Lemma 60.17.5. In Situation 60.5.1. Let $A \to P' \to C$ be ring maps with $A \to P'$ smooth and $P' \to C$ surjective with kernel $J'$. Let $D'$ be the $p$-adic completion of $D_{P', \gamma }(J')$. There are homomorphisms of divided power $A$-algebras

$a : D \longrightarrow D',\quad b : D' \longrightarrow D$

compatible with the maps $D \to C$ and $D' \to C$ such that $a \circ b = \text{id}_{D'}$. These maps induce an equivalence of categories of pairs $(M, \nabla )$ satisfying (1), (2), (3), and (4) over $D$ and pairs $(M', \nabla ')$ satisfying (1), (2), (3), and (4)1 over $D'$. In particular, the equivalence of categories of Proposition 60.17.4 also holds for the corresponding functor towards pairs over $D'$.

Proof. First, suppose that $P' = A[y_1, \ldots , y_ m]$ is a polynomial algebra over $A$. In this case, we can find ring maps $P \to P'$ and $P' \to P$ compatible with the maps to $C$ which induce maps $a : D \to D'$ and $b : D' \to D$ as in the lemma. Using completed base change along $a$ and $b$ we obtain functors between the categories of modules with connection satisfying properties (1), (2), (3), and (4) simply because these these categories are equivalent to the category of quasi-coherent crystals by Proposition 60.17.4 (and this equivalence is compatible with the base change operation as shown in the proof of the proposition).

Proof for general smooth $P'$. By the first paragraph of the proof we may assume $P = A[y_1, \ldots , y_ m]$ which gives us a surjection $P \to P'$ compatible with the map to $C$. Hence we obtain a surjective map $a : D \to D'$ by functoriality of divided power envelopes and completion. Pick $e$ large enough so that $D_ e$ is a divided power thickening of $C$ over $A$. Then $D_ e \to C$ is a surjection whose kernel is locally nilpotent, see Divided Power Algebra, Lemma 23.2.6. Setting $D'_ e = D'/p^ eD'$ we see that the kernel of $D_ e \to D'_ e$ is locally nilpotent. Hence by Algebra, Lemma 10.138.17 we can find a lift $\beta _ e : P' \to D_ e$ of the map $P' \to D'_ e$. Note that $D_{e + i + 1} \to D_{e + i} \times _{D'_{e + i}} D'_{e + i + 1}$ is surjective with square zero kernel for any $i \geq 0$ because $p^{e + i}D \to p^{e + i}D'$ is surjective. Applying the usual lifting property (Algebra, Proposition 10.138.13) successively to the diagrams

$\xymatrix{ P' \ar[r] & D_{e + i} \times _{D'_{e + i}} D'_{e + i + 1} \\ A \ar[u] \ar[r] & D_{e + i + 1} \ar[u] }$

we see that we can find an $A$-algebra map $\beta : P' \to D$ whose composition with $a$ is the given map $P' \to D'$. By the universal property of the divided power envelope we obtain a map $D_{P', \gamma }(J') \to D$. As $D$ is $p$-adically complete we obtain $b : D' \to D$ such that $a \circ b = \text{id}_{D'}$.

Consider the base change functors

$F : (M, \nabla ) \longmapsto (M \otimes ^\wedge _{D, a} D', \nabla ') \quad \text{and}\quad G : (M', \nabla ') \longmapsto (M' \otimes ^\wedge _{D', b} D, \nabla )$

on modules with connections satisfying (1), (2), and (3). See Remark 60.6.9. Since $a \circ b = \text{id}_{D'}$ we see that $F \circ G$ is the identity functor. Let us say that $(M', \nabla ')$ has property (4) if this is true for $G(M', \nabla ')$. A formal argument now shows that to finish the proof it suffices to show that $G(F(M, \nabla ))$ is isomorphic to $(M, \nabla )$ in the case that $(M, \nabla )$ satisfies all four conditions (1), (2), (3), and (4). For this we use the functorial isomorphism

$c_{\text{id}_ D, b \circ a} : M \otimes _{D, \text{id}_ D} D \longrightarrow M \otimes _{D, b \circ a} D$

of the proof of Proposition 60.17.4 (which requires the topological quasi-nilpotency of $\nabla$ which we have assumed). It remains to prove that this map is horizontal, i.e., compatible with connections, which we omit.

The last statement of the proof now follows. $\square$

 This condition is tricky to formulate for $(M', \nabla ')$ over $D'$. See proof.

Comment #5099 by Lei Zhang on

If $A\to P$ is only a smooth map of rings what does it mean by pairs $(M,\nabla)$ satisfying https://stacks.math.columbia.edu/tag/07JE ? In 07JE you use a fixed local coordinate system, can you make a definition of 07JE without the choice of a local coordinate? From 07JH we can conclude that the category of topological quasi-nilpotent connections do not depend on the choice of the local coordinate, but if the notion of a topologically quasi-nilpotent connection depends on the choice of the local coordinate a priori, then maybe it's better to fix the local coordinate before talking about them?

Comment #5308 by on

@#5099. Yes, OK, thanks. I have added the necessary definition in the proof. See here.

Comment #5814 by Manuel Hoff on

Small typo in the second sentence of the proof. "$P$-module" has to be replaced by "$P'$-module".

Comment #6329 by Lei Zhang on

The anonymous referee of our paper https://arxiv.org/abs/1812.05153 pointed out that the definition of quasi-nilpotency that was recently added is not correct, at least not in this form. This is what could go wrong: choose $A=W_k, P'=A[x,x^{-1}]$ and $C=k[x,x^{-1}]$, so that $D'=\hat{P}'$ is the completion. Consider the isomorphism $\Omega_{P'/A}\simeq P'$ mapping $dx$ to $x$ (and not 1). If $(D',d)$ is the trivial connection then $\theta_1' \colon D' \to \Omega_{D'} \to D'$ maps $x$ to $x$, so that $(\theta')^k(x)=x \notin pD'$ for all $k$. Probably this problem disappears if $\Omega_{P'/A} \to P'^n$ is a section of a map $P'^n \to \Omega_{P'/A}$ induced by generators $y_1,\dots,y_n\in P'$ as $A$-algebra. In any case, it seems a good idea to write some details in the proof to compare those notions of quasi-nilpotency, as it is not clear a priori that they coincide. More precisely, why does $a^*$ preserve quasi-nilpotency.?

Comment #6336 by on

OK, good point. I will look at this carefully the next time I go through all the comments

Comment #6433 by on

OK, I cheated and I defined the top q-nilp condition exactly so that the lemma is true. But to everybody: is there a short easy manner in which to define this notion when you just work with a smooth algebra? Thanks again and see changes here.

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