Group Cohomology learning seminar

Collin Roberts, Mathematics Department, University of Waterloo

“⊗ and Hom Functors”

In this talk we will prove many useful properties of the ⊗ and Hom functors, including:

1. ⊗ is right exact.
2. Hom is left exact.
3. The co-invariants functor is naturally equivalent to a ⊗ functor (and this is also right exact).
4. The invariants functor is naturally equivalent to a Hom functor (and this is also left exact).
5. For any ring R and left R-module M, we have HomR(R,M) ∼= M as left R-modules.
6. For any ring R and left R-module M, we have R ⊗R M ∼= M as left R-modules.

Time permitting, I will compute the cohomology groups of a finite cyclic group, to complete the picture that was started last week with the computation of the homology groups.