Loading [MathJax]/extensions/tex2jax.js

The Stacks project

Situation 21.30.1. With $\mathcal{C}$, $\tau $, and $\tau '$ as in Section 21.27. Assume we are given a subset $\mathcal{P} \subset \text{Arrows}(\mathcal{C})$ and for every object $X$ of $\mathcal{C}$ we are given a weak Serre subcategory $\mathcal{A}'_ X \subset \textit{Ab}(\mathcal{C}_{\tau '}/X)$. We make the following assumption:

  1. given $f : X \to Y$ in $\mathcal{P}$ and $Y' \to Y$ general, then $X \times _ Y Y'$ exists and $X \times _ Y Y' \to Y'$ is in $\mathcal{P}$,

  2. $f_{\tau '}^{-1}$ sends $\mathcal{A}'_ Y$ into $\mathcal{A}'_ X$ for any morphism $f : X \to Y$ of $\mathcal{C}$,

  3. given $X$ in $\mathcal{C}$ and $\mathcal{F}'$ in $\mathcal{A}'_ X$, then $\mathcal{F}'$ satisfies the sheaf condition for $\tau $-coverings, i.e., $\mathcal{F}' = \epsilon _{X, *}\epsilon _ X^{-1}\mathcal{F}'$,

  4. if $f : X \to Y$ in $\mathcal{P}$ and $\mathcal{F}' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}'_ X)$, then $R^ if_{\tau ', *}\mathcal{F}' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}'_ Y)$ for $i \geq 0$.

  5. if $\{ U_ i \to U\} _{i \in I}$ is a $\tau $-covering, then there exist

    1. a $\tau '$-covering $\{ V_ j \to U\} _{j \in J}$,

    2. a $\tau $-covering $\{ f_ j : W_ j \to V_ j\} $ consisting of a single $f_ j \in \mathcal{P}$, and

    3. a $\tau '$-covering $\{ W_{jk} \to W_ j\} _{k \in K_ j}$

    such that $\{ W_{jk} \to U\} _{j \in J, k \in K_ j}$ is a refinement of $\{ U_ i \to U\} _{i \in I}$.


Comments (0)


Your email address will not be published.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.