Definition 26.2.1. Locally ringed spaces.

1. A locally ringed space $(X, \mathcal{O}_ X)$ is a pair consisting of a topological space $X$ and a sheaf of rings $\mathcal{O}_ X$ all of whose stalks are local rings.

2. Given a locally ringed space $(X, \mathcal{O}_ X)$ we say that $\mathcal{O}_{X, x}$ is the local ring of $X$ at $x$. We denote $\mathfrak {m}_{X, x}$ or simply $\mathfrak {m}_ x$ the maximal ideal of $\mathcal{O}_{X, x}$. Moreover, the residue field of $X$ at $x$ is the residue field $\kappa (x) = \mathcal{O}_{X, x}/\mathfrak {m}_ x$.

3. A morphism of locally ringed spaces $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ is a morphism of ringed spaces such that for all $x \in X$ the induced ring map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is a local ring map.

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