The Stacks project

Remark 15.89.10. 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 ( 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.

Comments (0)

There are also:

  • 2 comment(s) on Section 15.89: Formal glueing of module categories

Post a comment

Your email address will not be published.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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05EL. Beware of the difference between the letter 'O' and the digit '0'.