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  "body": "\n## Definition\n\nA *ringed space* is a pair $(X, \\mathcal O_X)$ where $X$ is a [[topological space|0CQHFPGJ45]] and $\\mathcal O_X$ is a [[sheaf|TB6TM6R4MD]] of [[commutative rings|0310REJJY8]] on $X$.\nThe sheaf $\\mathcal O_X$ is called the *structure sheaf*.\nA *morphism of ringed spaces* $(f, f^\\#) \\colon (X, \\mathcal O_X) \\to (Y, \\mathcal O_Y)$ consists of two parts.\nThe first is a [[continuous map|0VGFHTM75Y]] $f \\colon X \\to Y$.\nThe second is a ring homomorphism $f^\\#_V \\colon \\mathcal O_Y(V) \\to \\mathcal O_X(f^{-1}(V))$ for each open set $V \\subseteq Y$, compatible with restriction to smaller open sets.\n\n## Remarks\n\nExample: let $X$ be a topological space and let $\\mathcal O_X(U)$ be the ring of continuous functions $U \\to \\R$; then $(X, \\mathcal O_X)$ is a ringed space.\nThe [[stalk|JTAKAPTT8H]] at each point $x$ is a [[local ring|MZ02BKPHJA]], whose maximal ideal consists of the germs that vanish at $x$.\n\nExample: an [[algebraic variety|HV6YBS0YDH]] $X$ over $k$, with $\\mathcal O_X(U)$ the ring of [[regular functions|4M94FZH9Y8]] on $U$, is a ringed space.\n\nExample: any topological space $X$ with the sheaf of locally constant functions to a commutative ring $A$ is a ringed space.\n\nNon-example: let $\\mathcal B(U)$ be the ring of bounded continuous functions on an open set $U \\subseteq \\R$.\nThen $(\\R, \\mathcal B)$ is not a ringed space, because $\\mathcal B$ is not a sheaf: the sections $x \\mapsto x$ on the intervals $(-m, m)$ do not glue.\n\nA ringed space whose stalks are all local rings is a locally ringed space; schemes and varieties are of this kind.\n",
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