The dimension vectors $\Euvect(L_{1}), \Euvect(L_{2}), \ldots, \Euvect(L_{p})$ are linearly independent in $\kk Q_{0}$.

We can check the statement for extended Dynkin quivers with particular orientations from
the table given in <cit.>.
Using the APR-tilting equivalences, we can reduce the general case to this case.

Is there a direct proof which does not rely on any  explicit description of regular modules on the mouth?

Does the same statement holds true for  $n$-tame algebras in  the sense of Herschend-Iyama-Oppermann [28]?

Let $u_{1}, u_{2}, \ldots, u_{p} \in \kk^{r}$ be linear independent elements
and $x \in \kk^{\times}$.
Then there exists $v \in \kk^{r}$ such that ${}^{t} u_{i} v = x$ for all $i =1,2, \ldots, p$.

Let $\xi := (x, x, \ldots, x)^{t} \in \kk^{p}$.
Since the map $F: \kk^{r} \to \kk^{p}, F(w) := ({}^{t}u_{1}w, \ldots, {}^{t}u_{p}w)^{t}$ is linear and surjective,
the inverse image $F^{-1}(\xi)$ is non-empty.

It follows from Lemma <ref> and Lemma <ref> that
there exists $v \in \kk Q_{0}$ such that
all the weighted Euler characteristics ${}^{v}\!\Euch(L_{1}), {}^{v}\!\Euch(L_{2}), \ldots, {}^{v}\!\Euch(L_{p})$ have the same non-zero  value.
If we set $x := {}^{v}\!\Euch(L_{1})=  {}^{v}\!\Euch(L_{2}) = \cdots = {}^{v}\!\Euch(L_{p})$, then
the numbers ${}^{v}\!\Euch(N)$ for $N \in \cC$ are given in the diagram below

\[
\begin{xymatrix}@R=4pt@C=8pt{
& 4x \ar[dr] & & 4x \ar[dr] &&&\cdots & 4x \ar[dr]& \\
3x \ar[ur] \ar[dr] & & 3x \ar[ur] \ar[dr] & & 3x \ar[ur] \ar[dr] & &  \cdots & &3x\\
& 2x \ar[ur] \ar[dr] & & 2x\ar[ur] \ar[dr] & & & \cdots  & 2x \ar[ur] \ar[dr] &  \\
x \ar[ur] && x \ar[ur] && x \ar[ur] && \cdots && x
\]

We have ${}^{v}\!\Euch(\PPi_{1} \lotimes_{\Pa} N ) ={}^{v}\!\Euch(N)$ and

\[
\frac{{}^{v}\!\Euch(\PPi_{1} \lotimes_{\Pa} N) }{{}^{v}\!\Euch(N) } = 1
\]

for any indecomposable modules $N$ belonging to $\cC$.

Thus we see that for any indecomposable module $K$ belonging to $\cC$
the  weight  $v \in \kk Q_{0}$ has the property (II)${}_{K, 1}$.

Now we have completed the proofs of Proposition <ref> and Proposition <ref>.
Hence we have completed the proofs of Theorem <ref> and Theorem <ref> as well.

§ QHA OF DYNKIN TYPE $A_{N}$

We study the case where $Q$ is  an $A_{N}$-quiver and the base field $\kk$ is of arbitrary characteristic.

Let $N \geq1$ and  $Q$  an $A_{N}$-quiver.
Assume that $\kk$ has a primitive $(N+1)$-th root of unity.
Then for a generic $v \in \kk^{\times} Q_{0}$,
the morphism

\[
{}^{v}\!\varrho_{M}^{n} :M \to {}^{v}\!\YYM_{n} \lotimes_{\Pa} M
\]

is a minimal left $\rad^{n}$-approximation
for all $M \in \Dbmod{\Pa}$ and $n = 1,2, \ldots, N$.

By Theorem <ref> and Theorem <ref>,
it is enough to show that
the matrix $\Psi= \Phi^{-t}$ has a regular eigenvector $v\in \kk Q_{0}$.
whose eigenvalue $\lambda$ satisfies

\[
V_{m}(\lambda) \neq 0 \ \textup{ for } m = 2, \ldots, N-1.
\]

Using APR-tilting equivalences, we may assume that $Q$ is   a directed $A_{N}$-quiver $Q$.

\[
Q: 1 \to 2 \to 3 \to \cdots \to N.
\]

It is straight forward to check

\[
\Psi=
\begin{pmatrix}
-1 & -1 & -1& \cdots & -1 & -1 \\
1 & 0 & 0 & \cdots & 0 & 0\\
0 & 1 & 0 & \cdots & 0 & 0\\
0 & 0 & 1 & \cdots & 0 & 0\\
\vdots & \vdots & \vdots & \ddots &\vdots & \vdots \\
0 & 0 & 0 & \cdots & 1 & 0\\
\end{pmatrix}
\]

and the eigenpolynomial $F_{\Psi}(t)$ is

\[
F_{\Psi}(t) = t^{N}+ t^{N-1} + \cdots +  t + 1 = V_{N +1}(t).
\]

Let $\lambda$  be  a primitive $(N+1)$-th  root unity, then it is a root   of $F_{\Phi}(t)$.
the vector  $v := (\lambda^{N-1}, \lambda^{N-2}, \ldots,   \lambda, 1)^{t} \in \kk Q_{0}$
is an eigenvector of $\Psi$  belonging to the eigenvalue $\lambda$.

