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Notes on prismatic cohomology

Kiran S. Kedlaya

Section 3 Witt vectors

References.

As in Section 2. The original paper of Witt is [127]. See also [116], chapter II, section 6.
We now relate the discussion of \(\delta\)-rings to the older construction of \(p\)-typical Witt vectors. Our main goal is to relate this construction to perfect \(\delta\)-rings Proposition 3.3.6; this only involves evaluating the Witt functor on perfect rings of characteristic \(p\text{,}\) but to develop the theory it is easier to remember that it defines a functor on arbitrary commutative rings.

Subsection 3.1 \(p\)-typical Witt vectors via adjunction

We introduce the \(p\)-typical Witt vectors, building upon our work with truncated Witt vectors in Subsection 2.3 and the adjunction between rings and \(\delta\)-rings. However, to make this unorthodox development compatible with the more standard treatment (and the big Witt vectors to follow), we must introduce a key change of coordinates.

Definition 3.1.1.

As indicated in Definition 2.4.5, the forgetful functor \(\Ring_\delta \to \Ring\) admits a right adjoint \(W\text{.}\) To identify the image of a ring \(A\) under this functor, we use the set-theoretic identifications
\begin{align*} W(A) &= \Hom_{\Ring}(\ZZ[y], W(A))\\ &= \Hom_{\Ring_\delta}(\ZZ\{y\}, W(A))\\ &= \Hom_{\Ring}(\ZZ[y_0, y_1, \dots], A)\\ &= A \times A \times \cdots. \end{align*}
This means that each element of \(W(A)\) has a unique expansion \((y_0, y_1, \dots)\) with each \(y_n \in A\text{;}\) we call the \(y_n\) the \(y\)-coordinates (or Joyal coordinates) of this element of \(W(A)\text{.}\) (This presentation does not directly describe the ring structure on \(W(A)\text{;}\) see Remark 4.2.6.)
In Lemma 3.1.3 below. we will give a second set of generators \(x_0, x_1, \dots\) of the polynomial ring \(\ZZ[y_0, y_1, \dots]\text{.}\) This means that each element of \(W(A)\) has a unique expansion \((x_0, x_1, \dots)\) with each \(x_n \in A\text{;}\) we call the \(x_n\) the \(x\)-coordinates (or Witt coordinates) of this element of \(W(A)\text{.}\) In these coordinates, \(W(A)\) will become none other than the ring of \(p\)-typical Witt vectors over \(A\) via the translation described in Definition 3.2.1.

Remark 3.1.2.

Before continuing, we record a statement which will come up repeatedly: for elements \(x,y\) of a commutative ring,
\begin{equation} (x + py)^p \equiv x^p \pmod{p^2 y}.\tag{3.1} \end{equation}
In particular,
\begin{equation} x \equiv y \pmod{p^m} \Rightarrow x^p \equiv y^p \pmod{p^{m+1}}.\tag{3.2} \end{equation}

Proof.

Given \(x_0,\dots,x_{n-1} \in \ZZ\{y\}\) satisfying (3.3) with \(n\) replaced by \(n-1\text{,}\) use (3.1) to write
\begin{align*} \phi^n(x_0) &= \phi(\phi^{n-1}(x_0)) \\ &= \phi(x_0)^{p^{n-1}} + \cdots + p^{n-1} \phi(x_{n-1}) \\ &= (x_0^p + p*)^{p^{n-1}} + \cdots + p^{n-2} (x_{n-2}^p + p*)^p + p^{n-1} \phi(x_{n-1}) \\ &= x_0^{p^n} + \cdots + p^{n-2} x_{n-2}^{p^2} + p^{n-1} (x_{n-1}^p + p \delta(x_{n-1})) + p^n * \end{align*}
where each \(*\) denotes a quantity in \((y_1,\dots,y_{n-1})\ZZ[y_0,\dots,y_{n-1}]\text{.}\) We thus can (and must) take
\begin{equation*} x_n = \delta(x_{n-1}) + *. \end{equation*}
Since \(x_{n-1} - y_{n-1} \in \ZZ[y_0,\dots,y_{n-2}]\text{,}\) we have
\begin{equation*} \delta(x_{n-1}) = \delta(y_{n-1}) + \delta(x_{n-1}-y_{n-1}) + * = y_n + * \end{equation*}
and so \(x_n - y_n \in (y_1,\dots,y_{n-1})\ZZ[y_0,\dots,y_{n-1}]\text{.}\)

Proof.

