The Stacks project

Remark 35.3.11. Let $R$ be a ring. Let $A_\bullet $ be a cosimplicial $R$-algebra. In this setting a descent datum corresponds to an cosimplicial $A_\bullet $-module $M_\bullet $ with the property that for every $n, m \geq 0$ and every $\varphi : [n] \to [m]$ the map $M(\varphi ) : M_ n \to M_ m$ induces an isomorphism

\[ M_ n \otimes _{A_ n, A(\varphi )} A_ m \longrightarrow M_ m. \]

Let us call such a cosimplicial module a cartesian module. In this setting, the proof of Proposition 35.3.9 can be split in the following steps

  1. If $R \to R'$ and $R \to A$ are faithfully flat, then descent data for $A/R$ are effective if descent data for $(R' \otimes _ R A)/R'$ are effective.

  2. Let $A$ be an $R$-algebra. Descent data for $A/R$ correspond to cartesian $(A/R)_\bullet $-modules.

  3. If $R \to A$ has a section then $(A/R)_\bullet $ is homotopy equivalent to $R$, the constant cosimplicial $R$-algebra with value $R$.

  4. If $A_\bullet \to B_\bullet $ is a homotopy equivalence of cosimplicial $R$-algebras then the functor $M_\bullet \mapsto M_\bullet \otimes _{A_\bullet } B_\bullet $ induces an equivalence of categories between cartesian $A_\bullet $-modules and cartesian $B_\bullet $-modules.

For (1) see Lemma 35.3.8. Part (2) uses Lemma 35.3.2. Part (3) we have seen in the proof of Lemma 35.3.5 (it relies on Simplicial, Lemma 14.28.5). Moreover, part (4) is a triviality if you think about it right!

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