Symmetry of hom-space dimensions for semisimple modules

For finite-dimensional semisimple modules M and N over a k-algebra A, the spaces Hom_A(M, N) and Hom_A(N, M) have the same dimension over k. No splitting-field hypothesis is needed: each pair of isomorphic simple constituents contributes the dimension of its endomorphism division algebra, rather than necessarily one.

This comparison lets one compute maps into a one-dimensional character by instead counting its occurrences in the source. In particular, it is the semisimple comparison used when reading degree-two local Galois cohomology through Tate duality.

Main results

• TauCeti.finrank_linearMap_comm_of_isSemisimpleModule: hom-space dimension is symmetric for finite-dimensional semisimple modules.

References

• J.-P. Serre, Linear Representations of Finite Groups, §2 (simple constituents and multiplicities).
• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, proof of (7.3.1) (the semisimple comparison in the local Euler characteristic).

theorem TauCeti.finrank_linearMap_comm_of_isSemisimpleModule {k : Type u_1} {A : Type u_2} [Field k] [Ring A] [Algebra k A] (M : Type u_3) (N : Type u_4) [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M] [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N] [FiniteDimensional k M] [FiniteDimensional k N] [IsSemisimpleModule A M] [IsSemisimpleModule A N] : Module.finrank k (M →ₗ[A] N) = Module.finrank k (N →ₗ[A] M)

Hom-space dimension is symmetric between finite-dimensional semisimple modules over an arbitrary field. The algebra itself need not be semisimple or finite-dimensional.