By Lemma 3.1.3, both sides are equal to \(\phi^{n+m}(x_0)\text{.}\)

Proof.

The first assertion is a consequence of the fact that \(\phi\) is a Frobenius lift (because \(\ZZ\{y\}\) is a \(\delta\)-ring). The second assertion is a direct consequence of the first, but let us spell this out for clarity as the mechanism of the argument will recur in what follows. For a general ring \(A\text{,}\) each of the elements \(x_0, x_1, \dots \in \ZZ\{x\}\) defines a function \(W(A) \to A\) which is natural in \(A\text{.}\) Similarly, every element of \(\ZZ\{x\}\) can be viewed as a “polynomial function” on \(W(A)\) valued in \(A\) which is again natural in \(A\text{;}\) that is, we have a map of sets \(h\colon \ZZ\{x\} \to \Hom_{\Set}(W(A), A)\text{.}\) This map has the property that
\begin{equation*} h(\phi(t))(u) = \phi(h(t)(u)) \qquad (t \in \ZZ\{x\}, u \in W(A)). \end{equation*}
In the case where \(A\) is of characteristic \(p\text{,}\) we have \(\phi(x_n) = x_n^p + p \delta(x_n)\) and so for any \(u \in W(A)\text{,}\)
\begin{align*} \phi(h(x_n)(u)) &= h(\phi(x_n))(u) \\ &= h(x_n^p + p \delta(x_n))(u)\\ &= h(x_n^p)(u) + p h(\delta(x_n))(u)\\ &= h(x_n)(u)^p. \end{align*}
This shows that the \(\phi\) acts via the functorial Frobenius.
We record some consequences of the adjunction between \(\Ring\) and \(\Ring_\delta\text{.}\)

Definition 3.1.6.

The identity map in \(\Hom_{\Ring_\delta}(W(A), W(A))\) corresponds via adjunction to a morphism \(W(A) \to A\) of rings. In coordinates, this is the map \((x_0,x_1,\dots) \mapsto x_0\text{.}\)
The identity map in \(\Hom_{\Ring}(W(A), W(A))\) corresponds via adjunction to a morphism \(\Delta\colon W(A) \to W(W(A))\) in \(\Ring_\delta\) which is moreover functorial in \(A\text{.}\) This map is sometimes called the diagonal.

Definition 3.1.7.

Recall that the action of \(\delta\) on \(\ZZ[y_0, y_1, \dots]\) satisfies \(\delta(y_n) = y_{n+1}\text{;}\) consequently, if we express an element of \(W(A)\) in the \(y\)-coordinates as \((y_0, y_1, \dots)\text{,}\) it is \(\delta\)-constant if and only if \(y_1 = y_2 = \cdots = 0\text{.}\) By Lemma 3.1.3, in the usual coordinates, an element \((x_0,x_1,\dots)\) of \(W(A)\) is \(\delta\)-constant if and only if \(x_1 = x_2 = \cdots = 0\text{.}\)
That is, the \(\delta\)-constants are the image of the multiplicative (but not additive; see Exercise 3.6.3) section \([\bullet]\colon A \to W(A)\) of the projection \(W(A) \to A\) given by \([x] = (x, 0, 0, \dots)\text{.}\) We call \([x]\) the constant lift (or the multiplicative lift) of \(x \in A\text{.}\)

Subsection 3.2 Ghost coordinates

The description of \(W(A)\) we are using does not make it especially clear how the addition and multiplication operations work. To clarify this, we relate back to the more standard presentation of Witt vectors.

Definition 3.2.1.

