Let $X\to S$ be a morphism of schemes.
Proposition 2.11 of the book "Notes on crystalline cohomology" by Berthelot and Ogus states that a stratified $\mathcal{O}_X$-module $(E,\{\varepsilon_n\colon P^n_{X/S}\otimes E\to E\otimes P^n_{X/S}\}_n)$ gives rise to a crystal on the infinitesimal site $\mathrm{Inf}_{X/S}$ (and vice versa).
However, their (sketch of) proof only treats infinitesimal thickenings $U→T $ for $U\subset X$ which locally admit a retraction.
How can we define the value $E(T)$ for an infinitesimal thickening which does not admit a retraction even locally?
Is this proposition really true?
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3$\begingroup$ In 2.11 the morphism $X \to S$ is assumed smooth so all thickenings locally have retractions. $\endgroup$– JohanCommented Jul 27, 2021 at 15:30
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$\begingroup$ @Johan I overlooked it. Thank you so much! $\endgroup$– Jun KoizumiCommented Jul 27, 2021 at 15:53
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