Quasi-coherent sheaf

A sheaf of modules locally defined by generators and relations. More precisely, let $ X $ be a topological space and let $ {\mathcal A} $ be a sheaf of rings on $ X $; a sheaf $ {\mathcal F} $ of $ {\mathcal A} $- modules is called quasi-coherent if for any point $ x \in X $ there is an open neighbourhood $ U $ and an exact sequence of sheaves of $ ( {\mathcal A} \mid _ {U} ) $- modules

$$ {\mathcal A} | _ {U} ^ {(} I) \rightarrow {\mathcal A} | _ {U} ^ {(} J) \rightarrow \ {\mathcal F} \mid _ {U} \rightarrow 0 , $$

where $ I $ and $ J $ are certain sets, $ \mid _ {U} $ denotes the restriction of a sheaf to $ U $ and $ {\mathcal A} ^ {(} I) $ is the direct sum of $ I $ copies of $ {\mathcal A} $. A quasi-coherent sheaf is similarly defined on a topologized category with a sheaf of rings.

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

References

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