Define the elements \(w_n \in \ZZ\{y\}\) as in Corollary 3.1.4. These define a set-theoretic map
\begin{equation*} w\colon W(A) \to A \times A \times \cdots, \qquad (x_n)_{n=0}^\infty \mapsto \left( \sum_{m=0}^n p^m x_m^{p^{n-m}} \right)_{n=0}^\infty. \end{equation*}
which we call the ghost map.
Note that in general, this map is neither injective (unless \(A\) is \(p\)-torsion-free) nor surjective (unless \(A\) is \(p\)-divisible). Nonetheless, for \(x \in W(A)\text{,}\) it will be convenient to refer to the terms of \(w(x) = (w_0, w_1, \dots)\) as the ghost coordinates of \(x\text{.}\) By Corollary 3.1.4, the ghost coordinates of \(\phi^n(x)\) are \((w_n, w_{n+1}, \dots)\text{.}\)
Now recall the map \(x \mapsto w_0 = x_0\) is the homomorphism \(W(A) \to A\) obtained by adjunction. It follows that the map \(W(A) \stackrel{\phi^n}{\to} W(A) \to A\) is given by \(x \mapsto w_n\text{.}\) That is, the ghost map is a natural transformation of functors of rings!

Remark 3.2.2.

Using the ghost map, we can now see that \(W(A)\) agrees with the usual definition of the ring of \(p\)-typical Witt vectors of \(A\text{,}\) in which the arithmetic operations on Witt vectors are given by certain universal polynomials in the entries of the Witt vectors. We may read off properties of these polynomials using functoriality; this is similar to a more typical proof of the existence of the functor \(W\) (see for example [73], section 8.10), except that now we don’t need to worry about its existence! This means that we can freely pass from general rings to \(p\)-torsion-free rings to \(\ZZ[p^{-1}]\)-algebras.

Definition 3.2.3.

For any ring \(A\text{,}\) the Verschiebung map \(V\colon W(A) \to W(A)\) is defined by
\begin{equation*} V(x_0, x_1, \dots) = (0, x_0, x_1, \dots). \end{equation*}
Note that this corresponds to the map on ghost coordinates given by
\begin{equation*} (w_0, w_1, \dots) \mapsto (0, pw_0, pw_1, \dots). \end{equation*}
Using the ghost map as per Remark 3.2.2, we may deduce that \(V\) is additive (but not multiplicative) and that \(\phi \circ V\) acts via multiplication by \(p\text{.}\)

Definition 3.2.4.

Using the method of Remark 3.2.2, we may show that for each positive integer \(n\text{,}\) there is a natural transformation from \(W\) to another functor \(W_n\) on \(\Ring\) which on sets corresponds to the projection
\begin{equation*} (x_0, x_1, \dots) \mapsto (x_0, \dots, x_{n-1}) \end{equation*}
(and similarly for ghost components). The ring \(W_n(A)\) is called the ring of truncated \(p\)-typical Witt vectors of length \(n\) over \(A\text{;}\) for \(n=1\) we get \(A\) itself, while for \(n=2\) we recover the construction of Definition 2.3.1. Note that the natural transformation \(W \to \lim_n W_n\) is an isomorphism.
The action of \(\phi\) on \(W(A)\) does not induce an endomorphism of \(W_n(A)\) in general (unless \(p=0\) in \(A\text{,}\) in which case Corollary 3.1.5 applies). However, it does induce a homomorphism \(W_{n+1}(A) \to W_n(A)\) (the Witt vector Frobenius), from which we can recover \(\phi\) as the induced map
\begin{equation*} \lim_n W_n(A) = \lim_n W_{n+1}(A) \to \lim_n W_n(A). \end{equation*}

Remark 3.2.5.

In [127], what we call ghost coordinates were instead called Nebenkomponenten, or secondary components. The terminology we use here is quite commonplace but its origins are unclear; the earliest reference we were able to find is Barsotti’s Mathematical Reviews synopsis of [130], but it seems likely that the terminology was in circulation before that.

