The Stacks project

Remark 60.24.9 (Base change isomorphism). The map (60.24.8.1) is an isomorphism provided all of the following conditions are satisfied:

  1. $p$ is nilpotent in $A'$,

  2. $\mathcal{F}'$ is a crystal in quasi-coherent $\mathcal{O}_{X'/S'}$-modules,

  3. $X' \to S'_0$ is a quasi-compact, quasi-separated morphism,

  4. $X = X' \times _{S'_0} S_0$,

  5. $\mathcal{F}'$ is a flat $\mathcal{O}_{X'/S'}$-module,

  6. $X' \to S'_0$ is a local complete intersection morphism (see More on Morphisms, Definition 37.62.2; this holds for example if $X' \to S'_0$ is syntomic or smooth),

  7. $X'$ and $S_0$ are Tor independent over $S'_0$ (see More on Algebra, Definition 15.61.1; this holds for example if either $S_0 \to S'_0$ or $X' \to S'_0$ is flat).

Hints: Condition (1) means that in the arguments below $p$-adic completion does nothing and can be ignored. Using condition (3) and Mayer Vietoris (see Remark 60.24.2) this reduces to the case where $X'$ is affine. In fact by condition (6), after shrinking further, we can assume that $X' = \mathop{\mathrm{Spec}}(C')$ and we are given a presentation $C' = A'/I'[x_1, \ldots , x_ n]/(\bar f'_1, \ldots , \bar f'_ c)$ where $\bar f'_1, \ldots , \bar f'_ c$ is a Koszul-regular sequence in $A'/I'$. (This means that smooth locally $\bar f'_1, \ldots , \bar f'_ c$ forms a regular sequence, see More on Algebra, Lemma 15.30.17.) We choose a lift of $\bar f'_ i$ to an element $f'_ i \in A'[x_1, \ldots , x_ n]$. By (4) we see that $X = \mathop{\mathrm{Spec}}(C)$ with $C = A/I[x_1, \ldots , x_ n]/(\bar f_1, \ldots , \bar f_ c)$ where $f_ i \in A[x_1, \ldots , x_ n]$ is the image of $f'_ i$. By property (7) we see that $\bar f_1, \ldots , \bar f_ c$ is a Koszul-regular sequence in $A/I[x_1, \ldots , x_ n]$. The divided power envelope of $I'A'[x_1, \ldots , x_ n] + (f'_1, \ldots , f'_ c)$ in $A'[x_1, \ldots , x_ n]$ relative to $\gamma '$ is

\[ D' = A'[x_1, \ldots , x_ n]\langle \xi _1, \ldots , \xi _ c \rangle /(\xi _ i - f'_ i) \]

see Lemma 60.2.4. Then you check that $\xi _1 - f'_1, \ldots , \xi _ n - f'_ n$ is a Koszul-regular sequence in the ring $A'[x_1, \ldots , x_ n]\langle \xi _1, \ldots , \xi _ c\rangle $. Similarly the divided power envelope of $IA[x_1, \ldots , x_ n] + (f_1, \ldots , f_ c)$ in $A[x_1, \ldots , x_ n]$ relative to $\gamma $ is

\[ D = A[x_1, \ldots , x_ n]\langle \xi _1, \ldots , \xi _ c\rangle /(\xi _ i - f_ i) \]

and $\xi _1 - f_1, \ldots , \xi _ n - f_ n$ is a Koszul-regular sequence in the ring $A[x_1, \ldots , x_ n]\langle \xi _1, \ldots , \xi _ c\rangle $. It follows that $D' \otimes _{A'}^\mathbf {L} A = D$. Condition (2) implies $\mathcal{F}'$ corresponds to a pair $(M', \nabla )$ consisting of a $D'$-module with connection, see Proposition 60.17.4. Then $M = M' \otimes _{D'} D$ corresponds to the pullback $\mathcal{F}$. By assumption (5) we see that $M'$ is a flat $D'$-module, hence

\[ M = M' \otimes _{D'} D = M' \otimes _{D'} D' \otimes _{A'}^\mathbf {L} A = M' \otimes _{A'}^\mathbf {L} A \]

Since the modules of differentials $\Omega _{D'}$ and $\Omega _ D$ (as defined in Section 60.17) are free $D'$-modules on the same generators we see that

\[ M \otimes _ D \Omega ^\bullet _ D = M' \otimes _{D'} \Omega ^\bullet _{D'} \otimes _{D'} D = M' \otimes _{D'} \Omega ^\bullet _{D'} \otimes _{A'}^\mathbf {L} A \]

which proves what we want by Proposition 60.21.3.

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