Elliptic Curves and Arithmetic Invariants by Haruzo Hida

Author:Haruzo Hida
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

with . Show that the corresponding morphism is given by m(a,b) = ab for .

4.3.3 Functorial Representations

Consider a group scheme G  ∕ B . We have a notion of a module over a given group. We generalize this to a notion of an action of a group scheme on a functor having values in MOD  ∕ B . We give only a brief outline of the theory (see [RAG] I.2) for general theory). For any B-module M, we define a group functor from ALG  ∕ B into MOD  ∕ B by . Let G  ∕ B be a group scheme. A B-module M is called a functorial G-module or schematic G-module if we have a functorial action that is R-linear. This means that we have a morphism of covariant functors such that for each R ∈ ALG  ∕ B , the induced map G(R) ×M(R) → M(R) is an action of the group G(R) on the R-module . In particular, if G = Spec(A) for an B-bialgebra A, G(A) = Hom S (G, G) acts on . Thus, id G  ∈ G(A) acts on [here id G is not the identity e A of the group G(A) but the identity map in Hom S (G, G) = G(A)].

Define a map by Δ(μ) = id G (μ ⊗ 1) (μ ∈ M), where 1 is the identity element of the ring A. Then we claim that Δ determines the G-module structure on . In fact, for , we have G(g)(id G ) = g and a commutative diagram:

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