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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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