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Notes on prismatic cohomology

Kiran S. Kedlaya

Section 4 Big Witt vectors and \(\lambda\)-rings

References.

In addition to the references given in Section 2, see [30] and [12] for the perspective of \(\lambda\)-rings and [128] for a comprehensive treatment. (An interesting historical reference, oriented towards characteristic classes of vector bundles, is [14].)
We take a bit of a digression to relate the \(p\)-typical Witt vector functor to the big Witt vector functor and to the theory of \(\lambda\)-rings. This is not used anywhere in [18] or [25], but we prefer to provide a broader context with an eye towards potential future developments.
In this section, we do not fix a prime \(p\text{.}\)

Subsection 4.1 The big Witt vector functor

We start with some context from [31]. See Remark 4.2.6 for more of the story.

Definition 4.1.1.

We started our description of the \(p\)-typical Witt vector functor with the fact that the underlying functor to sets is represented by the ring \(A = \ZZ\{y\}\text{,}\) but crucially we already had produced a functor valued in rings (and even in \(\delta\)-rings). If we had needed to construct from scratch a functor valued in rings, we would have needed structures on \(A\) giving rise to the addition and multiplication maps. These structures are:
  • a coaddition morphism \(\Delta^+\colon A \to A \otimes_{\ZZ} A\text{;}\) and
  • a comultiplication morphism \(\Delta^\times\colon A \to A \otimes_{\ZZ} A\text{.}\)
A ring \(A\) equipped with these structures represents a functor from rings to sets equipped with two binary operations \(+, \times\text{.}\) A biring is a ring equipped with coaddition and comultiplication operators which are further subject to the axioms that correspond to the ring axioms on \(+, \times\text{.}\) Namely, the coaddition map is cocommutative, coassociative, and admits a counit and an antipode (giving rise to additive inverses); the comultiplication map is cocommutative, coassociative, codistributive over coaddition, and admits a counit.
A shorter way to say this is that a biring is a commutative ring object in the category of affine schemes. (Remember that the functor \(\Spec\colon \Ring \to \Sch\) is contravariant!)

Proof.

It suffices to produce a unique biring structure on \(\ZZ[x_1, x_2, \dots]\) representing the desired functor. To begin with, since the ghost map is an isomorphism whenever \(A\) is a \(\QQ\)-algebra, we obtain a biring structure on \(\QQ[w_0,w_1,\dots] = \QQ[x_0, x_1,\dots]\text{;}\) this already implies uniqueness. For existence, it suffices to check that for each prime \(p\text{,}\) this biring structure descends to \(\ZZ_{(p)}[x_1, x_2, \dots]\text{;}\) this will imply that the coaddition and comultiplication maps act on \(\bigcap_p \ZZ_{(p)}[x_1,x_2,\dots] = \ZZ[x_1,x_2,\dots]\text{.}\)
Define the family of elements \(y_n\) of \(\QQ[x_1, x_2, \dots]\) as follows: for each positive integer \(m\) coprime to \(p\) and each nonnegative integer \(i\text{,}\)
\begin{equation*} w_{mp^i} = \sum_{j=0}^i p^j y_{mp^j}^{p^{i-j}}. \end{equation*}
By a calculation which we omit (see Exercise 4.3.1), we see that \(\ZZ_{(p)}[y_1,y_2,\dots] = \ZZ_{(p)}[x_1,x_2,\dots]\text{.}\) In the \(y\)-coordinates, \(\WW(A)\) splits into a collection of copies of \(W(A)\) indexed by positive integers coprime to \(p\text{;}\) hence we obtain a biring structure on \(\ZZ_{(p)}[y_1,y_2,\dots] = \ZZ_{(p)}[x_1,x_2,\dots]\) as needed.

Definition 4.1.3.