Subsection 3.3 Witt vectors and perfect \(\delta\)-rings

We now focus more closely on Witt vectors valued in a perfect ring of characteristic \(p\text{,}\) and obtain their more familiar ring-theoretic properties.

Definition 3.3.1.

A \(\delta\)-ring \(A\) is perfect if \(\phi\) is an isomorphism. By the same token, a ring of characteristic \(p\) is perfect if \(\phi\) is an isomorphism; in this case, injectivity of \(\phi\) is equivalent to \(A\) being reduced.

Proof.

By Corollary 3.1.5, \(\phi\) is an automorphism of \(W(A)\text{.}\) By Lemma 2.2.8, the ring \(W(A)\) is \(p\)-torsion-free. Since \(\phi \circ V\) is multiplication by \(p\) (Definition 3.2.3) and \(\phi\) is bijective, the ideal \(pW(A)\) coincides with the image of \(V\text{,}\) which in turn equals the kernel of the map \(W(A) \to A\text{;}\) hence \(W(A)/(p) \cong A\text{.}\) By similar logic, for each positive integer \(n\text{,}\) the ideal \(p^n W(A)\) coincides with the image of \(V^n\text{;}\) from this, we see that \(W(A)\) is \(p\)-adically complete.

Example 3.3.3.

We have \(W(\FF_p) \cong \ZZ_p\text{.}\) More generally, for any finite extension \(\FF_q\) of \(\FF_p\text{,}\) \(W(\FF_q) = \ZZ_p[\zeta_{q-1}]\text{.}\)

Definition 3.3.4.

For \(A\) a perfect ring of characteristic \(p\text{,}\) Lemma 3.3.2 implies that each element \(x\) of \(W(A)\) can be written uniquely as a \(p\)-adically convergent sum \(\sum_{n=0}^\infty p^n [x_n]\) with \(x_n \in A\text{,}\) where \([x_n]\) denotes the constant lift (Definition 3.1.7). We call this the series representation of \(x\text{.}\)

Proof.

We first use (3.2) to lift \(R \to S/(p)\) to a multiplicative map \(R \to S\text{.}\) Using the series representations from Definition 3.3.4, we then obtain a set-theoretic map \(W(R) \to S\) which we must show is an homomorphism; it is enough to check that it induces a homomorphism \(W(R) \to S/p^n\) for each \(n\text{.}\) This is not too onerous to prove by direct computation; see for example [80], Lemma 1.1.7.
A second, more conceptual approach is to apply the fact that if \(A\) is a perfect ring of characteristic \(p\text{,}\) then the cotangent complex \(L_{A/\FF_p}\) vanishes; we will revisit this comment once we have introduced the cotangent complex in Subsection 17.1. See Exercise 17.5.1 (and [18], Lecture II, Lemma 3.5).

Proof.

The composition from (3) to (1) to (2) to (3) is an equivalence by Lemma 3.3.2. In particular, the functor from (2) to (3) is essentially surjective. By Lemma 3.3.5, the composition from (3) to (1) to (2) is also essentially surjective; hence (2) and (3) are equivalent. We can now use Lemma 2.2.8 and Corollary 3.1.5 to add (1) to the loop.

Subsection 3.4 Beyond the perfect case in characteristic \(p\)

It is not the case that Proposition 3.3.6 can be extended to relate \(p\)-torsion-free, \(p\)-adically complete rings whose reductions modulo \(p\) are not perfect with the image of the functor \(W\) on nonperfect rings of characteristic \(p\text{.}\) We record some assorted remarks here.

Definition 3.4.1.

The inclusion of the full subcategory of perfect rings of characteristic \(p\) into arbitrary rings of characteristic \(p\) has both left and right adjoints. The left adjoint maps \(A\) to \(\colim_\phi A\text{,}\) which we call the coperfection of \(A\text{.}\) The right adjoint maps \(A\) to \(\lim_\phi A\text{,}\) which we call the perfection of \(A\text{.}\)
The following examples show that the relationship between the perfection and coperfection can be a bit subtle.

