# Quasi-coherent sheaf

A sheaf of modules locally defined by generators and relations. More precisely, let be a topological space and let be a sheaf of rings on ; a sheaf of -modules is called quasi-coherent if for any point there is an open neighbourhood and an exact sequence of sheaves of -modules

where and are certain sets, denotes the restriction of a sheaf to and is the direct sum of copies of . A quasi-coherent sheaf is similarly defined on a topologized category with a sheaf of rings.

If is an affine scheme, then the association gives rise to an equivalence of the category of quasi-coherent sheaves of -modules and the category of -modules. As a result of this, quasi-coherent sheaves find broad application in the theory of schemes (see also Coherent sheaf; Scheme).

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#### References

[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 111–115; 126 MR0463157 Zbl 0367.14001 |

**How to Cite This Entry:**

Quasi-coherent sheaf.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Quasi-coherent_sheaf&oldid=23945