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?
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.