OX modules Scheme (mathematics)

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just r-modules central in commutative algebra when studying commutative ring r, ox-modules central in study of scheme x structure sheaf ox. (see locally ringed space definition of ox-modules.) category of ox-modules abelian. of particular importance coherent sheaves on x, arise finitely generated (ordinary) modules on affine parts of x. category of coherent sheaves on x abelian.


the sections of structure sheaf ox of x called regular functions, defined on each open subsets u in x. invertible subsheaf of ox, denoted ox, consists of invertible germs of regular functions under multiplication. in situations, sheaf kx defined on open affine subset spec of x total quotient rings q(a) (though there cases definition more complicated). sections of kx called rational functions on x. invertible subsheaf of kx denoted kx. equivalent class of invertible sheaf turns abelian group tensor products , isomorphic h(x, ox), called picard group. on projective varieties sections of structure sheaf ox defined on each open subsets u of x called regular functions though there no global sections except constants.








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