Example 3.4.2.

For \(A = \FF_p[x]\text{,}\) the coperfection equals \(\FF_p[x^{p^{-\infty}}]\) while the perfection equals \(\FF_p\text{.}\)

Example 3.4.3.

For \(A = \FF_p[x^{p^{-\infty}}]/(x)\text{,}\) the coperfection equals \(\FF_p\) while the perfection equals the \(x\)-adic completion of \(\FF_p[x^{p^{-\infty}}]\text{.}\)

Remark 3.4.4.

Let \(A\) be a \(p\)-torsion-free, \(p\)-adically complete \(\delta\)-ring. Let \(R\) be the coperfection of \(A/(p)\) (Definition 3.3.1). Using Proposition 3.3.6, we obtain a morphism \(A \to W(R)\) in \(\Ring_\delta\text{;}\) this map is injective if \(A/(p)\) is reduced. If we fix \(A\) as an underlying ring while varying its \(\delta\)-ring structure, the target \(W(R)\) remains fixed while the morphism \(A \to W(R)\) varies.

Example 3.4.5.

Put \(A = \ZZ[x]\text{.}\) As in Example 2.2.5, for each \(y \in A\) there is a unique \(\delta\)-ring structure on \(A\) for which \(\delta(x) = y\text{.}\) Each of these gives rise to an injective morphism \(A \to W(\colim_\phi \FF_p[x])\) of \(\delta\)-rings.

Proof.

See [18], Lecture II, Lemma 2.9. See also [104] for a supplemental argument that can be used to eliminate the \(p\)-torsion-free hypothesis.

Remark 3.4.7.

Corollary 3.1.5 implies that when \(A\) is a reduced ring of characteristic \(p\text{,}\) the map \(\phi\) on \(W(A)\) is injective. By contrast, if \(A\) is a nonreduced ring of characteristic \(p\text{,}\) then \(\phi\) is not injective: for any nonzero \(x \in A\) with \(x^p = 0\text{,}\) we have \([x] \neq 0\) but \(\phi([x]) = [x^p] = 0\text{.}\)
If \(A\) is a ring not of characteristic \(p\text{,}\) then the map \(\phi\) on \(W(A)\) is not injective either, but this is somewhat more subtle. See Exercise 3.6.7. (One case which is not subtle: if \(p\) is invertible in \(W(A)\text{,}\) then the ghost map is an isomorphism and so we may see the kernel of \(\phi\) on the ghost side, remembering that \(\phi\) acts here as the left shift.)

Remark 3.4.8.

For \(A\) of characteristic \(p\text{,}\) the map \(\phi\) on \(W(A)\) is surjective if and only if it is surjective on \(A\text{,}\) i.e., if and only if \(A\) is semiperfect. However, by contrast with Remark 3.4.7, there are many rings \(A\) not of characteristic \(p\) for which \(\phi\) is surjective on \(W(A)\text{.}\) There are even more rings for which \(\phi\colon W_{n+1}(A) \to W_n(A)\) is surjective for each \(n\text{;}\) these rings are said to be Witt-perfect in [40], which see for additional characterizations.

Subsection 3.5 Additional remarks

Proof.

This was originally shown by van der Kallen ([122], (2.4)); see also [28], Theorem B. (Both of these references also cover the truncated big Witt vector functors; see Definition 4.1.3.) For the case of a localization, see also Exercise 3.6.5.

Remark 3.5.2.

By Proposition 3.5.1, we may apply the functors \(W_n\) also to schemes. See [29] for some discussion of this construction.

Exercises 3.6 Exercises

1.

Describe the ring \(W(A)\) explicitly for \(A = \FF_p[x]/(x^p)\text{,}\) and show that it is a \(\delta\)-ring with nontrivial \(p\)-torsion. (This provides a nontrivial example of Lemma 2.2.8.)
Hint.
Use the fact that \(\phi \circ V\) acts as multiplication by \(p\text{.}\)

2.

