I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1 , and I have a question on its proof of main properties of transpose $Tr$, appearing in Auslander-Reiten theory.
Here we assume $A$ is a finite dimensional algebra over an algebraically closed field $K$, and $mod A$ is a category of finitely generated modules of $A$. Also, consider a functor $(−)^t = Hom_A(−,A) : mod A → mod A^{op}$.
For an object $M$ in $mod A$, we define $Tr M$ as follows:
If $P_1→P_0→M→0$ is a minimal projective presentation, then $Tr M=Coker(P_0^t→P_1^t)$. In other words, $P_0^t→P_1^t→TrM→0$ is exact.
Proposition 2.1.(b) in the textbook p.107 is about showing that
If M is not projective, then the exact sequence $P_0^t→P_1^t→TrM→0$ is a minimal projective presentation of an $A^{op}$ module $Tr M$.
In the proof, author assumes $P_0^t→P_1^t→TrM→0$ is not a minimal projective presentation. Then there exist non-trivial decompositions (by projective modules) $P_0^t=E_0'⊕E_0''$, $P_1^t=E_1'⊕E_1''$ and an isomorphism $v:E_0''→E_1''$ and a homomorphism $u:E'_0→E'_1$ such that the previous exact sequence is expressed as
$E_0'⊕E_0''\overset{\begin{pmatrix} u & 0 \\ 0 & v \end{pmatrix}}{\rightarrow}E_1'⊕E_1''→Tr M→0$.
(I think the author assumed $E'_0\overset{u}{\rightarrow}E'_1→Tr M→0$ is a minimal projective presentation.)
But I couldn't quite understand the next step. It is written as,
But then applying $(−)^t$ yields a projective presentation of M of the form ${E'_1}^t\overset{u^t}{\rightarrow}{E'_0}^t→M→0$
But I couldn't figure out why we get $M$ as a cokernel of $u^t$. Any helps will be appreciated. Thank you for reading.
Best Answer
For finitely generated projective modules $X$, the evaluation map $X\to X^{tt}$ (defined by $x\mapsto[\varphi\mapsto\varphi(m)]$) is a natural isomorphism.
So $P_1\to P_0$ is isomorphic to $P_1^{tt}\to P_0^{tt}$, which in turn is isomorphic to the direct sum of $E_1'^t\xrightarrow{u^t}E_0'^t$ and $E_1''^t\xrightarrow{v^t}E_0''^t$.
But $v$ is an isomorphism, so $v^t$ is an isomorphism and so its cokernel is zero. So $M$, which is the cokernel of $P_1\to P_0$, is also the cokernel of the first summand $E_1'^t\xrightarrow{u^t}E_0'^t$.