What's the meaning of the underlined sentence in page 1 of GTM 102?

Question

What's the meaning of the underlined sentence in page 1 of GTM 102?

Answer 1

Let me start with the explicit definition of $\mathcal D(U)$, i.e. $$ \mathcal D(U) = \{ f: U \to \mathbb C \mid f \text{~is smooth} \}. $$

Now, an algebra over the field $k=\mathbb C$ is a vector space with an bilinear from $$ \cdot: \mathcal D(U) \times \mathcal D(U) \to \mathcal D(U): (f,g) \mapsto f \cdot g. $$ This bilinear form assigns a ring structure to $\mathcal D(U)$ and is also a multiplication in this sense, but it is not the same as the scalar multiplication!

In our case, this map is defined pointwise. For $f,g \in \mathcal D(U)$, we define $f \cdot g \in \mathcal D(U)$ by $$ f \cdot g : U \to \mathbb C: x \mapsto (f\cdot g)(x) := f(x) \cdot g(x), $$ where the last addition is the addition of complex numbers. In addition, this map has to satisfy the axioms of a bilinear map, e.g. stuff like linearity with respect to scalars of the field $\lambda (f\cdot g) = (\lambda f) \cdot g$, for $\lambda \in k = \mathbb C$ and the other axioms.

Since the functions of the differentiable structure are complex valued, these properties are satisfied.

The last statement $1 \in \mathcal D(U)$ asserts, that the constant function $1: U \to \mathbb C: x \mapsto 1$ is an element of the algebra $\mathcal D(U)$. It is also the unity element with respect to the ring multiplication $\cdot$.

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If you know a bit of algebraic geometry, you can see that the axioms in the book point a little bit in the direction of sheafs.

Hence, topological objects like open set $U, V$ are assigned to rings $\mathcal D(U)$ and $\mathcal D(V)$, an we if $\mathcal U \subseteq \mathcal V$, then we can find a (restriction) morphism $\vert_U: \mathcal D(V) \to \mathcal D(U)$, which is in this case just the restriction of the functions onto the smaller domain. (See the wikipedia article for the complete definitions of sheafs.)

Hence, these a sheaf is a link between topology and algebraic strucutres. And they allow to relate purely local with global data, therefore it is no surprise, that they are also important for manifolds. (But I am not an expert about that! Don't trust me.)

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Details about the vector space structure:

The addition $$ + : \mathcal D(U) \times \mathcal D(U) \to \mathcal D(U): (f,g) \mapsto f+g $$ is also defined pointwise by $(f+g) (x) = f(x) + g(x) \in \mathbb C$.

And the scalar multiplication $$\mathbb C \times \mathcal D(U) \to \mathcal D(U): (\lambda,f) \mapsto \lambda f, $$ is defined via $(\lambda f)(x) = \lambda \cdot f(x) \in \mathbb C$.