# On the Nakayama functor

Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite dimensional. Denote by $D$ the duality with respect to $k$, i.e. $D(M)=\mathrm{Hom}_k(M,k)$. Then the following functor is a functor from $_AP$ to $_AI$:

$\nu=D\mathrm{Hom}_A(\;.\;,_AA)$

how can I prove it?

Define also

$\nu^{-1}=\mathrm{Hom}_A(D(A_A),\;.\;)$

how can I prove that this functor goes from $_AI$ into $_AP$?

I want to prove that $\nu$ and $\nu^{-1}$ are quasi-inverse, how can I do it?

Here they say there is an invertible natural transformation $\alpha_P:D\mathrm{Hom}(P,\;.\;)\rightarrow\mathrm{Hom}(\;.\;,\nu P)$ but I don't understand why it exists, why it is invertible and why this implies $\nu$ and $\nu^{-1}$ are quasi-inverse. Any help?

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Using the definition of what being quasi-inverse means, the definitions of the two fuunctors, and the various adjunction properties of $\hom$ and $\otimes$ should do it. Why don't you tell us what you have tried and where you got stuck? – Mariano Suárez-Alvarez Nov 6 '12 at 5:50
You have not explained some of the notation you are using (like ${}_AP$ and so on). Also, you speak of «here», presumably having a specific textbook in mind: be explicit about what «here» is, as we are not all «there» with you to know! – Mariano Suárez-Alvarez Nov 6 '12 at 5:51
I explained what $_AP$ means. The book is "triangulated categories in the representation theory of finite dimensional algebras" of Happel – Nick Nov 6 '12 at 6:17