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\begin{document}

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\section*{coherent sheaf}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{over_a_ringed_topos}{Over a ringed topos}\dotfill \pageref*{over_a_ringed_topos} \linebreak
\noindent\hyperlink{over_a_structured_topos}{Over a structured $(\infty,1)$-topos}\dotfill \pageref*{over_a_structured_topos} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{historical_note_and_definition_variants}{Historical note and definition variants}\dotfill \pageref*{historical_note_and_definition_variants} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{coherent sheaf of modules} is a geometric globalization of the notion of [[coherent module]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{over_a_ringed_topos}{}\subsubsection*{{Over a ringed topos}}\label{over_a_ringed_topos}

Let $(X,\mathcal{O})$ be a [[ringed space]] or, more generally, a [[ringed site]].

A [[sheaf]] $\mathcal{E}$ on $X$ of $\mathcal{O}$-[[module]]s is

\begin{itemize}%
\item \textbf{finitely generated}, or of \textbf{finite type} , if every point $x \in X$ has an open neighbourhood such that there is a surjective morphism

\begin{displaymath}
\mathcal{O}^n|_U \to \mathcal{E}|_U
\end{displaymath}
from a [[free module]] to $\mathcal{E}|_{U}$, where $n$ is finite.


\item \textbf{coherent} if it is

\begin{enumerate}%
\item finitely generated


\item for every open $U$ in the base space (resp. every object $U$ in the base site), every finite $p \in \mathbb{N}$ and every morphism

\begin{displaymath}
\mathcal{O}^p|_U\to \mathcal{E}|_U
\end{displaymath}
of $\mathcal{O}|_U$-modules has a finitely generated [[kernel]].



\end{enumerate}

\item \textbf{finitely presented} if there is an exact sequence of the form

\begin{displaymath}
\mathcal{O}^p\to\mathcal{O}^n\to\mathcal{E}\to 0
\end{displaymath}
with $p$ and $n$ finite.

Every finitely presented $\mathcal{O}$-module is finitely generated.


\item \textbf{[[quasicoherent sheaf|quasi coherent]]} if it is \emph{locally} -- on a cover $\{U_i\}$ -- \emph{presentable}, i.e. for each $i$ there is an [[exact sequence]]s

\begin{displaymath}
\mathcal{O}^{I_i}|_{U_i} \to \mathcal{O}^{J_i}|_{U_i}
  \to \mathcal{E}|_{U_i}
  \to 0\,,
\end{displaymath}
where $I_i$ and $J_i$ may be infinite, i.e. $\mathcal{E}$ is locally the [[cokernel]] of [[free module]]s. For more see [[quasicoherent sheaf]].



\end{itemize}
\hypertarget{over_a_structured_topos}{}\subsubsection*{{Over a structured $(\infty,1)$-topos}}\label{over_a_structured_topos}

Over a [[spectral Deligne-Mumford stack]]:

([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|Lurie QCoh, def. 2.6.20]])

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

For a coherent sheaf $\mathcal{E}$ over a [[ringed space]], for every point $y$ in the base space $X$ there is a neighborhood $V$ such that the $\mathcal{O}_X(V)$-module $\mathcal{E}(V)$ of sections of $\mathcal{E}$ over $V$ is finitely presented. On a [[noetherian scheme]] the notions of finitely presented and coherent sheaves of $\mathcal{O}$-modules agree, but this is not true on a general scheme or general analytic space; sometimes even the structure sheaf $\mathcal{O}$ itself is a counterexample (not coherent while finitely presented).

