Proposition 35.3.9 (023N) - The Stacks Project

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Effective descent for modules along faithfully flat ring maps.

Proposition 35.3.9. Let $R \to A$ be a faithfully flat ring map. Then

1. any descent datum on modules with respect to $R \to A$ is effective,

2. the functor $M \mapsto (A \otimes _ R M, can)$ from $R$-modules to the category of descent data is an equivalence, and

3. the inverse functor is given by $(N, \varphi ) \mapsto H^0(s(N_\bullet ))$.

Proof. We only prove (1) and omit the proofs of (2) and (3). As $R \to A$ is faithfully flat, there exists a faithfully flat base change $R \to R'$ such that $R' \to A' = R' \otimes _ R A$ has a section (namely take $R' = A$ as in the proof of Lemma 35.3.6). Hence, using Lemma 35.3.8 we may assume that $R \to A$ has a section, say $\sigma : A \to R$. Let $(N, \varphi )$ be a descent datum relative to $R \to A$. Set

\[ M = H^0(s(N_\bullet )) = \{ n \in N \mid 1 \otimes n = \varphi (n \otimes 1)\} \subset N \]

By Lemma 35.3.7 it suffices to show that $A \otimes _ R M \to N$ is an isomorphism.

Take an element $n \in N$. Write $\varphi (n \otimes 1) = \sum a_ i \otimes x_ i$ for certain $a_ i \in A$ and $x_ i \in N$. By Lemma 35.3.2 we have $n = \sum a_ i x_ i$ in $N$ (because $\sigma ^0_0 \circ \delta ^1_1 = \text{id}$ in any cosimplicial object). Next, write $\varphi (x_ i \otimes 1) = \sum a_{ij} \otimes y_ j$ for certain $a_{ij} \in A$ and $y_ j \in N$. The cocycle condition means that

\[ \sum a_ i \otimes a_{ij} \otimes y_ j = \sum a_ i \otimes 1 \otimes x_ i \]

in $A \otimes _ R A \otimes _ R N$. We conclude two things from this:

1. applying $\sigma $ to the first $A$ we get $\sum \sigma (a_ i) \varphi (x_ i \otimes 1) = \sum \sigma (a_ i) \otimes x_ i$,

2. applying $\sigma $ to the middle $A$ we get $\sum _ i a_ i \otimes \sum _ j \sigma (a_{ij}) y_ j = \sum a_ i \otimes x_ i$.

Part (1) shows that $\sum \sigma (a_ i) x_ i \in M$. Applying this to $x_ i$ we see that $\sum \sigma (a_{ij})y_ i \in M$ for all $i$. Multiplying out the equation in (2) we conclude that $\sum _ i a_ i (\sum _ j \sigma (a_{ij}) y_ j) = \sum a_ i x_ i = n$. Hence $A \otimes _ R M \to N$ is surjective. Finally, suppose that $m_ i \in M$ and $\sum a_ i m_ i = 0$. Then we see by applying $\varphi $ to $\sum a_ im_ i \otimes 1$ that $\sum a_ i \otimes m_ i = 0$. In other words $A \otimes _ R M \to N$ is injective and we win. $\square$

Comments (11)

Suggested slogan: One has effective descent for modules along faithfully flat ring maps.

Two typos in the proof: "we may assume that as a section" should be replaced by "we may assume that has a section", and "(because in any cosimplicial object)" should be replaced by "(because in any cosimplicial object)"

THanks, fixed here.

Could you elaborate on the "Hence by the first conclusion we see that..." part? why is in M?

The first conclusion was that is in . Apply this to to get in . OK?

Could you please tell me how to deduce that is in the image of ? I cannot see it from the fact that and .

Because is in by the first conclusion applied to . But I think we need to rewrite the proof as it is too succint.

OK, I tried to improve the expostion, see this commit.

I am sorry that I cannot get from the fact . I know that from the conclusion by (1). And as , I only can conclude that .

Oh I am sorry. "Apply to x_inx_i$ and use the same argument.

I think this means you agree now?

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