Confusion about what it means for a section to vanish on a subscheme Let $i: V \rightarrow \mathbb{A}^n_k$ be a subscheme of the affine space of dimension $n$ over a field $k$. Let $f \in k[T_1, \dots, T_n]$.
What does it mean for $f$ to vanish on $V$?
Does it mean that for all $P \in V$, the image of $f$ in $\mathcal{O}_{\mathbb{A}^n_k, i(P)} / \mathfrak{m}_{\mathbb{A}^n_k,i(P)}$ is zero.
Or does it mean that the $f$ lies in the kernel of the following map
$$  \mathcal{O}_{\mathbb{A}^n_k, i(P)}  \rightarrow \mathcal{O}_{V, P}.$$
 A: This is maybe a little tricky because there actually are two different reasonable ways to interpret this, as you have noted. Which sort of meaning the author intends should be made clear via either an explicit statement in the text or a collection of context clues.
Let us explain a little bit about when these definitions are the same and when they are different. These definitions agree when $V$ is reduced: any function which evaluates to $0$ in the residue field at every point of a reduced scheme must actually be the zero function. They do not agree when $V$ is nonreduced: if $V=V(x^2)\subset \Bbb A^1$, then $x$ evaluates to $0$ in the residue field but $x$ is not zero in $\mathcal{O}_{V,P}$ for the unique point $P\in V$.
The latter definition is also equivalent to $f$ belonging to $I_V$, the ideal cutting out $V$, which is evidence it's the correct thing to say (talking about a function which vanishes on $V$ but isn't in the ideal of functions vanishing on $V$ is troubling). As a result, I would submit that the latter definition is the correct one while the first one should rather be called "vanishing at every point of a subscheme". From my experience reading the literature, I think authors are generally careful enough to state reducedness assumptions when they say #1 but need to upgrade it to #2, though this is obviously not a guarantee for all texts or papers.
