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Hartshorne defined locally ringed space $(X,O_X)$ as for every $p\in X$, $O_{X,p}$ stalk is local ring where $O_X$ is the structural sheaf over $X$.

For $X=\operatorname{Spec}(A)$ for unital rings, $O_{X,p}$ is always local as $p\in \operatorname{Spec}(A)$ and $O_{X,p}\cong A_p$.

Q1. Why do I need definition of locally ringed space? For classical variety, this holds trivially. For $X=\operatorname{Spec}(A)$ where $A$ is an unital ring, this holds trivially as well.

Q2. Where can I find the case such that $O_{X,p}$ is not local? It seems that structural sheaf is going to be weird.

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Every scheme has a corresponding locally ringed space. Furthermore, morphisms between schemes are in one-to-one correspondence with morphisms between their corresponding locally ringed spaces.

Thus, schemes can be understood in terms of their corresponding locally ringed spaces. In fact, one of the main approaches to the subject outright defines a scheme to be a locally ringed space of a certain type.


An example of a ringed space that is not a locally ringed space is almost any example where $O_X$ is the constant sheaf $\mathbb{Z}$.

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