Ringed space but not locally ringed space

Question

Give an example of a ringed space which is not locally ringed space.Why we need to go to locally ringed space if all ringed spaces are locally ringed spaces before defining schemes in Hartshone?Please clarify the difference in locally ringed spaces and ringed spaces?

Answer 1

There are many examples of ringed spaces that are not locally ringed. For instance, take a nice topological space $X$ (say a manifold) with the constant sheaf $\mathcal{F}=\underline{\Bbb{Z}}$. Then choose $p\in X$. Now, $$\varinjlim_{U\ni p} \mathcal{F}(U)=\varinjlim \Bbb{Z}=\Bbb{Z}$$ but this is not a local ring. Locally ringed spaces are important because they give us the ability to evaluate functions abstractly. Indeed, if $(X,\mathcal{O}_X)$ is a locally ringed space, then we can take $f\in \mathcal{O}_X(U)$ and define its evaluation at $p\in U$ by $$f\mapsto (f_p\in \mathcal{O}_{X,p})\mapsto (f(p)\in \mathcal{O}_{X,p}/\mathfrak{m}_p=:k(p)).$$ We call $k(p)$ the residue field at $p$. The first map is the stalk map, and the second is the map to the quotient.