Examples of locally ringed space A locally ringed space is a pair (X,$\mathcal{O}_X$) where $\mathcal{O}_X$ is a structure sheaf over X and the stalks $\mathcal{O}_{x,X}$ are local rings. 
What is a natural example of a locally ringed space?


*

*Let X=$\mathbb{C}^n$ and $\mathcal{O}_X$= germs of holomoprhic functions on X

*Let X=$\mathbb{C}^n$ and $\mathcal{O}_X$= germs of regular functions on X


Are the standard examples 1 and 2 locally ringed spaces? (obviuosly ringed spaces). How do you show the stalks have unique maximal ideals?
 A: Yes, these are both examples of locally ringed spaces.  Some more examples:


*

*$X$ is any topological spaces, and $\mathscr{O}_X$ is the sheaf of real-valued continuous functions on $X$.  (The sheaf of complex-valued continuous functions will also work, of course.)

*$X$ is any differential manifold, and $\mathscr{O}_X$ is the sheaf of real-valued $C^\infty$ functions on $X$.

*Of course, algebraic geometry gives the construction of a locally ringed space $\operatorname{Spec} R$ for any commutative ring $R$ - which I won't go into much detail on here.


The way I tend to think about a locally ringed space is as a ringed space where there is a coherent notion of a "non-zero" set of each section, such that each section becomes a unit when restricted to its non-zero set.  To be specific, suppose we have a ringed space $(X, \mathscr{O}_X)$ along with an operation $D$ which takes each $f \in \Gamma(U, \mathscr{O}_X)$ over an open subset $U$ to an open subset $V \subseteq U$, satisfying these conditions:


*

*$D(0_U) = \emptyset$, $D(1_U) = U$ for each open $U$.

*$D(f+g) \subseteq D(f) \cup D(g)$.

*$D(fg) = D(f) \cap D(g)$.

*If $V \subseteq U$ and $f \in \Gamma(U, \mathscr{O}_X)$, then $D(f |_V) = D(f) \cap V$.

*For each $f$, $f |_{D(f)}$ is a unit of $\Gamma(D(f), \mathscr{O}_X)$.


Then, $(X, \mathscr{O}_X)$ is a locally ringed space.  For $x \in X$, the unique maximal ideal of $\mathscr{O}_{X, x}$ is essentially the set of germs which "are zero" at $x$; this is (informally) the set of germs $f$ such that for some neighborhood $U$ of $x$, $f \in \Gamma(U, \mathscr{O}_X)$ and $x \notin D(f)$.  Then (4) above implies that this is independent of the choice of $U$, and also independent of the choice of representative of the germ.  Conditions (1) through (3) imply that this set is a proper ideal, and condition (5) ensures that it is a unique maximal ideal.
(Conversely, if $(X, \mathscr{O}_X)$ is locally ringed, then if we define $D(f) := \{ x \in U \mid f_x \notin \mathfrak{m}_x \}$, then $D$ will satisfy the above conditions.)
Now, in all the cases given, if we simply define $D(f) := \{ x \in U \mid f(x) \ne 0 \}$, then it is straightforward to check that this $D$ operation satisfies the conditions above.  (The exception is $\operatorname{Spec} R$, which has a slightly different construction for $D$.)
A: I'm not particularly well versed in this field, but we can consider the following germs to check that there is only one maximal ideal in each of the rings:
If $\mathbb C\{X\}$ is the set of power series with positive radius of convergence, then $f^{-1}$ is also analytic if and only if $a_0=0$ (the first term in series expension.) Hence, $\mathbb C\{X\}$ is a local ring with maximal ideal $(X)$.
To see this, we note that an element in the formal power series is invertible whenever its first term is nonzero.
Similarly, consider $0 \in \mathbb R$ and take an equivalence class on an arbitrarily small neighborhood $U$ of $0$. Then, if there exists some neighborhood where two functions agree on the restriction, they are considered equivalent. This ring of germs, $R_{\epsilon}$ (this is nonstandard notation) is local since all functions such that vanish on the origin form a maximal ideal, and any other function is invertible.
