[100] for the basic setup of \(q\)-de Rham cohomology in the \(\lambda\)-ring context. Beyond that, we are into terra incognita.

We conclude with some wild speculation about a potential link back to \(\lambda\)-rings (Section 4), particularly with regard to \(q\)-de Rham cohomology in the context of \(\lambda\)-rings.

Notational warning: in this lecture, we use the notation \(R\{J\}\) to denote a free object in the category of \(\lambda\)-rings, not \(\delta\)-rings.

Subsection29.1Divided power envelopes of \(\lambda\)-rings

We revisit the discussion of divided power envelopes of \(\delta\)-rings from Section 14, starting by formulating a variant of Lemma 14.3.2, in which we start with a larger ideal but do not require the base ring to be \(p\)-local.

Lemma29.1.1.

Let \(R_0 \in \Ring\) be \(\ZZ\)-torsion-free. Let \(R\) be the free \(\delta\)-ring over \(R_0\) on a single generator \(x\text{,}\) and let \(J\) be the ideal of \(R\) generated by \(\delta^m(x)\) for all \(m \geq 0\text{.}\) Then the map from \(R\) to the divided power envelope of \((R, J)\) promotes to a morphism of \(\delta\)-rings.

Let \(D\) be the divided power envelope; it is the smallest subring of \(R \otimes_{\ZZ} \QQ\) containing \(R\) and \(\gamma_n(\delta^m(x))\) for all \(m \geq 0\text{,}\)\(n \geq 1\text{.}\) The maximal ideal on which \(D\) admits divided powers includes \(\delta^m(x)\) for all \(m \geq 0\) by construction, and hence also \(\phi(\delta^m(x)) = \delta^m(x)^p + p \delta^{m+1}(x)\) for all \(m \geq 0\text{;}\) consequently, for all \(m \geq 0\text{,}\)\(n \geq 1\text{,}\)

\begin{equation*}
\phi(\gamma_n(\delta^m(x))) = \gamma_n(\phi(\delta^m(x))) \in D.
\end{equation*}

Hence \(\phi\) induces an endomorphism of \(D\text{.}\)

We next check that \(\phi\) induces a Frobenius lift on \(D\text{;}\) this amounts to checking that for all \(m \geq 0\text{,}\)\(n \geq 1\text{,}\)

We will see that in fact both sides are divisible by \(p\text{.}\) For \(\phi(\gamma_n(\delta^m(x))) = \gamma_n(\phi(\delta^m(x)))\text{,}\) this holds by writing

\begin{equation*}
\phi(\delta^m(x))) = p(\delta^m(x)^p/p) + p \delta^{m+1}(x) \in pD
\end{equation*}

For \(\gamma_n(\delta^m(x))^p\text{,}\) this holds by writing \(\gamma_n(\delta^m(x))^p = p! \gamma_p(\gamma_n(\delta^m(x)))\) and applying (14.4).

Since \(D\) is \(p\)-torsion-free, by Lemma 2.1.3 we obtain a \(\delta\)-structure compatible with \(R\text{,}\) as desired.

This statement and its proof transpose naturally to the \(\lambda\)-ring setting.

Lemma29.1.2.

Let \(R_0 \in \Ring\) be \(\ZZ\)-torsion-free. Put \(R = R\{x\}\) (the free \(\lambda\)-ring over \(R\) on a single generator \(x\)), and let \(J\) be the ideal of \(R\) generated by \(\lambda^m(x)\) for all \(m \geq 1\text{.}\) Then the map from \(R\) to the divided power envelope of \((R, J)\) promotes to a morphism of \(\lambda\)-rings.

Let \(D\) be the divided power envelope; it is the smallest subring of \(R \otimes_{\ZZ} \QQ\) containing \(R\) and \(\gamma_n(\lambda^m(x))\) for all \(m,n \geq 1\text{.}\) The maximal ideal on which \(D\) admits divided powers includes \(\delta^m(x)\) for all \(m \geq 0\) by construction, and hence also \(\psi^p(\delta^m(x)) = \delta^m(x)^p + p \delta^{m+1}(x)\) for each prime \(p\) and all \(m \geq 0\text{;}\) consequently, for each prime \(p\) and all \(m,n \geq 1\text{,}\)

\begin{equation*}
\psi^p(\gamma_n(\delta^m(x))) = \gamma_n(\psi^p(\delta^m(x))) \in D.
\end{equation*}

We thus obtain a commuting family of endomorphisms \(\psi^j\) for all \(j \geq 1\) satisfying \(\psi^{j_1j_2} = \psi^{j_1} \circ \psi^{j_2}\text{.}\)

As in the proof of Lemma 29.1.1, we verify that for each prime \(p\text{,}\)\(\psi^p\) induces a \(p\)-Frobenius lift on \(D\text{.}\) Since \(D\) is \(\ZZ\)-torsion-free, by Wilkerson's criterion (Remark 4.2.3) we obtain a \(\lambda\)-structure compatible with \(R\text{,}\) as desired.

