## Tag `039Y`

Chapter 34: Descent > Section 34.3: Descent for modules

Remark 34.3.11. 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 34.3.9 can be split in the following steps

- If $R \to R'$ is faithfully flat, $R \to A$ any ring map, then descent data for $A/R$ are effective if descent data for $(R' \otimes_R A)/R'$ are effective.
- Let $A$ be an $R$-algebra. Descent data for $A/R$ correspond to cartesian $(A/R)_\bullet$-modules.
- If $R \to A$ has a section then $(A/R)_\bullet$ is homotopy equivalent to $R$, the constant cosimplicial $R$-algebra with value $R$.
- 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 34.3.8. Part (2) uses Lemma 34.3.2. Part (3) we have seen in the proof of Lemma 34.3.5 (it relies on Simplicial, Lemma 14.28.4). Moreover, part (4) is a triviality if you think about it right!

The code snippet corresponding to this tag is a part of the file `descent.tex` and is located in lines 719–753 (see updates for more information).

```
\begin{remark}
\label{remark-homotopy-equivalent-cosimplicial-algebras}
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 {\it cartesian module}.
In this setting, the proof of Proposition \ref{proposition-descent-module}
can be split in the following steps
\begin{enumerate}
\item If $R \to R'$ is faithfully flat, $R \to A$ any ring map,
then descent data for $A/R$ are effective if
descent data for $(R' \otimes_R A)/R'$ are effective.
\item Let $A$ be an $R$-algebra. Descent data for $A/R$ correspond
to cartesian $(A/R)_\bullet$-modules.
\item If $R \to A$ has a section then $(A/R)_\bullet$ is homotopy
equivalent to $R$, the constant cosimplicial
$R$-algebra with value $R$.
\item 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.
\end{enumerate}
For (1) see Lemma \ref{lemma-descent-descends}.
Part (2) uses Lemma \ref{lemma-descent-datum-cosimplicial}.
Part (3) we have seen in the proof of Lemma \ref{lemma-with-section-exact}
(it relies on Simplicial,
Lemma \ref{simplicial-lemma-push-outs-simplicial-object-w-section}).
Moreover, part (4) is a triviality if you think about it right!
\end{remark}
```

## Comments (0)

## Add a comment on tag `039Y`

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

All contributions are licensed under the GNU Free Documentation License.

There are no comments yet for this tag.

There are also 2 comments on Section 34.3: Descent.