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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

  1. 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.
  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 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}

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