my beamer presentation - Departament de matemà tiques

Outline
Definitions and examples
Homological properties of kC-mod
An example
A closed symmetric monoidal category
Adjoint functors and a spectral sequence
Two categorical constructions
Hochschild cohomology
Right Kan extension and internal hom
Let M, N ∈ kC-mod. Then M ⊗ N is a kC ⊗ kC-module.
M ˆ⊗N = Res ∆ (M ⊗ N) where Res ∆ : kC ⊗ kC-mod
→ kC-mod.
Thus
Hom kC (L ˆ⊗M, N) = Hom kC (Res ∆ (L ⊗ M), N)
∼ = HomkC⊗kC (L ⊗ M, RK ∆ N)
∼ = HomkC (L, Hom kC (M, RK ∆ N)).
The internal hom is hom(M, N) = Hom kC (M, RK ∆ N).
The dual module of M is defined to be hom(M, k).
When C = G is a group, hom(M, N) ∼ = Hom k (M, N).
Fei Xu
Finite category algebras