Locally ringed space

A locally ringed space is a pair $(X,\mathcal O_X)$ consisting of a topological space $X$ and a sheaf of commutative rings $\mathcal O_X$ such that every stalk $\mathcal O_{X,x}$ is a local ring. The sheaf $\mathcal O_X$ is called the structure sheaf.

The unique maximal ideal in $\mathcal O_{X,x}$ distinguishes functions that vanish at $x$ from those locally invertible near $x$. This extra condition makes points and local algebra interact correctly.

For example, if $A$ is a commutative ring, then $(\operatorname{Spec}A,\mathcal O_{\operatorname{Spec}A})$ is locally ringed because the stalk at a prime ideal $\mathfrak p$ is the local ring $A_{\mathfrak p}$. Such examples are precisely the affine schemes.