Function field question from Silverman's AEC Just before Proposition 1.7 on page 5 of AEC (2nd ed), Silverman defines $M_P$ as an ideal in the affine coordinate ring.
Then he states Proposition 1.7 (the intrinsic characterization of smoothness):
$\dim(M_P/M_P^2) = \dim(V)$.
His proof is an appeal to Theorem I.5.1 of Hartshorne. And this is how my question arises.
Hartshorne uses the maximal ideal in the local ring, not a maximal ideal in the affine coordinate ring and this seems to be the standard way of doing so.
Since the local ring is the localization of the affine coordinate ring by the maximal ideal associated with the point, using results like Theorems 4.1 and 4.2 (on the behavior of ideals with regard to localization) in Matsumura's "Commutative Ring Theory", I suspect one can make this all work out.
So to ask specific questions:
(1) is there a reason or benefit to defining $M_P$ as Silverman has done here?
(as opposed to Hartshorne and others)
(2) if so, then why does he change to the "conventional" definition of $M_P$ in Chapter II?
(see the notation at the start of that chapter)
(3) is Proposition 1.7 actually true with Silverman's definition of $M_P$? And if so, how does one step from the use of Silverman's definitions to Hartshorne's definitions?
 A: *

*I would guess his motivation comes from the fact he is discussing affine varieties in this section, so it makes sense to quickly note that $M_P$ (as he originally defines it) is a maximal ideal of the affine coordinate ring. Depending on the situation we're in, it can be useful to work either in the coordinate ring or the local ring, and maybe this way he covers both cases (albeit not with an explicit explanation of the equivalence).

*But, in general when we are interested in local behaviour near a specific point, we first localize, i.e. we compute in the local ring at that point. It is somewhat unfortunate that the notation is inconsistent here.

*Yes, it is still true. One way to verify this is to note the natural vector space structures these quotients have: $M_P/M_P^2$ is a $\overline K[V]/M_P$-vector space of finite dimension, while $(M_P\cdot\overline K[V]_P)/(M_P\cdot\overline K[V]_P)^2$ is a $\overline K[V]_P/(M_P\cdot\overline K[V]_P)$-vector space. But, since localization commutes with quotients by ideals, these two fields are equal (they are the residue field $\kappa(P)$ at $P$). It should not be too hard to find an isomorphism between these two vector spaces. For example, given a basis for $M_P/M_P^2$ can you find a basis for the quotient in the local ring?

