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It wouldn't be too difficult to cook up some contrived examples of ringed spaces that aren't locally ringed spaces; however, are there any such examples that appear "in the wild," i.e. that are somehow natural or interesting?

Edit: perhaps one natural example is any space with the constant sheaf of a non-local ring; is there anything a bit more interesting than that?

J. T. B.
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    For general reasons if you have a continuous map $f : X \to Y$ and a sheaf $O_Y$ of local rings on $Y$, then the inverse image sheaf $f^{-1} O_Y$ is a sheaf of local rings on $X$. On the other hand this fails for the direct image sheaf. I don't know if this is interesting for you. – Zhen Lin Oct 01 '20 at 04:19
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    One natural construction of a locally ringed space is to take a topological space $X$ and equip it with the sheaf of continuous functions to $\mathbb{R}$ (or $\mathbb{C}$, say). This is locally ringed since inside the stalk at $x$, a function is invertible iff $f(x) \neq 0$, and the non-invertible functions can be made invertible by adding $1$. If $f(x) \neq 0$, then $f \neq 0$ on some open neighbourhood around $x$ by continuity, therefore the germ of $f$ is invertible. A natural way to mess up this construction is to simply take the sheaf of all functions to $\mathbb{R}$. – Joppy Oct 01 '20 at 04:39
  • @Joppy Do you happen to have a concrete example of a maximal ideal other than $\mathfrak{m}={f\in\mathcal{F}_p:f(x)=0}$ in the stalk $\mathcal{F}_x$ (where $\mathcal{F}=\operatorname{Func}(-,\mathbb{R})$ is the sheaf of all not-necessarily-continuous real-valued functions, and $x\in X$ is a point of our topological space)? – Oskar Henriksson Nov 05 '22 at 23:37
  • @OskarHenriksson Let $\delta_x : X \to \mathbb{R}$ be the indicator function which is 1 at $x$ and zero elsewhere. Then $\delta_x$ is nonzero and not invertible in the stalk $F_x$, hence is contained in some maximal ideal. But $\delta_x$ is not in $\mathfrak{m}_x$. – Joppy Nov 10 '22 at 01:13
  • @OskarHenriksson I suppose my answer works only if $x$ is not a closed point - take any traditional space like $X = \mathbb{R}^n$ etc. Of course all functions on a discrete topological space are continuous, so one has to rule out stuff like this to get a counterexample. – Joppy Nov 10 '22 at 01:38

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