View source for Quasi-coherent sheaf

A [[Sheaf|sheaf]] of modules locally defined by generators and relations. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764101.png" /> be a topological space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764102.png" /> be a sheaf of rings on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764103.png" />; a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764104.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764105.png" />-modules is called quasi-coherent if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764106.png" /> there is an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764107.png" /> and an exact sequence of sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764108.png" />-modules

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764109.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641011.png" /> are certain sets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641012.png" /> denotes the restriction of a sheaf to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641014.png" /> is the direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641015.png" /> copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641016.png" />. A quasi-coherent sheaf is similarly defined on a [[Topologized category|topologized category]] with a sheaf of rings.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641017.png" /> is an affine scheme, then the association <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641018.png" /> gives rise to an equivalence of the category of quasi-coherent sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641019.png" />-modules and the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641020.png" />-modules. As a result of this, quasi-coherent sheaves find broad application in the theory of schemes (see also [[Coherent sheaf|Coherent sheaf]]; [[Scheme|Scheme]]).



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)  pp. 111–115; 126</TD></TR></table>