By analogy with Remark 3.2.2, we can detect various additional structures on \(\WW(A)\) using the ghost map. We leave the details to the reader. (Another approach is to use the splitting principle; see Exercise 4.3.5.)
  • For any nonempty subset \(S\) of the positive integers which is closed under taking divisors, there is a natural transformation from \(\WW\) to another functor \(\WW_S\) on \(\Ring\) (the \(S\)-truncated Witt vectors) which on sets corresponds to the projection
    \begin{equation*} (x_1, x_2, \dots) \mapsto (x_n)_{n \in S} \end{equation*}
    (and similarly for ghost components). In the case where \(S = \{1,p,p^2,\dots\}\) for some prime \(p\text{,}\) this yields a projection \(\WW(A) \to W(A)\text{.}\)
  • There is a family of commuting endomorphisms \(\phi_n\colon \WW(A) \to \WW(A)\) indexed by positive integers \(n\text{,}\) which are natural in \(A\) and correspond via the ghost map to
    \begin{equation*} (w_1, w_2, \dots) \mapsto (w_n, w_{2n}, \dots). \end{equation*}
    The map \(\phi_n\) induces a map \(\WW_S(A) \to \WW_{S'}(A)\) on truncated Witt vectors whenever \(nS' \subseteq S\text{.}\)
  • The map \([\bullet]\colon A \to \WW(A)\) given by \([x] = (x, 0, 0, \dots)\) is multiplicative; it corresponds via the ghost map to \(x \mapsto (x, x^2, x^{3}, \dots)\text{.}\) We again refer to \([x]\) as the constant lift of \(x \in A\) (see Exercise 4.3.4).
  • The Verschiebung maps \(V_n\colon \WW(A) \to \WW(A)\text{,}\) for \(n\) a positive integer, defined by
    \begin{equation*} V_n((x_m)_{m=1}^\infty) = (y_m)_{m=1}^\infty, \qquad y_m = \begin{cases} x_{m/n} & m \equiv 0 \pmod{n} \\ 0 & m \not\equiv 0 \pmod{n} \end{cases} \end{equation*}
    form a commuting family of additive maps such that \(\phi_n \circ V_n\) acts by multiplication by \(n\text{.}\)
  • There is a natural transformation \(\Delta\colon \WW \to \WW \circ \WW\) (the diagonal) such that \(\Delta([x]) = [[x]]\) for all \(x \in A\text{,.}\)

Subsection 4.2 \(\lambda\)-rings

Remark 4.2.1.

Another interpretation of \(\WW(A)\) (also due to Witt) can be given starting with the bijection of \(\WW(A)\) with \(1 + T A \llbracket T \rrbracket\) given by
\begin{equation} (x_1, x_2, \dots) \mapsto \prod_{n=1}^\infty (1 - x_n T^n)^{-1}.\tag{4.1} \end{equation}
When \(A\) is a \(\QQ\)-algebra, the addition operation on \(\WW(A)\) imposed by the ghost map corresponds to the multiplication of formal power series in \(1 + T A \llbracket T \rrbracket\text{.}\) This gives us the underlying additive group on \(\WW(A)\text{.}\) One then shows that there is a \(T\)-adically continuous map \(\otimes\) which is natural in \(A\text{,}\) distributes over addition, and satisfies
\begin{equation} (1-aT)^{-1} \otimes (1-bT)^{-1} = (1-abT)^{-1}.\tag{4.2} \end{equation}
A more conceptual way to express (4.2) is that given two finite projective \(A\)-modules \(M_1, M_2\) equipped with \(A\)-linear endomorphisms \(S_1, S_2\text{,}\)
\begin{equation*} \det(1 - T S_1, M_1)^{-1} \otimes \det(1 - T S_2, M_2)^{-1} = \det(1 - T (S_1 \otimes S_2), M_1 \otimes M_2)^{-1}. \end{equation*}
This point of view appears in work of Almkvist [2] and Grayson [57], [58] in the context of \(K\)-theory of endomorphisms. See also Exercise 4.3.6.

Definition 4.2.2.

The interpretation from Remark 4.2.1 leads naturally to the related notion of a \(\lambda\)-ring. This consists of a ring \(A\) together with operations \(\lambda^n\colon A \to A\) for \(n=0,1,\dots\) satisfying various conditions. To begin with,
\begin{equation*} \lambda^0(x) = 1, \lambda^1(x) = x \quad (x \in A). \end{equation*}
To state the remaining conditions, define the object
\begin{equation*} \Lambda(x) = (1 - \lambda_1(x) T + \lambda_2(x) T^2 - \cdots)^{-1} \in 1 + T A \llbracket T \rrbracket. \end{equation*}
In this notation, we impose the conditions
\begin{gather*} \Lambda(x + y) = \Lambda(x) \Lambda(y)\\ \Lambda(xy) = \Lambda(x) \otimes \Lambda(y)\\ \Lambda(\lambda^m(x)) = \wedge^m \Lambda(x) \end{gather*}
where \(\otimes\) is the map described in Remark 4.2.1 and \(\wedge^m\) is similar; it is the \(T\)-adically continuous map characterized by
\begin{equation*} \wedge^m \det(1 - T S, M)^{-1} = \det(1 - T (\wedge^m S), \wedge^m M)^{-1}. \end{equation*}
(The last condition implies that \(\lambda^n(1) = 0\) for all \(n \geq 2\text{.}\)) We define in the obvious way a morphism of \(\lambda\)-rings (as a morphism of underlying rings which commute with the maps \(\lambda^n\)), and hence the category \(\Ring_\lambda\) of \(\lambda\)-rings. (One can also express the conditions on the \(\lambda^n\) in terms of certain operations on symmetric functions.)
With this definition, we can show that there is a unique way to promote \(\WW\) to a functor from \(\Ring\) to \(\Ring_\lambda\) such that \(\Lambda([x]) = (1-xT)^{-1}\) for every ring \(A\) and every element \(x \in A\text{.}\) We omit details here.
The analogue of the adjunction property of the functor \(W\) is that \(\WW\) is a right adjoint of the forgetful functor from \(\Ring_\lambda\) to \(\Ring\text{.}\) This follows from the existence of the diagonal transformation \(\Delta\colon \WW \to \WW \circ \WW\text{.}\)

Remark 4.2.3.

