I am reading the book “ Elements of the Representation Theory of Associative Algebras:Volume 1” written by Ibrahim Assem, Daniel Simson, Andrzej Skowronski and I have a question about chapter 2 proposition $2.10.$
Let $A=KQ/I$ be a finite,connected basic $K$-algebra, so we have all simple modules $S(a)$, indecomposable projective modules $P(a)=e_aA$ and indecomposable injective modules $I(a)\cong D(Ae_a)$ (up to isomorphic classes) where $a\in Q_0$.
Here is the proposition:
The restriction of the Nakayama functor $\nu: $mod $A$ → mod $A$ to the full subcategory proj $A$ of mod $A$ whose objects are the projective modules induces an equivalence between proj $A$ and the full subcategory inj $A$ of mod $A$ whose objects are the injective modules. The quasi-inverse of this restriction is given by $\nu^{-1}= \mathrm{Hom}_A(D(_A A),−) $: inj $A$ → proj $A$ where mod $A$ is the finitely generated right-modules category, Nakayama functor $\nu=D \mathrm{Hom}_A(-,A)$, $\nu^{-1}= \mathrm{Hom}_A(D(_AA),-)$ and $D$ is the standard duality.
I can understand why the author just check $\nu(P(a))\cong I(a)$ since every finitely generated projective right-$A$ module is the direct sums of $P(a)$ and the Nakayama functor is additive.
From this, I can get that $\nu$ sends projective modules to injective modules.
The the author just check $\nu^{-1}(I(a))\cong P(a)$.
I do not understand it. I think we should check $\nu^{-1}(I)$ is a projective modules where $I$ is an arbitrary injective module.
To get it, I want to prove that $D\nu^{-1}(I))$ is a projective module since $M$ is projective in mod $A$ if and only if $D(M)$ is a injective module in mod $A^{op}$.
$D\nu^{-1}(I)=D \mathrm{Hom}_A(D(_AA),I)= D\mathrm{Hom}_A(D(_AA),DDI)=D \mathrm{Hom}_{A^{op}}(DI,_AA).$
I think we can get $Ae_a(a\in Q_0)$ are all indecomposable projective modules and $D(e_aA)$ are all indecomposable injective modules similarly.
Since $D(I) $ is projective, we have $D(I)$ is the direct sums of $Ae_a(a\in Q_0)$, for example $D(I)\cong Ae_a\oplus Ae_b$.
By the functors $\mathrm{Hom}_{A^{op}}$ and $D$ are additive, we have $D \mathrm{Hom}_{A^{op}}(DI,_AA)\cong D \mathrm{Hom}_{A^{op}}(Ae_a\oplus Ae_b,_AA)\cong D(e_aA\oplus e_bA)\cong D(e_aA)\oplus D(e_bA) $
is an injective module.
Is the above opinion true and is it why the author just check $\nu^{-1}(I(a))\cong P(a)$?
Any help and references are greatly appreciated.
Thanks!
Here is the picture: