Chromotopy

What are Witt vectors?

The ring of Witt vectors has shown up at multiple points in my reading, and I’ve been pretty dissatisfied with the descriptions of this object. Looking into things more seriously, I’ve found that this is such an immensely complicated construction that no single description is going to do it justice (Hazewinkel tries for 300ish pages!), but I’d like to share a couple viewpoints that I find important and/or enlightening.

Let’s first figure out a starting point with global Witt vectors. Fix your favorite ring $R$, and let $\mathcal{PS}(R) = R [\![ x ]\!]$ denote the multiplicative semigroup of power series with maximal subgroup

$$\mathcal{PS}^{gp}(R) = R^\times + x \cdot R [\![ x ]\!] = R^\times \times (1 + x \cdot R [\![ x ]\!]).$$

The group $R^\times$ is already understood as $\mathbb{G}_m(R) = \mathsf{Rings}(\mathbb{Z}[x^{\pm 1}], R)$, so let’s focus attention on the other factor.

Definition: Write $\mathbb{W}(R) = 1 + x \cdot R [\![x]\!]$. The functor $\mathbb{W}$ is called the “(global / big) Witt scheme”.

Our first goal should be to understand the product structure on $\mathbb{W}(R)$. Of course one has the usual formulas for the product of series, but these are unsatisfyingly complicated; what we really want is to try to unknot the product of power series with something like a logarithm: a homomorphism $\mathbb{W} \to \mathbb{G}_a^\infty$. Let’s start by restricting to the case where $R$ is a $\mathbb{Q}$-algebra, where we have an actual analytic logarithm to work with, and see what happens.

We should first rewrite our power series in a form amenable to this property. By successive divisions and induction on bottom nonzero degree, it’s seen that we can write a power series $1 + a_1 x + a_2 x^2 + \cdots$ in the form $\prod_{i=1}^\infty (1 - c_i t^i)$ for a unique sequence of coefficients $c_i \in R$. Now, let’s fuss around with our logarithm:

$$\begin{array}{rl} \prod_{i=1}^\infty (1 - c_i t^i) & = \exp \log \prod_{i=1}^\infty (1 - c_i t^i) \\ & = \exp \sum_{i=1}^\infty \log (1 - c_i t^i) \\ & = \exp \sum_{i=1}^\infty \left( -\sum_{j=1}^\infty \frac{(c_i t^i)^j}{j} \right) \\ & = \exp \sum_{N=1}^\infty \frac{t^N}{N} \left( \sum_{d \mid N} d c_d^{N/d} \right). \end{array}$$

Write $w_N(c_*)$ for the parenthesized coefficient of $t^N$, a polynomial in the $c_i$s. By performing these same steps for the product $\prod_{i=1}^\infty (1 - c_i t^i) \cdot \prod_{j=1}^\infty (1 - d_j t^j)$, we find that these polynomials are homomorphic in the sense that $w_N(c_* +_{\mathbb{W}} d_*) = w_N(c_*) + w_N(d_*)$, and so all together they give a homomorphism called the “Witt map”:

$$\mathbb{W} \xrightarrow{(w_1, \ldots, w_N, \ldots)} \mathbb{G}_a^\infty.$$

The image of a point of $\mathbb{W}$ under this map is called its “ghost components”. Now, what good is this map? Here’s an important lemma:

Lemma: Write $\sigma_p$ for the endomorphism of $\mathbb{Z}[x_1, x_2, \ldots]$ determined by $x_i \mapsto x_i^p$. The sequence $(c_1, c_2, \ldots)$ is in the image of the Witt map if and only if for all indices $N$ and primes $p$, $c_N = \sigma_p(c_{N/p}) \pmod{p^j}$, where $p^j$ is the maximal power of $p$ dividing $N$.

This is proven one prime at a time, where the Witt polynomials $w_N$ simplify drastically, allowing an induction on $N$. I won’t go into the details of the proof, but I do want to note two things about the setup: this case of $\mathbb{Z}[x_1, x_2, \ldots]$ is the universal case, and in this universal case the Witt map is injective. Hence, if a preimage exists, it is unique, and we have in this sense completely embedded power series multiplication into ring addition.^†

There are several other incarnations of the Witt vectors worth mentioning. First, we can truncate our coefficient sequence $c_*$ after some fixed $N$; this gives a truncated Witt scheme, written $\mathbb{W}_N$, and there’s an isomorphism $\mathbb{W} = \lim_N \mathbb{W}_N$. Musing about the Witt polynomials $w_N$ and the proof of the Lemma causes one to notice that the polynomials $w_{p^j}(c_*) = \sum_{i=0}^j p^i c_{p^i}^{p^{j-i}}$ depend only upon the sequence of coefficients $c_{p^i}$ with $i < j$. The p-typical Witt scheme $\mathbb{W}_p$ is defined as the quotient of the global Witt scheme which forgets all but these coefficients. By restricting to $\mathbb{Z}_{(p)}$-algebras, the Lemma shows that the quotient map admits a unique section inducing the action

$$s(c_*)_N = \begin{cases} c_i & \mathrm{for}\,N = mp^i, \\ 0 & \mathrm{otherwise} \end{cases}$$

on ghost components for any natural $m$ prime to $p$. This demonstrates a splitting

$$\mathrm{spec}\,\mathbb{Z}_{(p)} \times_{\mathrm{spec}\,\mathbb{Z}} \mathbb{W} \cong \prod_{\gcd(n, p) = 1} (\mathrm{spec}\,\mathbb{Z}_{(p)} \times_{\mathrm{spec}\,\mathbb{Z}} \mathbb{W}_p).$$

Finally, the finite schemes $\mathbb{W}_N$ have formal completions at the identity $\hat{\mathbb{W}}_N$, defined on an $R$-algebra $A$ by $\hat{\mathbb{W}}_N(A) = \ker [\mathbb{W}_N(A) \to \mathbb{W}_N(R)]$. The Witt formal scheme is defined as $\hat{\mathbb{W}} = \mathrm{colim}_N\,\hat{\mathbb{W}}_N$ or as above through kernels.

This last object is actually rather exceptional:

Theorem: The Witt formal group $\hat{\mathbb{W}}$ is the free formal group on the formal affine line $\hat{\mathbb{A}}^1$.

Let’s explore that statement now.

One idea for trying to study the structure of formal groups is to make use of the analogy between schemes and manifnews and construct, as one would attach a Lie algebra to a Lie group, some module of curves passing through the origin of the formal group. Write $CG$ for the Cartier module of the commutative formal $(\mathrm{spec}\, R)$-group $G$, defined as $CG = \mathsf{FormalVarieties}(\hat{\mathbb{A}}^1, G)$. To rigidify this object, we introduce some operations on it, i.e., collections of natural transformations $C \to C$.

• Homothety The affine line hnews two kinds of “scaling” operations. First, for any element $r \in R$, we get a map $\hat{\mathbb{A}}^1 \xrightarrow{t \mapsto rt} \hat{\mathbb{A}}^1$. For a curve $\gamma: \hat{\mathbb{A}}^1 \to G$, we get an induced curve $[r]\gamma: \hat{\mathbb{A}}^1 \to \hat{\mathbb{A}}^1 \to G$, and so an operation $[r]$ on $C$.
• Verschiebung: For a natural $n$, we can also use multiplication to get a kind of scaling. Write $V_n \gamma$ for the composite $\hat{\mathbb{A}}^1 \xrightarrow{t \mapsto t^n} \hat{\mathbb{A}}^1 \xrightarrow{\gamma} G$.
• Frobenius: We have not yet used multiplication in $G$. Since $G$ is commutative, the composite $(\hat{\mathbb{A}}^1)^n \xrightarrow{\gamma^n} G^n \to G$ descends along the quotient $(\hat{\mathbb{A}}^1)^n \to (\hat{\mathbb{A}}^1)^n / \Sigma_n$. The Frobenius operation $F_n \gamma$ is defined by the composite $\hat{\mathbb{A}}^1 \xrightarrow{(0, \ldots, 0, t)} (\hat{\mathbb{A}}^1)^n \xrightarrow{\cong} (\hat{\mathbb{A}}^1)^n / \Sigma_n \to G$. To give a sense of what this means, if we (base-extend so that we can) select a primitive $n$th root of unity $\zeta$, then one computes $F_n \gamma(t) = \gamma(t^{1/n} \zeta^0) +_G \cdots +_G \gamma(t^{1/n} \zeta^{n-1})$. For example, in the case $G = \mathbb{W}$, consider the path $\gamma_1(t) = (t, 0, 0, \ldots)$. Then, we can compute $F_n \gamma_1(t) = (1 - t^{1/n} x) \cdots (1 - t^{1/n} \zeta^{n-1} x) = (1 - tx^n)$, corresponding to the point $(0, \ldots, 0, t, 0, \ldots)$ with $t$ in the $n$th position.