This in turn leads to a \(\lambda\)-analogue of Corollary 14.3.3.

Corollary29.1.3.

In Lemma 29.1.2, the divided power envelope is generated as a \(\lambda\)-ring over \(R_0\) by the elements \(\psi^p(x)/p\) for \(p\) prime.

Let \(D\) be the divided power envelope and let \(D'\) be the subring of \(D \otimes_\ZZ \QQ = R \otimes_\ZZ \QQ\) which is the \(\lambda\)-subring over \(R_0\) generated by the elements \(\psi^p(x)/p\) for \(p\) prime. By Lemma 29.1.2, we have \(D' \subseteq D\text{;}\) it thus remains to check the converse.

We check that \(\gamma_n(\delta^m(x)) \in D'\) for all \(m \geq 0\text{,}\)\(n \geq 1\) by induction on \(n\text{,}\) with trivial base case \(n=1\text{.}\) The induction step is also trivial if \(n\) is not a prime power, as in this case \(1/n!\) belongs to the fractional ideal of \(\ZZ\) generated by \(1/i!\) for \(i \in \{1,\dots,n-1\}\text{.}\) If on the other hand \(n = p^\ell\text{,}\) then the difference between the fractional ideal of \(\ZZ\) generated by \(1/n!\) and the one generated by \(1/i!\) for \(i \in \{1,\dots,n-1\}\) is a factor of \(n\text{;}\) consequently, we may reduce to the case where \(R_0\) is \(p\)-local, in which case Corollary 14.3.3 implies the claim.

Remark29.1.4.

At this point, we can formulate a striking but not immediately meaningful analogue of Lemma 14.4.8 as follows. Let \(R_0 \in \Ring\) be arbitrary and put \(R = R_0[x_1,\dots,x_r]\text{,}\) viewed as a \(\lambda\)-ring with \(x_1,\dots,x_r\) constant. For \(n \geq 0\text{,}\) put

(remembering that now we mean the free \(\lambda\)-ring) and let \(J^n\) be the kernel of the morphism \(P^n \to R\) in \(\Ring_\lambda\) taking \(x_{ij}\) to \(x_i\) and \(\lambda^m(x_{ij})\) to \(0\) for all \(m \gt 1\text{.}\) Now consider the double complex displayed in Figure 29.1.5.

In the diagram, we may argue as in Lemma 14.4.1 to see that each row except the first is homotopic to zero. We may then apply the Poincaré lemma (Proposition 14.2.6) to deduce that each column is quasi-isomorphic to \(\Omega^\bullet_{R/R_0}\text{,}\) and in particular each of the \(n+1\) morphisms between the \(n\)-th column and the \((n+1)\)-st column induces the same isomorphism on cohomology groups. By Corollary 13.3.8, we deduce that the top row of Figure 29.1.5 is quasi-isomorphic to \(\Omega^\bullet_{R/R_0}\text{.}\)

Moreover, by Corollary 29.1.3, we may identify \(D_{J^n}(P^n)\) with \(P^n\{\psi^p(J^n)/p\}\) (running over all primes \(p\)). Here is where we get stuck: the latter object does not have any evident site-theoretic interpretation. However, in the \(q\)-analogue of this setup we will be able to provide such an interpretation.

Subsection29.2\(q\)-divided powers for \(\lambda\)-rings

Definition29.2.1.

In the following discussion, we view \(A = \ZZ \llbracket q-1 \rrbracket\) as a \(\lambda\)-ring (Definition 4.2.2) with \(q\) constant; that is, \(\lambda^i(q) = 0\) for all \(i \gt 1\) (and so \(\psi^i(q) = q^i\) for all \(i \gt 0\)).

For \(D\) an \(A\)-torsion-free \(\lambda\)-ring over \(A\text{,}\) for \(p\) prime and \(x \in D\) with \(\psi^p(x) \in [p]_qD\text{,}\) write

\begin{equation*}
\gamma_{p,q}(x) = \frac{\psi^p(x)}{[p]_q} - \delta_p(x) \in D.
\end{equation*}

Lemma29.2.3.

Let \(D\) be an \(A\)-torsion-free \(\lambda\)-ring over \(A\text{.}\) Then for each prime \(p\text{,}\) the ideal \((\psi^p)^{-1}([p]_q D)\) is stable under \(\gamma_{p,q}\text{.}\)

is itself an ideal of \(D\text{;}\) hence to check that \(J = I\text{,}\) it suffices to check that \(J\) contains a generating set of \(I\text{.}\)

Definition29.2.5.

A \(\lambda\)-pair over \(A\) is a pair \((D,I)\) in which \(D\) is a \(\lambda\)-ring over \(A\) and \(I\) is an ideal of \(D\text{.}\) A morphism \((D,I) \to (E,J)\) of \(\lambda\)-pairs is a morphism \(D \to E\) of \(\lambda\)-rings carrying \(I\) into \(J\text{.}\)

Definition29.2.6.

