I'm trying to read Frank Warner's Foundations of Differentiable Manifolds and Lie Groups and got confused with Theorem $1.17$ as some who does not have a pure mathematics background.
Let $F_m$ be the set of germs that vanish at $m$ and let $F_m^k$ be the ideal of $\tilde{F}_m,$ the set of germs which vanish at $m,$ consisting of all finite linear combinations of $k$-fold products of elements of $F_m.$
He wishes to show that $\dim (F_m / F_m^2) = \dim M$ and here's how the proof begins:
I'm good with the first three sentences, but have no idea what he means by the second equation. What does he mean to take the "mod" of that expression? I know that $F_m/F_m^2$ is just a quotient vector space, and I'm certain the "mod" has something to do with that. Also, how does he switch from talking about functions to germs (denoted by boldface)?
Many thanks! I suppose this might be rather obvious if I took an abstract algebra class, but I have not (nor do I have the space in my schedule!).
Im not fully sure of what it could mean in this context, but as i have experienced it in number theory, a value lets say x, mod m is the remainder after dividing x by m.
As pointer out-by a comment on this post ive realized I have made this an answer and not a comment, as im not fully sure of what “mod” means in this case take my answer with a grain of salt.