In any \(\lambda\)-ring, we can define additional ring homomorphisms \(\psi^n\) for \(n =0,1,\dots\) called Adams operations. In the case of \(\WW(A)\text{,}\) these are characterized by \(T\)-adic continuity and the property
\begin{equation*} \psi^n \det(1 - T S, M)^{-1} = \det(1 - T S^n, M)^{-1}; \end{equation*}
this implies that
\begin{equation*} \psi^n([x]) = [x^n], \end{equation*}
from which we can deduce that in fact \(\psi^n = \phi_n\text{.}\)
In general, the maps \(\psi^p\) for \(p\) prime form a family of pairwise commuting Frobenius lifts; moreover, a \(\lambda\)-ring is a \(\delta\)-ring for every prime \(p\text{.}\) Conversely (and analogously to Lemma 2.1.3), for a \(\ZZ\)-torsion-free ring any family of pairwise commuting Frobenius lifts gives rise to a unique \(\lambda\)-ring structure (see [126]).

Example 4.2.4.

Equip the ring \(A = \ZZ [q]\) with the endomorphisms \(\psi^p\) sending \(q\) to \(q^p\) for each prime \(p\text{.}\) By the criterion of Remark 4.2.3, these occur as the Adams operations for a unique \(\lambda\)-ring structure on \(A\text{.}\) Similarly, the rings \(\ZZ \llbracket q-1 \rrbracket\) and \(\ZZ \llbracket q-1 \rrbracket [(q-1)^{-1}]\) admit \(\lambda\)-ring structures.
If one wishes to avoid the \((q-1)\)-adic completion, the ring
\begin{equation*} \ZZ[q, (q-1)^{-1}, (q^2-1)^{-1}, \dots] \end{equation*}
also admits a \(\lambda\)-ring structure.

Remark 4.2.5.

Some additional examples of \(\lambda\)-rings occurring “in nature” include:
  • the ring of symmetric polynomials over \(\ZZ\) (see Remark 4.2.7);
  • the representation ring of a finite group (see [86] for more on the relationship with the previous example);
  • the Grothendieck ring of the category of finite projective modules over a commutative ring;
  • the \(K\)-theory of a topological space (or a connective spectrum).

Remark 4.2.6.

In [31] one finds the concept of a plethory, which is a monoid in the category of birings; the functors \(W\) and \(\WW\) are represented by such objects. (The name comes from the operation of plethysm from representation theory or the corresponding operation in the theory of symmetric polynomials.) The systematic study of plethories, which builds upon ideas from the subject of universal algebra (see especially [120]), provides a natural context in which to talk about variant constructions. For example, for a prime \(p\) and a finite extension \(E\) of \(\QQ_p\text{,}\) one can define a functor of ramified Witt vectors valued in \(\frako_E\)-algebras. (See any of [41], section 1; [62], (18.6.13); or [36]. See also [34] for the corresponding version of \(p\)-derivations.)

Remark 4.2.7.

The category \(\Ring_\lambda\) admits all limits and colimits, and these are compatible with the forgetful functor to \(\Ring\) (either by direct calculation, or using the interpretation from [31]). Consequently, the forgetful functor from \(\lambda\)-rings to rings admits a left adjoint; as in Definition 2.4.5, the value of the left adjoint on the free polynomial ring \(\ZZ[S]\) is the free \(\lambda\)-ring on \(S\text{.}\) (The free \(\lambda\)-ring on a single element is the \(\lambda\)-ring of symmetric polynomials over \(\ZZ\text{.}\))

Remark 4.2.8.

Circling back to the original interpretation of a \(\delta\)-ring as a ring in which one can “differentiate with respect to \(p\)”, one can think of a \(\lambda\)-ring as a ring equipped with descent data from \(\Spec \ZZ\) to something “below”. That putative object shares some of the expected characteristics of a mythical object called the field with one element; another (nonmythical) object that does likewise is the sphere spectrum in algebraic topology.

Exercises 4.3 Exercises

1.

