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Outline Definitions and examples Homological properties of kC-mod An example Finite categories and their algebras Re<strong>presentation</strong>s and modules Motivation A proper setting for ordinary cohomology The natural functor π : G ⋉ P → G induces a ring homomorphism, the restriction, H ∗ (G; k) → H ∗ (G ⋉ P; k). Becker and Gottlieb constructed a transfer map H ∗ (G ⋉ P; k) → H ∗ (G; k) such that tr ◦ res = χ(P), the Euler characteristic of P. Applying a spectral sequence to P ↩→ G ⋉ P → G one can show H ∗ (G ⋉ P; k) is a finitely generated H ∗ (G; k)-module, and hence is finitely generated as a ring. Fei Xu Finite category algebras
Outline Definitions and examples Homological properties of kC-mod An example Finite categories and their algebras Re<strong>presentation</strong>s and modules Motivation Hochschild cohomology rings of transporter category algebras Let G be a finite group, P a finite G-poset and G ⋉ P the transporter category. The previously mentioned spectral sequence for P ↩→ G ⋉ P → G can be used to improve our observation on the finite generation of H ∗ (G ⋉ P; k) ∼ = Ext ∗ k(G⋉P)(k, k). Given a k(G ⋉ P)-module N, Ext ∗ k(G⋉P)(k, N) is finitely generated as a Ext ∗ k(G⋉P)(k, k)- and Ext ∗ kG(k, k)-module. With a bit extra work, one shows that the Hochschild cohomology ring HH ∗ (k(G ⋉ P)) is finitely generated as a Ext ∗ k(G⋉P)(k, k)-module, and hence is finitely generated as a ring. Fei Xu Finite category algebras
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my beamer presentation - Departament de matemÃ tiques

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