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Sheaves of Modules
Source of package stacks.sheaf_cohomology · The Stacks Project, Chapter «Sheaves of Modules» (tag 01AC)
- work
- urn:stacks:clir:sheaf-cohomology#STACKS_MODULES
- number
- The Stacks Project, Chapter «Sheaves of Modules» (tag 01AC)
- kind
- urn:stacks:clir:sheaf-cohomology#treatise
modules.tex at commit a04446e (master, 2026-07-28)
- edition
- urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER
- language
- en
- officiality
- official
- materialisation
- PINNED_OFFICIAL_BYTES
- life cycle
- adopted: 2026-07-28
Publications
| URI | Type | documentHash | Retrieved |
|---|---|---|---|
| https://raw.githubusercontent.com/stacks/stacks-project/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex | application/x-tex; charset=utf-8 | sha256:7bd3e9e096717ef6fd458492d8ad91fc9ffe428cfd7dd80e386fae377bb7cb0d | 2026-09-06T00:00:00Z |
Fragments — 2
tag/01AD · section · referenced by: 11
en · official · sha256:3237316f9dec6bd87194ad43833111dbdf86cade5633082fbc0e7d50b3e2e7cf
\section{Introduction}
\label{section-introduction}
\noindent
In this chapter we work out basic notions of sheaves of modules.
This in particular includes the case of abelian sheaves, since
these may be viewed as sheaves of $\underline{\mathbf{Z}}$-modules.
Basic references are \cite{FAC}, \cite{EGA} and \cite{SGA4}.
\medskip\noindent
We work out what happens for sheaves of modules on ringed topoi
in another chapter (see
Modules on Sites, Section \ref{sites-modules-section-introduction}),
although there we will mostly just duplicate the discussion
from this chapter.Referencing norms: AbAdditive, AbCoimageImage, AbCokernels, AbKernels, AbXAdditive, AbXCoimageImage, AbXCokernels, AbXKernels, FlasqueIsAcyclic, GodementResolutionExists, InjectiveSheafIsFlasque
tag/01AG · lemma · referenced by: 8
en · official · sha256:d40ca7777a52e10948b9c7e8da68432632fc0b8f4bb33c427370acaf48a4b19f
\begin{lemma}
\label{lemma-abelian}
Let $(X, \mathcal{O}_X)$ be a ringed space. The category
$\textit{Mod}(\mathcal{O}_X)$ is an abelian category. Moreover
a complex
$$
\mathcal{F} \to \mathcal{G} \to \mathcal{H}
$$
is exact at $\mathcal{G}$ if and only if for all $x \in X$ the
complex
$$
\mathcal{F}_x \to \mathcal{G}_x \to \mathcal{H}_x
$$
is exact at $\mathcal{G}_x$.
\end{lemma}Referencing norms: AbAdditive, AbCoimageImage, AbCokernels, AbKernels, AbXAdditive, AbXCoimageImage, AbXCokernels, AbXKernels
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