Locally Ringed Space

Algebraic Geometry
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Tsakanikas Nickos
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Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

Locally Ringed Space

#1

Post by Tsakanikas Nickos »

Let \( X \) be a topological space and let \( \mathscr{C}_{X} \) be the sheaf of continuous function on \( X \). Show that \( \left( X , \mathscr{C}_{X} \right) \) is a locally ringed space and describe (for each \( x \in X \)) the maximal ideal \( \mathfrak{m}_{x} \) of \( \mathscr{C}_{X,x} \).
Papapetros Vaggelis
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Posts: 426
Joined: Mon Nov 09, 2015 1:52 pm

Re: Locally Ringed Space

#2

Post by Papapetros Vaggelis »

Let \(\displaystyle{U}\) be an open subset of \(\displaystyle{X}\) and \(\displaystyle{O_{X}(U)}\) the

set of all \(\displaystyle{\mathbb{K}}\)-valued and continuous functions on \(\displaystyle{U}\).

1. Obviously, \(\displaystyle{O_{X}(U)}\) is a \(\displaystyle{\mathbb{K}}\) - subalgebra of the \(\displaystyle{\mathbb{K}}\)

algebra \(\displaystyle{C(U,\mathbb{K})}\).

2. If \(\displaystyle{f\in O_{X}(U)}\) and \(\displaystyle{U'}\) is an open subset of \(\displaystyle{U}\),

then \(\displaystyle{f|_{U'}\in O_{X}(U')}\).

3. Let \(\displaystyle{f\in O_{X}(U)}\). Then, for each \(\displaystyle{p\in U}\), if \(\displaystyle{U_{p}}\)

is an open neighbourhood of \(\displaystyle{p}\), then, \(\displaystyle{f|_{U_{p}}\in O_{X}(U_{p})}\).

Conversely, if \(\displaystyle{U=\bigcup_{i\in I}U_i}\) is an open subcovering of \(\displaystyle{U}\)

then, \(\displaystyle{f|_{U_i}\in O_{X}(U_i)\,,\forall\,i\in I}\) and then, \(\displaystyle{f\in O_{X}(U)}\).

According to 1,2,3 , \(\displaystyle{\left(X,C_{X}\right)}\) is a locally \(\displaystyle{\mathbb{K}}\) - ringed space.

Now, let \(\displaystyle{x\in X}\). If \(\displaystyle{U\,,U'}\) are open neighbourhoods of \(\displaystyle{x}\)

and \(\displaystyle{f\in O_{X}(U)\,,f'\in O_{X}(U'}\), then,

\(\displaystyle{(U,f)\equiv (U',f')}\) iff \(\displaystyle{f=f'}\) on some neighbourhood of \(\displaystyle{x\in U\cap U'}\).

Therefore,

\(\displaystyle{m_{x}=\left\{f\in C_{X}: f(x)=0\right\}}\).
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