my beamer presentation - Departament de matemà tiques

Outline
Definitions and examples
Homological properties of kC-mod
An example
Finite categories and their algebras
Representations and modules
Motivation
Ordinary cohomology, an algebraic definition
For any M, N ∈ kC-mod, we can consider Ext ∗ kC(M, N).
There is a constant functor k ∈ Vectk
C such that
k(x) = k and k(α) = Id k for any α ∈ Mor C. It is also
called the trivial kC-module.
There is an augmentation map ɛ : kC → k.
The groups Ext ∗ kC(k, N) have different interpretations.
∗
They are isomorphic to both lim N, the higher limits
←− C
of N, and H ∗ (C; N), the cohomology of C with
coefficients in N.
Ext ∗ kC(k, k) possesses a cup product and is isomorphic to
H ∗ (BC, k), the cohomology ring of the classifying
space BC of C. It is called the ordinary cohomology ring
of kC.
Fei Xu
Finite category algebras