A global \(q\)-pd pair is a \(\lambda\)-pair \((D,I)\) in which \(D\) is derived \((q-1)\)-complete, \(q-1 \in I\text{,}\) and for each prime \(p\text{,}\)\(\psi^p(I) \subseteq [p]_q D\) (so that \(\gamma_{p,q}\) is defined on \(I\)) and \(\gamma_{p,q}(I) \subseteq I\text{.}\)

Example29.2.7.

The \(\lambda\)-pair \((A, (q-1))\) is the initial object in the category of global \(q\)-pd pairs.

Proposition29.2.8.

Let \(P\) be a \(\lambda\)-ring over \(A\) equipped with a surjection \(P \to R = \ZZ[x_1,\dots,x_r]\) with kernel \(J = (q-1,y_1,y_2,\dots)\) where each initial segment \(y_1,\dots,y_s\) is a regular sequence in \(P/(q-1)\text{.}\) Define the \(\lambda\)-ring

\begin{equation*}
D = P\{\psi_p(y_i)/[p]_q\colon i=1,2,\dots; p \mbox{ prime}\}^\wedge_{(q-1)}.
\end{equation*}

Let \(I\) be the kernel of \(D \to D/(q-1) \to R\text{.}\)

The map \(P \to D\) induces an isomorphism \(P/J \cong D/I\text{.}\)

The ring \(D/(q-1)\) identifies with the divided power envelope of the map \(P/(q-1) \to R\text{.}\)

The map \((P, J) \to (D,I)\) of \(\lambda\)-pairs is universal for the target being a global \(q\)-pd pair.

We now globalize the earlier discussion of \(q\)-crystalline cohomology.

Definition29.3.1.

Put \(R = \ZZ[x_1,\dots,x_r]\llbracket q-1 \rrbracket\text{,}\) viewed as a \(\lambda\)-ring over \(A\) with \(q,x_1,\dots,x_r\) all constant. For \(n \geq 0\text{,}\) put

and let \(J^n\) be the kernel of the morphism \(P^n \to R\) in \(\Ring_\lambda\) taking \(x_{ij}\) to \(x_i\) and \(\lambda^m(x_{ij})\) to \(0\) for all \(m \gt 1\text{.}\) Let \(D_{J^n,q}(P^n)\) be the ring \(D\) obtained by applying Proposition 29.2.8 to the pair \((P^n, J^n)\text{.}\)

Lemma29.3.2.

In Figure 29.3.3, the first row of the diagram is quasi-isomorphic to the left column, which is in turn quasi-isomorphic to \(q\widehat{\Omega}^\bullet_{R/\ZZ,\square}\text{.}\) (Here all derived completions are with respect to \(q-1\text{.}\))

In the following argument, all applications of derived Nakayama (Remark 6.6.6) will be modulo \(q-1\text{.}\)

In the diagram, by derived Nakayama plus Lemma 14.4.1 (or a direct argument), each row except the first is homotopic to zero. By derived Nakayama plus the Poincaré lemma (Proposition 14.2.6, each column is quasi-isomorphic to \(q\Omega^\bullet_{R/\ZZ,\square}\text{,}\) and in particular each of the \(n+1\) morphisms between the \(n\)-th column and the \((n+1)\)-st column induces the same isomorphism on cohomology groups. By Corollary 13.3.8, we deduce that the top row of Figure 29.3.3 is quasi-isomorphic to \(q\widehat{\Omega}^\bullet_{R/\ZZ,\square}\text{.}\)

Definition29.3.4.

Put \(R = \ZZ[x_1,\dots,x_r]\text{.}\) Define the global \(q\)-crystalline site to be the opposite category to the category of global \(q\)-pd-pairs \((P, J)\) equipped with isomorphisms \(P/J \cong R\text{,}\) equipped with the indiscrete Grothendieck topology. By Proposition 29.2.8, the ring \(D_{J^0,q}(P^0)\) from Definition 29.3.1 yields a weakly final object in this category, so by Lemma 11.1.7 we can compute the cohomology of this site using the associated Čech-Alexander complex. This is precisely the top row of Figure 29.3.3, so Lemma 29.3.2 gives us a quasi-isomorphism with \(q\widehat{\Omega}^\bullet_{R/\ZZ,\square}\text{.}\)

Subsection29.4Okay, now what?

Remark29.4.1.

One would like to pursue the analogy with the \(p\)-local situation further, e.g., by comparing étale localization with the left Kan extension. One dangerous point is that the Hodge-Tate isomorphism is going to be more subtle due to the lack of a conjugate filtration when not working in characteristic \(p\text{;}\) compare Remark 17.2.8.

Nonetheless, one could try to define the analogue of a prism in this context, at least relative to the pair \((A, (q-1))\text{,}\) and see where this leads. We shall see...

Exercises29.5Exercises

1.

View \(\ZZ[x,y,q]\) as a \(\Lambda\)-ring in such a way that \(x,y,q\) are all constant. Show that in the \(\Lambda\)-localization \(\ZZ[x,y,q]\{(q-1)^{-1}\}\text{,}\) we have