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Structure Sheaf

The structure sheaf O_X of a scheme X is the sheaf of rings that assigns to each open set U subset= X the ring O_X(U) of regular functions on U.

For an affine scheme X=Spec(R), the structure sheaf is characterized on the principal open set

D(f)={p in Spec(R):f not in p},

by O_X(D(f))=R_f. Together, the underlying topological space and the structure sheaf make a scheme a locally ringed space.

See also

Affine Scheme, Ideal Sheaf, Locally Ringed Space, Scheme, Sheaf

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References

Hartshorne, R. Algebraic Geometry. New York: Springer-Verlag, 1977.The Stacks Project Authors. "Affine Schemes." §26.5 in The Stacks Project, Tag 01HR, 2026. https://stacks.math.columbia.edu/tag/01HR.

Cite this as:

Weisstein, Eric W. "Structure Sheaf." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StructureSheaf.html

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