Let \(A\) be a \(p\)-torsion-free, \(p\)-adically complete ring. Let \(R\) be the perfection of \(A/(p)\) (Definition 3.3.1). Show that the natural maps
\begin{equation*} \lim_{\phi} W(A) \to \lim_{\phi} W(A/(p)) \to \lim_{\phi} W(R) \to W(R) \end{equation*}
are all isomorphisms.

3.

Show that for any ring \(A\text{,}\) the map \([\bullet]\colon A \to W(A)\) is multiplicative.
Hint.
Use the fact that the \(\delta\)-constant elements both form a multiplicative subset and coincide with the image of \([\bullet]\text{.}\)

4.

Let \(R\) be a perfect ring of characteristic \(p\text{.}\) Prove that \(R\) is noetherian if and only if \(R\) is a finite (possibly empty) direct product of fields. Consequently, \(W(R)\) is noetherian if and only if the same conditions hold.

5.

Let \(A\) be a ring and let \(S\) be a multiplicative subset. Let \([S]\) be the image of \(S\) under the constant section. Prove that for each positive integer \(n\text{,}\) there is a natural isomorphism \([S]^{-1} W_n(A) \to W_n(S^{-1} A)\text{.}\) By contrast, the natural map \([S]^{-1} W(A) \to W(S^{-1} A)\) is not an isomorphism.
Hint.
The natural map exists because elements of \([S]\) become units in \(W_n(S^{-1} A)\text{.}\) To show that it is surjective, first use the ghost map (and naturality) to figure out how multiplication by a constant lift acts on the Witt components of a general vector.

6.

Show that in Example 3.4.5, the image of \(\ZZ[x]\) in \(W(\colim_\phi \FF_p[x])\) need not be generated (as a \(\ZZ\)-algebra) by multiplicative lifts.

7.

Let \(A\) be a ring. Show that if \(\phi\colon W(A) \to W(A)\) is injective, then \(p=0\) in \(A\text{.}\)
Hint.
Show that any multiple of \(p\) occurs as \(x_0\) in some \(x = (x_0, x_1,\dots) \in W(A)\) with \(\phi(x) = 0\text{.}\) For more details, see [40], Corollary 2.6.

8.

Let \(R\) be a perfect ring of characteristic \(p\text{.}\) Show that \(V \circ \phi\) acts on \(W(R)\) by multiplication by \(p\) (just as \(\phi \circ V\) does for arbitrary \(R\)).
Hint.
Use the fact that \(\phi\) agrees with the functorial Frobenius to see that it commutes with multiplication by \(p\text{.}\)

9.

Prove that for any nonzero ring \(A\text{,}\) the characteristic of the ring \(W(A)\) is either 0 or not divisible by \(p\text{.}\)
Hint.
If any maximal ideal of \(A\) has characteristic \(p\text{,}\) then \(W(A)\) maps to a ring of characteristic 0. Otherwise, \(W(A)\) splits as a product of copies of \(A\text{.}\)

10.

Prove that Figure 3.6.1 is a pullback square.
Figure 3.6.1.
Hint.
The corresponding statement with \(\ZZ_p\) replaced by \(\widehat{\ZZ}\) holds because the latter is faithfully flat over \(\ZZ\text{.}\) Now rewrite \(\widehat{\ZZ}\) as the product of \(\ZZ_p\) with a \(\ZZ[p^{-1}]\)-algebra and recall that for the latter, the ghost map is an isomorphism.

11.

Show that for \(A \in \Ring\text{,}\) the map
\begin{equation*} N\colon W(A) \to W(A), \qquad N(x) = x - V_p(\delta(x)) \end{equation*}
is multiplicative and satisfies
\begin{equation*} (F \circ N)(x) = x^p \qquad (x \in W(A)). \end{equation*}
This is called the norm map in [3]; see also Exercise 4.3.8.
Hint.
Show that the effect on ghost coordinates is given by
\begin{equation*} (w_0, w_1, w_2, \dots) \mapsto (w_0, w_0^p, w_1^p, w_2^p, \dots). \end{equation*}
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