Lemma 60.8.3 (07IA)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 60: Crystalline Cohomology
Section 60.8: The big crystalline site
Lemma 60.8.3 (cite)

numberstags

$\xymatrix{ (U_3, T_3, \delta _3) \ar[d] \ar[r] & (U_2, T_2, \delta _2) \ar[d] \\ (U_1, T_1, \delta _1) \ar[r] & (U, T, \delta ) }$

be a fibre square in the category of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma )$ . If $T_2 \to T$ is flat and $U_2 = T_2 \times _ T U$ , then $T_3 = T_1 \times _ T T_2$ (as schemes).

Proof. This is true because a divided power structure extends uniquely along a flat ring map. See Divided Power Algebra, Lemma 23.4.2. $\square$

Comments (2)

Comment #2313 by Daxin Xu on December 05, 2016 at 17:10

Should we add the condition $U_2\simeq T_2\times_T U$ ? (compare to Berthelot P, Breen L, Messing W. Théorie de Dieudonné cristalline. II, lemma 1.1.2)

Comment #2389 by Johan on February 17, 2017 at 00:17

Yes indeed, good catch! Fixed here.

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