Products of CM elliptic curves - Universität Duisburg-Essen

Indeed, (5) and (6) are clear from the definitions. Moreover, if f ∈ I, then Ker(f) ≥H(I) = ∩ g∈I Ker(g), so f ∈ I(H(I)) and hence I ⊂ I(H(I)), which proves (7).Similarly, since H ≤ Ker(f), for all f ∈ I(H), we have H ≤ ∩ f∈I(H) Ker(f) =H(I(H)), which yields (8).From the above properties we see immediately that(9)H(I) = H(I(H(I))) and I(H) = I(H(I(H))).Indeed, by (8) (applied to H = H(I)) we have H(I) ≤ H(I(H(I)). On the otherhand, since I ⊂ I(H(I)) by (7), it follows from (5) that H(I) ≥ H(I(H(I)), and sothe first equation of (9) follows. The second is proved similarly.Definition. A regular left R-ideal I is called a kernel ideal if we have that I = I(H(I)).A finite subgroup scheme H of A is called an ideal subgroup (scheme) if we haveH = H(I(H)).Remark 7 (a) Waterhouse[26], p. 533, calls an ideal I a kernel ideal if we haveI = {f ∈ R : fH(I) = 0}. Since fH(I) = 0 ⇔ H(I) ≤ Ker(f), it is clear that{f ∈ R : fH(I) = 0} = I(H(I)), and so his definition agrees with the one above. (Hedoes not define “ideal subgroups”.)(b) It follows from the definition and (9) that I is a kernel ideal if and only ifI = I(H), for some finite subgroup scheme H of A. Similarly, we see that H is anideal subgroup if and only if H = H(I) for some regular left R-ideal I.(10)(c) If f ∈ R is an isogeny, then it clear thatH(Rf) = Ker(f) and I(Ker(f)) = Rf(the latter by the universal property), and so Rf is a kernel ideal and Ker(f) is anideal subgroup. More generally, we have for any regular left R-ideal I, any finitesubgroup scheme H of A and isogeny f ∈ R that(11)H(If) = f −1 (H(I)) and I(f −1 (H)) = I(H)f.Indeed, the first equality follows from the fact that intersections commute with inverseimages, and the second follows because the faithful flatness of f implies that(12)H ≤ Ker(g) ⇔ f −1 (H) ≤ f −1 (Ker(g)) = Ker(gf), ∀g ∈ R.From (11) it thus follows that(13)I(H(If)) = I(H(I))f and H(I(f −1 (H))) = f −1 (H(I(H))).and so we see that if I is a kernel ideal, then so is If. Similarly, if H is an idealsubgroup, then so is f −1 (H). In fact, the converses of these assertions are also true:(14)(15)I is a kernel ideal ⇔ If is a kernel ideal;H is an ideal subgroup ⇔ f −1 (H) is an ideal subgroup.6