1.7. A THEOREM OF GROTHENDIECK AND A CONSTRUCTION OF SERRE 81

is injective and the target is free of rank 1 over L . Hence, the finitely generated

torsion-free OL-module M is invertible. Also, the injective map

Hom((A , i ), (A, i)) → Hom((AKs , i ), (AKs , i))

between finitely generated OL-modules has image with finite index since it becomes

an equality after applying Q ⊗Z (·) (for L-dimension reasons); let n be the index.

It follows that all L-linear Ks-homomorphisms f : AKs → AKs are Gal(Ks/K)-

invariant because nf is defined over K. This shows that the formation of M is

unaffected by ground field extension to Ks, and hence by any ground field extension

(due to Lemma 1.2.1.2).

Now we assume that char(K) = 0 and seek to prove that M ⊗OL

A → A is

an isomorphism. We may assume K is finitely generated, and then that K = C.

The OL-modules H1(A(C), Z) and H1(A (C), Z) are each invertible (due to being

Z-flat of rank [L : Q]). By Example 1.5.3, we get OL-linear isomorphisms A(C) =

(R ⊗Q L)Φ/a and A (C) = (R ⊗Q L)Φ/a for non-zero ideals a, a ⊂ OL. Hence,

elements of M are precisely multiplication on (R ⊗Q L)Φ by those c ∈ L such that

ca ⊆ a. We conclude that M = HomOL (a , a) = aa

−1

, with M ⊗OL A → A given

by the evident evaluation pairing on C-points. This is an isomorphism because the

induced map on homology lattices is the natural map M ⊗OL a → a that is clearly

an isomorphism.

The isomorphism property for the map M ⊗OL A → A in Example 1.7.4.1 fails

away from characteristic 0, even for elliptic curves over finite fields. For example, if

L is imaginary quadratic with class number 1 then the relative Frobenius isogeny

provides counterexamples (using elliptic curves whose j-invariant is not in the prime

field). More explicitly, for p ≡ −1 mod 4 and κ := Z[i]/(p) Fp2 with p 3,

consider the elliptic curves E± = E = {y2 = x3 − x} over κ with CM by OL

via the actions [i](x, y) = (−x, ±iy). These are not OL-linearly isomorphic (since

Aut(E) = μ4 ⊂ OL

×

, as p 3) but the Frobenius isogeny E →

E(p)

is an OL-linear

isogeny E+ → E−. Thus, the module M of OL-linear homomorphisms from E+ to

E− is non-zero but the OL-linear map M ⊗OL E+ → E− cannot be an isomorphism

(since M OL as OL-modules).

1.7.4.2. Example. Let (A, i, L) be as in Example 1.7.4.1 over a field K, so OL ⊂

End(A). For an invertible OL-module M, another such abelian variety is given by

M ⊗OL A. Any m ∈ M defines an OL-linear map em : A → M ⊗OL A via x → m⊗x.

For = char(K), the map T (em) induced by em on -adic Tate modules is the map

T (A) → M ⊗OL T (A) given by v → m ⊗ v, so em = 0 when m = 0. In particular,

the module HomOL (A, M ⊗OL A) of OL-linear homomorphisms is non-zero and

therefore invertible (by Example 1.7.4.1).

The natural map of invertible OL-modules

eA,M : M → HomOL (A, M ⊗OL A)

is injective, hence of finite index. We shall now show that this map is an isomor-

phism. It suﬃces to check the result after applying Z ⊗Z (·) for every prime

(allowing = char(K)). This scalar extension is the first map in the diagram

M → Z ⊗Z HomOL (A, M ⊗OL A) → HomOL, (A[

∞

],M ⊗OL, A[

∞

])