multiplication or composition?

Question

I am studying the ring homomorphism of the following function:

If $f: M \to N$ is a smooth function, then $$f^*: C^{\infty}(N) \to C^{\infty}(M), \textbf{ defined by } \phi \mapsto \phi \circ f$$ is a ring homomorphism.

I know that the given function $f^*$ should satisfy that for every $h,g \in C^{\infty}(M),$ we have that $f^*(hg) = f^*(g) f^*(h).$ my question is should the operation between $g,h$ be multiplication or composition and why?

Could someone explain this to me please?

There is a reason to use a different alphabet for the elements of $C^{\infty}$ and $f:M\to N.$ $f^{*}(hg)$ makes $f,g,h$ look similar. They are not. — Thomas Andrews, Feb 28 '22 at 23:32

score 1 · Accepted Answer · answered Feb 28 '22 at 23:18

1

It is multiplication. If $h$ and $g$ are maps from $N$ into $\Bbb R$ how could you possibly compose them? Only if $N=\Bbb R$.

When I write that it is multiplication, what I mean is that $hg$ is the map $n\mapsto h(n)g(n)$

answered Feb 28 '22 at 23:18

José Carlos Santos

427,504

but then it will not be a ring homomorphism which is what is required to be proved ... right? – Brain Feb 28 '22 at 23:33
After this $f^*(g h) = (gh) \circ f$ what should I do? – Brain Feb 28 '22 at 23:44
Is $(gh \circ f) = (g \circ f)(h \circ f)?$ – Brain Mar 01 '22 at 00:11
Sure it is! You have\begin{align}(gh\circ f)(x)&=(gh)\bigl(f(x)\bigr)\&=g\bigl(f(x)\bigr)h\bigl(f(x)\bigr)\&=(g\circ f)(x)\cdot(h\circ f)(x).\end{align} – José Carlos Santos Mar 01 '22 at 02:27
I have never seen this before .... multiplication in case of composition is like coset multiplication? – Brain Mar 01 '22 at 03:17
I fail to see a connection between those two concepts here. All I use is composition of functions and multiplication of real numbers. – José Carlos Santos Mar 01 '22 at 03:19
oh, so you are looking to $f(x)$ like a new variable say $y$ and you are evaluating $g$ and $h$ on it? – Brain Mar 01 '22 at 03:21
Sure. That's how function composition works, right?! – José Carlos Santos Mar 01 '22 at 03:22
yeah, you are correct but in the end it looks like coset multiplication – Brain Mar 01 '22 at 03:23

multiplication or composition?

1 Answers1