REMARKS: Corrections, supplements, etc.
[09.11.2012] As Holden pointed out to me, I wrongly said that the affine line modulo G_m has no categorical quotient. To the contrary it does have a categorial quotient but not a geometric quotient. See 4.6 of [vdGM].
[09.13.2012] Regarding the theorem on the quotient by a strictly free action (compare with Th 4.16 of [vdGM], I did not complete the proof but the rest of it is basically 4.25 of [vdGM]. You can find a relevant part in Sec 12 of [Mum].
[09.18.2012] Again I did not have time to finish the proof of the theorem. The missing details are found in the proof of Section 12, Theorem 1.(B) of [Mum]. In that proof, refer to the proof of his assertions (b) and (c). Even though our case is more general, exactly the same proof works, noting that A is faithfully flat and projective of finite rank over B.
[09.20.2012] For the standard reduction argument to the locally noetherian case, see [Lan] Th 1.3.1.3 for a summary and also the reference thereof (basically EGA IV-3). It works in the situation of Cor 1 as follows: first you reduce to the case where S is locally noetherian as explained in [Lan] Th 1.3.1.3 and the paragraph below it. Next, since p and q are of finite type and finite presentation, resp, you see that X and Y must be locally noetherian as well. If you wish, it would do you little harm to treat this reduction step as a black box.
I mentioned it in class but note that the connectedness assumption on S is gone in Cor 2 (and later) unlike in the rigidity lemma and Cor 1. This is because you can reduce the proof of Cor 2 to the case of connected S by pulling back to each connected component (after which the assumption that "identity maps to identity" remains true). Notice that Cor 2 applies to G = abelian scheme because an abelian scheme is of finite presentation and separated.
Finally, "fp" meant "of finite presentation" in class, but I slightly regret this for the possible danger of confusion. In the widely popular French terminology, "fp" is for "faithfully flat" (fidelement plat) and "pf" stands for (locally) "of finite presentation" (de presentation finie). E.g. "fppf" = faithfully flat and locally of finite presentation.
See [Stacks] for the precise statements on geometrically connected/irreducible etc mentioned during the lecture.
The non-example of a non-flat proper group scheme with geometrically connected fibers over Z_p was stupid and wrong. As a group scheme has identity section over S, it should surject onto S. In particular it can't be empty over the generic point of Spec(Z_p). Perhaps one can try to put the trivial group scheme over Spec(Q_p), rather than the empty set, and somehow glue it to the special fiber over F_p to construct a non-example. Let me know if you find a better way.
[09.25.2012] In the rigidity lemma, I should have asssumed p:X->S to be not only proper and flat but also have geometrically connected geometrically reduced fibers. More generally one could assume proper, flat, and H^0(X_s,O_{X_s})=k(s) for all s in S. The fact is that if Y is a geometrically connected and geometrically reduced k-scheme (k=field) then H^0(Y,O_Y)=k. (See [GW] Prop 12.66 for instance.) We have seen how the condition H^0=k(s) was used in the proof of Step 1.
I thank Bao for pointing out two inaccuracies. One is the additional condition in the rigidity lemma mentioned above. The other concerns the following assertion: "Let B be an A-algebra and finitely generated as an A-module, where A,B=rings. Then B is finitely presented as an A-algebra if and only if B is finitely presented as an A-module", which was implicitly used in showing that if f:A->A' is an isogeny then ker f is finite and locally free over S. (My argument showed only that ker f is l.f.p over S "as an algebra". However to conclude that finite flat implies finite locally free, we need ker f to be locally finitely presented over S "as a module".) The assertion is true: EGA IV, Prop 1.4.7. Alternatively see [GW] Prop B.12.
Here are some useful references: fiber criterion for flatness - EGA IV, Th 11.3.10; generic flatness - EGA IV, Th 6.9.1; finite = q-finite + proper - EGA IV. Cor 18.12.4. If you somehow have fear of French, you can find corresponding assertions in [GW].
[09.27.2012] The statement on fiber dimension is that if X and Y are varieties over a field k and f:X->Y is an open morphism (i.e. sending every open set of X to an open set of Y) then for every y in f(X), dim(X)=dim(Y)+dim(f^{-1}(y)). This is a special case of [GW] Th 14.114 (+ Prop 14.102). Note that in our case, we used generic flatness plus "flat+lfp=>open" (cf. [GW] Th 14.33) to ensure the openness condition. It appears to provide a counterexample to have Y a projective plane and X (together with f) a blow up of Y at a point (if so, the fiber dim bumps up from 0 to 1 over the point where Y is blown up) but it's actually not a counterexample because such a map is not open. (For instance one could apply the criterion for open morphisms in [GW] Cor 10.21 to the generic point in X for the exceptional divisor. I guess there's an easier way.)
[10.25.2012] For abelian schemes pi:A->S, it's known that pi_*(O_A)=O_S.
[11.15.2012] For g-dimensional abelian variety A over an algebraically closed field k and a prime l such that (l,char k)=1, we assert that the union of A[l^n](k) for all n is Zariski dense in A. In the proof we defined Z to be the Zariski closure of the union and remarked that the reduced closed subscheme for the connected part Z^0 is an abelian subvariety of A. Note that Z^0[l^n](k)=(Z/l^n)^{2dimZ^0}. Now let i:=[Z:Z^0]. For every n, Z[l^n](k) contains A[l^n](k)=(Z/l^n)^{2g} by definition, so the image of the multiplication by i map on A[l^n](k) is contained in Z^0[l^n](k). Hence #Z^0[l^n](k)=l^{2n dimZ^0} >= #A[l^n](k)/i = l^{2ng}/i. By taking n->infty, we obtain dim Z^0 >= g. Since Z^0 is an abelian subvariety of A of dim g, the only possibility is that Z^0=Z=A.
