Knowlpedia

Definition

Ringed space

A topological space equipped with a sheaf of rings.

Definition

A ringed space is a pair (X,OX)(X,\mathcal O_X) consisting of a XX and a of OX\mathcal O_X on XX. The sheaf OX\mathcal O_X is called the .

A morphism of ringed spaces

(f,f#):(X,OX)⟶(Y,OY)(f,f^\#):(X,\mathcal O_X)\longrightarrow(Y,\mathcal O_Y)

consists of a f:X→Yf:X\to Y and a of rings

f#:OY⟶f∗OX.f^\#:\mathcal O_Y\longrightarrow f_*\mathcal O_X.

The arrow on functions points opposite to the map on spaces: a function near f(x)f(x) is pulled back to a function near xx.

Stalkwise form

At every x∈Xx\in X, the sheaf morphism induces a

fx#:OY,f(x)⟶OX,xf_x^\#:\mathcal O_{Y,f(x)}\longrightarrow\mathcal O_{X,x}

between . Composition combines the continuous maps and these pullback homomorphisms.

Important refinements

A additionally requires every stalk to be a and its morphisms to induce local homomorphisms on stalks. Schemes, complex manifolds with their holomorphic functions, and smooth manifolds with their smooth functions carry such refinements. A bare ringed space imposes no local-ring condition.

Sheaves of modules and locally free sheaves are defined relative to OX\mathcal O_X; the topological space alone is not enough to specify their scalar multiplication.

References
  1. The Stacks Project Authors, The Stacks Project. Tag 01HY. Relevant: ringed spaces and morphisms of ringed spaces.
  2. Robin Hartshorne, Algebraic Geometry, Springer, 1977. DOI record. Relevant: Chapter II, §1, sheaves and ringed spaces.