Complete the proof of Proposition 4.1.2 by proving that
\begin{equation*} \ZZ_{(p)}[y_1,y_2,\dots] = \ZZ_{(p)}[x_1,x_2,\dots]. \end{equation*}
Hint.
Using the equality
\begin{equation*} \sum_{j=0}^i p^j y_{mp^j}^{p^{i-j}} = w_{mp^i} = \sum_{j=0}^i p^j \sum_{d|m} d x_{dp^j}^{p^{i-j} m/d}, \end{equation*}
show by induction on \(i\) that
\begin{equation*} y_{mp^i} = \sum_{d|m} d x_{dp^i}^{m/d} + * \end{equation*}
where \(* \in \ZZ_{(p)}[x_{dp^j}\colon d|m, j < i]\text{.}\)

2.

Check that the map (4.1) defines a homomorphism between the additive group of \(\WW(A)\) and the multiplicative group \(1 + T A \llbracket T \rrbracket\text{.}\)

3.

Let \(X,Y\) be two schemes of finite type over a finite field \(\FF_q\text{.}\) Let \(Z(X/\FF_q,T)\) and \(Z(Y/\FF_q,T)\) be the zeta functions of \(X\) and \(Y\text{,}\) respectively.
  1. Prove that
    \begin{equation*} Z((X \times_{\FF_q} Y)/\FF_q, T) = Z(X/\FF_q, T) \otimes Z(Y/\FF_q, T) \end{equation*}
    where \(\otimes\) is the operation on \(1 + T \ZZ \llbracket T \rrbracket\) corresponding to multiplication in \(\WW(\ZZ)\) via the isomorphism (4.1).
  2. Prove that for any positive integer \(n\text{,}\)
    \begin{equation*} Z((X \times_{\FF_q} \FF_{q^n})/\FF_{q^n}) = \psi^n(Z(X,T)) \end{equation*}
    where \(\psi^n\) is the \(n\)-th Adams operation (Remark 4.2.3).
Hint.
Note that the second statement is a special case of the first. To prove the first, write \(Z(X,T)\) as the product of \((1 - T^{\deg(x/\FF_q)})^{-1}\) as \(x\) varies over closed points of \(X\text{,}\) and similarly for \(Y\text{;}\) then describe the closed points of \(X \times_{\FF_q} Y\) and appeal to (4.2).

4.

Prove the following analogue of Definition 3.1.7: for any ring \(A\text{,}\) the elements of \(\WW(A)\) in the kernel of the \(p\)-derivation for all primes \(p\) are precisely the constant lifts. (Combined with Remark 4.2.1, this explains the terminology elements of rank 1 in [25] for what we call \(\delta\)-constant elements of a \(\delta\)-ring.)
Hint.
First show that the elements in the kernels of all of the \(p\)-derivations form a set stable under the Frobenius maps to reduce to checking the vanishing of the Witt components for all nontrivial prime powers. Then use the projection maps \(\WW(A) \to W(A)\) to reduce to the \(p\)-typical case.

5.

Let \(A\) be a ring and let \(x \in \WW(A)\) be an element. Prove that for each positive integer \(n\text{,}\) there exists a faithfully flat ring map \(A \to B\) such that the image of \(x\) in \(\WW(B) \cong 1 + T B \llbracket T \rrbracket\) is congruent modulo \(T^{n+1}\) to a sum of constant elements. This is sometimes called the splitting principle, as it allows various algebraic properties of the big Witt vectors (or more generally of \(\lambda\)-rings) to be verified using arithmetic on constant elements. (This occurs frequently in the theory of characteristic classes of vector bundles, as in [119].)

6.

Let \(A\) be a ring. Show that under the identification \(\WW(A) \cong 1 + T A \llbracket T \rrbracket\text{,}\) the power series which represent rational functions of \(T\) form a subring of \(\WW(A)\text{.}\) (Compare Remark 4.2.1.)

7.

Let \(p_1,\dots,p_n\) be distinct primes and let \(S\) be the set of positive integers of the form \(p_1^{e_1} \cdots p_n^{e_n}\) for some nonnegative integers \(e_1,\dots,e_n\text{.}\) Let \(W_{p_i}\) denote the \(p_i\)-typical Witt vector functor. Show that there exists a natural isomorphism
\begin{equation*} W_{p_1} \circ \cdots \circ W_{p_n}\cong \WW_S \end{equation*}
of functors from \(\Ring\) to \(\Ring\text{.}\)

8.

For \(A \in \Ring\) and \(p\) prime, define the \(p\)-norm map \(N_p\colon \WW(A) \to \WW(A)\) by
\begin{equation*} N_p(x) = x - V_p(\delta_p(x)), \end{equation*}
where \(\delta_p\) is the \(p\)-derivation associated functorially to the Frobenius lift \(\psi^p\) (see Remark 4.2.3). Prove that the maps \(N_p\) are multiplicative, commute with each other, and satisfy
\begin{equation*} (\psi_p \circ N_p)(x) = x^p \qquad (x \in \WW(A)). \end{equation*}
As in Exercise 3.6.11, see [3] for some discussion of the role of this construction in algebraic topology.
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