The Stacks project

Remark 15.84.10. In this remark we define a category of glueing data. Let $R \to S$ be a ring map. Let $f_1, \ldots , f_ t \in R$ and $I = (f_1, \ldots , f_ t)$. Consider the category $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ as the category whose

  1. objects are systems $(M', M_ i, \alpha _ i, \alpha _{ij})$, where $M'$ is an $S$-module, $M_ i$ is an $R_{f_ i}$-module, $\alpha _ i : (M')_{f_ i} \to M_ i \otimes _ R S$ is an isomorphism, and $\alpha _{ij} : (M_ i)_{f_ j} \to (M_ j)_{f_ i}$ are isomorphisms such that

    1. $\alpha _{ij} \circ \alpha _ i = \alpha _ j$ as maps $(M')_{f_ if_ j} \to (M_ j)_{f_ i}$, and

    2. $\alpha _{jk} \circ \alpha _{ij} = \alpha _{ik}$ as maps $(M_ i)_{f_ jf_ k} \to (M_ k)_{f_ if_ j}$ (cocycle condition).

  2. morphisms $(M', M_ i, \alpha _ i, \alpha _{ij}) \to (N', N_ i, \beta _ i, \beta _{ij})$ are given by maps $\varphi ' : M' \to N'$ and $\varphi _ i : M_ i \to N_ i$ compatible with the given maps $\alpha _ i, \beta _ i, \alpha _{ij}, \beta _{ij}$.

There is a canonical functor

\[ \text{Can} : \text{Mod}_ R \longrightarrow \text{Glue}(R \to S, f_1, \ldots , f_ t), \quad M \longmapsto (M \otimes _ R S, M_{f_ i}, \text{can}_ i, \text{can}_{ij}) \]

where $\text{can}_ i : (M \otimes _ R S)_{f_ i} \to M_{f_ i} \otimes _ R S$ and $\text{can}_{ij} : (M_{f_ i})_{f_ j} \to (M_{f_ j})_{f_ i}$ are the canonical isomorphisms. For any object $\mathbf{M} = (M', M_ i, \alpha _ i, \alpha _{ij})$ of the category $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ we define

\[ H^0(\mathbf{M}) = \{ (m', m_ i) \mid \alpha _ i(m') = m_ i \otimes 1, \alpha _{ij}(m_ i) = m_ j\} \]

in other words defined by the exact sequence

\[ 0 \to H^0(\mathbf{M}) \to M' \times \prod M_ i \to \prod M'_{f_ i} \times \prod (M_ i)_{f_ j} \]

similar to (15.84.8.1). We think of $H^0(\mathbf{M})$ as an $R$-module. Thus we also get a functor

\[ H^0 : \text{Glue}(R \to S, f_1, \ldots , f_ t) \longrightarrow \text{Mod}_ R \]

Our next goal is to show that the functors $\text{Can}$ and $H^0$ are sometimes quasi-inverse to each other.


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