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Tag 023W

Chapter 34: Descent > Section 34.31: Descent data for schemes over schemes

Definition 34.31.3. Let $S$ be a scheme. Let $\{X_i \to S\}_{i \in I}$ be a family of morphisms with target $S$.

  1. A descent datum $(V_i, \varphi_{ij})$ relative to the family $\{X_i \to S\}$ is given by a scheme $V_i$ over $X_i$ for each $i \in I$, an isomorphism $\varphi_{ij} : V_i \times_S X_j \to X_i \times_S V_j$ of schemes over $X_i \times_S X_j$ for each pair $(i, j) \in I^2$ such that for every triple of indices $(i, j, k) \in I^3$ the diagram $$ \xymatrix{ V_i \times_S X_j \times_S X_k \ar[rd]^{\text{pr}_{01}^*\varphi_{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi_{ik}} & & X_i \times_S X_j \times_S V_k\\ & X_i \times_S V_j \times_S X_k \ar[ru]^{\text{pr}_{12}^*\varphi_{jk}} } $$ of schemes over $X_i \times_S X_j \times_S X_k$ commutes (with obvious notation).
  2. A morphism $\psi : (V_i, \varphi_{ij}) \to (V'_i, \varphi'_{ij})$ of descent data is given by a family $\psi = (\psi_i)_{i \in I}$ of morphisms of $X_i$-schemes $\psi_i : V_i \to V'_i$ such that all the diagrams $$ \xymatrix{ V_i \times_S X_j \ar[r]_{\varphi_{ij}} \ar[d]_{\psi_i \times \text{id}} & X_i \times_S V_j \ar[d]^{\text{id} \times \psi_j} \\ V'_i \times_S X_j \ar[r]^{\varphi'_{ij}} & X_i \times_S V'_j } $$ commute.

    The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 7196–7236 (see updates for more information).

    \begin{definition}
    \label{definition-descent-datum-for-family-of-morphisms}
    Let $S$ be a scheme.
    Let $\{X_i \to S\}_{i \in I}$ be a family of morphisms with target $S$.
    \begin{enumerate}
    \item A {\it descent datum $(V_i, \varphi_{ij})$ relative to the
    family $\{X_i \to S\}$} is given by a scheme $V_i$ over $X_i$
    for each $i \in I$, an isomorphism
    $\varphi_{ij} : V_i \times_S X_j \to X_i \times_S V_j$
    of schemes over $X_i \times_S X_j$ for each pair $(i, j) \in I^2$
    such that for every triple of indices $(i, j, k) \in I^3$
    the diagram
    $$
    \xymatrix{
    V_i \times_S X_j \times_S X_k
    \ar[rd]^{\text{pr}_{01}^*\varphi_{ij}}
    \ar[rr]_{\text{pr}_{02}^*\varphi_{ik}} &
    &
    X_i \times_S X_j \times_S V_k\\
    &
    X_i \times_S V_j \times_S X_k
    \ar[ru]^{\text{pr}_{12}^*\varphi_{jk}}
    }
    $$
    of schemes over $X_i \times_S X_j \times_S X_k$ commutes
    (with obvious notation).
    \item A {\it morphism
    $\psi : (V_i, \varphi_{ij}) \to (V'_i, \varphi'_{ij})$
    of descent data} is given by a family
    $\psi = (\psi_i)_{i \in I}$ of morphisms of
    $X_i$-schemes $\psi_i : V_i \to V'_i$ such that all the diagrams
    $$
    \xymatrix{
    V_i \times_S X_j \ar[r]_{\varphi_{ij}} \ar[d]_{\psi_i \times \text{id}} &
    X_i \times_S V_j \ar[d]^{\text{id} \times \psi_j} \\
    V'_i \times_S X_j \ar[r]^{\varphi'_{ij}} & X_i \times_S V'_j
    }
    $$
    commute.
    \end{enumerate}
    \end{definition}

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