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