The notion of coherent sheaf behaves well on the category of noetherian schemes. On a general topological space, by a basic result of Serre, if two of the sheaves of $\mathcal{O}$-modules in a [[short exact sequence]]

\begin{displaymath}
0\to \mathcal{E}\to\mathcal{E}'\to \mathcal{E}''\to 0
\end{displaymath}
are coherent then so is the third. All this holds even if $\mathcal{O}$ is a sheaf of noncommutative [[rings]]. For commutative $\mathcal{O}$, the inner hom $Hom_{\mathcal{O}}(\mathcal{E},\mathcal{F})$ in the category of sheaves of $\mathcal{O}$-modules is coherent if $\mathcal{E},\mathcal{F}$ are coherent.

A theorem of Serre says that the category of coherent sheaves over a projective variety of the form $Proj R$ where $R$ is a graded commutative Noetherian ring is equivalent to the localization of the category of finitely generated graded $R$-modules modulo its (``torsion'') subcategory of (finitely generated graded) $R$-modules of finite length.

\hypertarget{historical_note_and_definition_variants}{}\subsection*{{Historical note and definition variants}}\label{historical_note_and_definition_variants}

First works on coherent sheaves in complex analytic geometry. Serre adapted their work to algebraic framework in his famous article [[FAC]]. Hartshorne's definitions are changed/adapted to the special setup of Noetherian schemes with the excuse that the coherence does not behave that well otherwise; thus they differ from the definitions in [[EGA]] and [[FAC]].

[[A. Vistoli]] commented at MathOverflow \href{http://mathoverflow.net/questions/68150}{here} that for some categorical purposes

\begin{quote}%
one should interpret ``coherent'' as meaning ``quasi-coherent of finite presentation''. The notion of coherent sheaf, as defined in EGA, is not functorial, that is, pullbacks of coherent sheaves are not necessarily coherent. Hartshorne's book defines ``coherent'' as ``quasi-coherent and finitely generated'', but this is a useless notion when working with non-noetherian schemes.


\end{quote}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[degree of a coherent sheaf]], [[rank of a coherent sheaf]] [[slope of a coherent sheaf]],


\item [[stable coherent sheaf]]


\item [[quasicoherent sheaf]]


\item [[triangulated categories of sheaves]]


\item [[Bondal-Orlov reconstruction theorem]]


\item [[coherent cohomology]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[J-P. Serre]], \emph{[[Faisceaux algébriques cohérents]]}, Ann. of Math. (2) \textbf{61}, (1955) 197--278, \href{http://dx.doi.org/10.2307/1969915}{doi}.


\item H. Grauert, R. Remmert, \emph{Coherent analytic sheaves}, Grundlehren der Math. Wissenschaften \textbf{265}, Springer 1984. xviii+249 pp.


\item M. M. [[Kapranov]], \emph{On the derived categories of coherent sheaves on some homogeneous spaces}, Invent. Math. 92 (1988), no. 3, 479--508, \href{http://dx.doi.org/10.1007/BF01393744}{doi}.


\item [[D. O. Orlov]], \emph{       } (\href{http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=629&what=fullt&option_lang=rus}{pdf}, Russian) Uspekhi Mat. Nauk \textbf{58} (2003), no. 3(351), 89--172; Engl. transl. \emph{Derived categories of coherent sheaves and equivalences between them}, Russian Math. Surveys \textbf{58} (2003), no. 3, 511--591.


\item V. D. Golovin, \emph{Homology of analytic sheaves and duality theorems}, Contemporary Soviet Mathematics (1989) viii+210 pp. transl. from Russian original      , Moskva, Nauka 1986. (192 pp.)


\item [[EGA]] 0.5.3.1


\item Qing Liu, \emph{Algebraic geometry and arithmetic curves}, 5.1.3



\end{itemize}
Categories of ind-coherent sheaves on schemes and stacks are studied in [[Dennis Gaitsgory]], \emph{Notes on Geometric Langlands: ind-coherent sheaves}, \href{http://arxiv.org/abs/1105.4857}{arxiv/1105.4857}

Discussion in [[(∞,1)-topos theory]]

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]}

\end{itemize}
category: sheaf theory, algebraic geometry

[[!redirects coherent sheaves]]



\end{document}