition, subtraction, and multiplication operations. What will happen if we use the above ‘expansion’ expression for these operations? That is to say, if a=\sum_n a_n\pi^n,b=\sum_n b_n\pi^n,c=a+b=\sum_n c_n\pi^n,then what are the relations between a_n,b_n,c_n? Is there some simple algebraic expression for this? If not, can we choose S to be as simple as possible such that the expression desired is very simple?

The above observation can be seen as a motivation for Witt vectors. Note that in the above proposition, we can show that as sets, we have, by construction, that A=\kappa^{\mathbb{N}}. So, the questions raised at the end of the last paragraph is just to find a ring structure on \kappa^{\mathbb{N}} such that this ring is isomorphic to A. Of course, conditions could be posed on A.

One general construction is as follows:

Fix a prime number p. For a countable set of indeterminates (X_0,X_1,X_2,...) and we set W_0=X_0,W_1=X_0^p+pX_1,...,W_k=X_0^{p^k}+pX_1^{p^{k-1}}+p^2X_2^{p^{k-2}}+...+p^kX_k. Then it is easy to see that

For any F\in\mathbb{Z}[X,Y], there is a unique sequence (f_0,f_1,...) in \mathbb{Z}[X_0,X_1,...,Y_0,Y_1,...] such that W_n(f_0,f_1,...)=F(W_n(X_0,X_1,...),W_n(Y_0,Y_1,...)) for all n=0,1,...

I will not prove this result. One method is just to use induction on n and some calculations. The interesting thing is that, when we express X_i in terms of W_k, the coefficients lie in \mathbb{Z}[1/p], yet this proposition tell us that these f_j have integer coefficients. This is due to the assumption that p is a prime number.

We shall apply this result to two polynomials, that is F=X+Y and G=XY, and we denote the corresponding sequence as (f_0,f_1,...),(g_0,g_1,...). Then a surprising result is:

Suppose that R is a commutative ring, and for any a=(a_0,a_1,...),b=(b_0,b_1,...)\in R^{\mathbb{N}}, if we set a+b=(f_0(a,b),f_1(a,b),...),a*b=(g_0(a,b),g_1(a,b),...), then with these two operations(addition and multiplication, as the symbols suggest), (R^{\mathbb{N}},+,*) is a commutative ring.

Here we simply write g_i(a,b)=g_i(a_0,a_1,...,b_0,b_1,...) and the same for others.

One little remark, in some sense, this result is part of the theory of formal groups.

We shall denote this ring as W(R), and a\in W(R) is called a Witt vector with coefficients in R. Note that since f_i contains only addition, subtraction and multiplication operations and divide p operation, so if this proposition is valid for a ring R, then it is also valid for subrings and quotient rings of R. As any ring is a quotient ring of some R=\mathbb{Z}^I, so it suffices to prove this proposition for these kinds of rings. If R=\mathbb{Z}^I, then we construct a map W:W(R)\rightarrow R^{\mathbb{N}}(the latter is equipped with the product ring structure), a\mapsto (W_0(a),W_1(a),...). This is a bijection, which is obvious. Now, it is rather clear that W respects the addition and multiplication operations of these two rings, thus proving that F is an isomorphism. Moreover, it is easy to see, (1,0,0,...) is the unit element in W(R).

Now we can put the first result and the last result above together to see what happens: from a discrete valuation ring A we get a \pi-expansion for each of its elements, and thus a bijection A\simeq \kappa^{\mathbb{N}}. And conversely, from \kappa, we construct a ring W(\kappa)\simeq\kappa^{\mathbb{N}}. Then what is the relation between A and W(\kappa)? In fact, they are isomorphic in some cases:

If \kappa is a perfect field of characteristic p, then A\approx W(A/m).

To prove this result, we need Teichmüller representatives:

Suppose that \kappa is a perfect field of characteristic p, then there exists a unique map f:\kappa\rightarrow A such that:(1) f(a^p)=f(a)^p;(2)for a\in A, a\in f(\kappa) if and only if a=b^p for some b\in A;(3)f(ab)=f(a)f(b) for any a,b\in \kappa;(4) if A is of characteristic p, then f(a+b)=f(a)+f(b) for any a,b\in \kappa.

The construction used in the proof is the same flavor as the one we have seen in Scholze’s paper, or in Fontaine-Wintenberg’s theorem. For any a\in\kappa, we consider (a_0,a_1,a_2,...)\in \kappa^{\mathbb{N}} such that a_{i+1}^p=a_i. And for each a_i we find a lift b_i\in A, and we set f(a)=\lim_n b_n^{p^n}. And the first three points are automatically satisfied(the uniqueness is a consequence of the fact that f commutes with the p-power operation). As for the last point, it is also very clear.

This map is called a Teichmüller map, and f(\kappa) are called the Teichmüller representatives of \kappa in A.

Now we can construct a map from W(\kappa) to A as follows: for any a\in W(\kappa), we send it to T(a)=\sum_n f(a_n)^{1/p^n}p^n, for some special rings, we can show that this is an isomorphism:

If A is a strict p-adic ring such that p is invertible in A, then T:W(\kappa)\rightarrow A is an isomorphism.

Here a strict p-adic ring A is a discrete valuation ring such that the (p)\supset (p^2)\supset (p^3)\supset... forms a system of open neighborhoods of 0 in A. Note that, under this assumption, the very first \pi-expansion is can be rephrased for p-expansion. And we can in the before last proposition replace the residue field \kappa by the quotient ring A/pA, everything is still correct since the essential point is of characteristic p. And then this proposition can be proved by calculations.

So, in this way, we find an algebraic expression for (A/pA)^{\mathbb{N}} such that the corresponding ring structure coincides with that of A.

One important remark: for any n>0, we can consider W_n(R)=\{(r_0,r_1,...,r_{n-1},0,0,...)\in W(R)\}. Note that this W_n(R) is a quotient of W(R) by W'_n(R)=\{(0,0,...,0,r_n,r_{n+1},...)\in W(R)\}. Indeed, W'_n(R) is easily seen to be an ideal of W(R), thus W_n(R)=W(R)/W'_n(R) is a quotient ring. And it is not hard to show that W(R)=\varprojlim_nW_n(R).

Another remark is about the topology on W(R). As above, we have defined some ideals W'_n(R) of W(R), and it is easy to see that W'_n(R)W'_m(R)\subset W'_n(R) if n\geq m. So, we can use these ideals to define a topology on W(R).

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