my beamer presentation - Departament de matemÃ tiques

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Outline Definitions and examples Homological properties of kC-mod An example Category algebras Finite categories and their algebras Representations and modules Motivation The k-category algebra of C, kC, is defined as a k-vector space k Mor C, in which the multiplication is given on base elements by { α ◦ β, if α and β are composable in C; α ∗ β = 0 , otherwise. The previous category 1 x x α 1 y y has the following k-category algebra with identity 1 kC = 1 x + 1 y Fei Xu Finite category algebras g
Outline Definitions and examples Homological properties of kC-mod An example B. Mitchell’s theorem Finite categories and their algebras Representations and modules Motivation Given a finite category algebra kC, we are interested in the homological properties of kC-mod of finitely generated left kC-modules. Let Vect k be the category of finite-dimensional k-vector spaces, and Vectk C the category of covariant functors. Each functor is called a k-representation of C. Mitchell’s Theorem: kC-mod ∼ = Vect C k . It simply says that each kC-module corresponds uniquely to a k-representation of C, and vice versa. F ↦→ ⊕ x∈Ob C F (x) and M ↦→ F M such that F M (x) = 1 x · M. Fei Xu Finite category algebras
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