These operations satisfy a whole mess of identities, including $F_n V_m = V_m F_n$ if $n$ and $m$ are coprime, $F_n V_m = n_{G}$, $F_n F_m = F_{nm}$, $V_n V_m = V_{nm}$, $F_n [r] = [r^n] F_n$, and $[r] V_n = V_n [r^n]$. None of these is difficult to prove.

The $\gamma_1$ curve in the example is what induces the natural isomorphism

$$\mathsf{FormalGroups}(\hat{\mathbb{W}}, G) \xrightarrow{\cong} CG.$$

I’ll demonstrate that this map is a set-theoretic isomorphism, but is also a homomorphism and it also respects the Frobenius and Verschiebung maps which can be independently defined on $\hat{\mathbb{W}}$. These facts are much harder and are not, as near as I can tell, worth regurgitating for you in the space of a blog post.

To show that this induced map is injective, let $\pi_i$ denote the projection $\hat{\mathbb{W}} \to \hat{\mathbb{A}}^1$ acting by $c_* \mapsto c_i$. Using our example computation of the Frobenius in the Witt formal group, we can write the identity $\hat{\mathbb{W}} \to \hat{\mathbb{W}}$ as the sum $\sum_{i = 1}^\infty F_i \gamma_1 \pi_i$. If $g: \hat{\mathbb{W}} \to G$ has $g \gamma_1 = 0$, then we check

$$g = g \circ \sum_{i=1}^\infty F_i \gamma_1 \pi_i = \sum_{i=1}^\infty F_i g \gamma_1 \pi_i = 0.$$

To show that it’s surjective, for a curve $\gamma: \hat{\mathbb{A}}^1 \to G$, we define a map $g: \hat{\mathbb{W}} \to G$ by $g = \sum_{i=1}^\infty F_i \gamma \pi_i$, which restricts to $\gamma$ as required. Hence, $\hat{\mathbb{W}}$ is the free formal group on the formal affine line.^‡

This barely scratches the surface of the geometry of Witt vectors, which can be gestured at by saying that the representing Hopf algebra for $\mathbb{W}$ is the Hopf algebra of symmetric functions. This algebra is self-dual, plethysm gives it a ring structure (and hence, by self-duality, a coring structure), it represents a ring scheme and a coring scheme. It occurs as the homology and cohomology of the classifying space for complex K-theory $BU$, and makes many other appearances in representation theory through its interpretation as controlling symmetric functions. It also occurs in the context of deformation theory; for a finite field $k$, $\mathbb{W}_p k$ is initial for complete local rings with residue field $k$. It really is a hole with no bottom.

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^† - Using this lemma, one can show that the product of two sequences in the image of the Witt map (using pointwise ring multiplication) is itself in the image of the Witt map, and hence ring multiplication lifts to some complicated operation “above” multiplication of power series, satisfying distributivity and so forth. This is what’s called “plethysm”, and it turns $\mathbb{W}$ into a ring scheme.

^‡ - The Witt ring described in the previous footnote is also involved in the Cartier functor. Namely, for a group $(\mathrm{spec}\,R)$-scheme $G$, $CG$ is a $\mathbb{W}(R)$-module, where the map $\mathbb{W}(R) \to \mathrm{End}(C)$ is described by $(c_1, \ldots) \mapsto \sum_{n=1}^\infty V_n[c_n]F_n$. This takes quite a bit of work to prove.

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