I saw in some notes on differential forms of varieties, the following definition: Let $X$ be a nonsingular variety and $U$ an open subset. Let $f \in \mathcal{O}(U)$ be a regular function. Then the differential form $df$ associated to $f$ is defined as follows: for any $P \in U$, $df(P)$ is an element of $m_P / m_P^2$, given by the equivalence class $<U,f-f(P)>$, where $m_P$ is the maximal ideal of the local ring $\mathcal{O}_{X,P}$. My question is the following: $<U,f-f(P)>$ is a germ, an element of the stalk $\mathcal{F}_P$. Why is it an element of $m_P / m_P^2$?
Here are the notes i am referring to, see Def. 1.2
http://www.math.ucdavis.edu/~osserman/classes/248A-F09/differentials.pdf
Best Answer
$f-f(P)$ is an element of $m_P$. You consider its class modulo $m_P^2$.