Recall that  isomorphism classes of indecomposable modules over $\Pa$ is
parameterized by pairs  $(i,j)$ of integers such that $1 \leq i \leq j \leq N$.
Namely, for such a pair $(i,j)$,
there exists a unique indecomposable module $M_{(i,j)}$ such that
$\dim e_{k} M= 1 \ ( i \leq k \leq j), \ \dim e_{k}M = 0 \ $(otherwise)
and these modules form a set of representatives of $\ind Q$.

\[
{}^{v}\!\Euch(M_{i,j} ) = \lambda^{N-i} + \lambda^{N -i-1} + \cdots + \lambda^{N -j},
\]

we conclude that ${}^{v}\!\Euch(M_{i,j}) \neq 0$ for all $(i,j)$.

§ BIMODULE VERSIONS AND THE UNIVERSAL LADDER

The aim of this section is to prove Theorem <ref> which is a bimodule version of Theorem <ref>. Throughout the section we again suppress the weight in our notation and write $\YYM = {}^{v}\!\YYM$.

First we establish a bimodule version of Theorem <ref>.

If the weight $v \in \kk^{\times} Q_{0}$  is a regular eigenvector of $\Psi= \Phi^{-t}$ with the eigenvalue $\lambda$.
Then, there exists a morphism $\xxi_{2}: \YYM_{1} \lotimes_{\Pa} \YYM_{1} \to \PPi_{1}$ in $\sfD(\Pa^{\mre})$
that satisfies the following conditions:

* We have the following equalities in $\Hom_{\Pa^{\mre}} (\YYM_{1}, \PPi_{1})$

\[
\xxi_{2} (\rrho_{\YYM_{1}}) = \ppi_{1}, \ \xxi_{2} ({}_{\YYM_{1}}\rrho) = -\lambda \ppi_{1}.
\]

* We have the following equality in $\End_{\Pa^{\mre}}(\PPi_{1}) \cong \kk$

\[
\xxi_{2} \eeta^{*}_{2}= - (1 + \lambda) \id_{\PPi_{1}}.
\]

* If $\lambda \neq -1$, then the composition $\xxi_{2} \eeta^{*}_{2}$ is an automorphism of $\PPi_{1}$ in $\sfD(\Pa^{\mre})$.

We need a preparation.
Let $X$ be an object of $\Dbmod{\Pa^{\mre}}$.
We provide a lemma that compares the endomorphism algebra $\End_{\Pa^{\mre}}(X)$ over $\Pa^{\mre}$ and
the endomorphism algebra $\End_{\Pa}(X)$ over $\Pa$.
Note that the forgetful functor $\sfD(\Pa^{\mre}) \to \sfD(\Pa)$ is not fully faithful.

Let $X \in \Dbmod{\Pa^{\mre}}$ be such an object  that  $\tuH^{< 0}(\RHom_{\Pa}(X, X)) = 0$.
Then the canonical map $i: \End_{\Pa^{\mre}}(X) \to \End_{\Pa}(X)$ is injective.
Moreover we have $\rad \End_{\Pa}(X) \cap \End_{\Pa^{\mre}}(X) \subset \rad\End_{\Pa^{\mre}}(X)$.

Recall that $\Hom_{\Pa^{\mre}}(X, X) \cong \RHom_{\Pa^{\mre}}(\Pa, \RHom_{\Pa}(X, X))$.
Applying the functor $\RHom_{\Pa^{\mre}}(-, \RHom_{\Pa}(X, X))$ to the exact sequence $0\to \Pa V\Pa \to \Pa^{\mre} \to \Pa \to 0$,
we obtain  the  exact sequence

\[
\Hom_{\Pa^{\mre}}(\Pa V \Pa,  \RHom_{\Pa}(X, X) [-1]) \to \Hom_{\Pa^{\mre}}(X, X) \xrightarrow{ i} \Hom_{\Pa}(X, X).
\]

Since the left term vanishes by the assumption, we conclude that $i$ is injective.

The second statement follows from the fact that inside a finite dimensional algebra $R$ the radical $\rad R$  is characterized as
the largest nilpotent left ideal (see <cit.>).

The idea of  the proof is to take the $\PPi_{2}$-dual $\RHom_{\Pa^{\op}}(\eeta^{*}, \PPi_{2})$ of $\eeta^{*}$.
To complete this idea, we need to fix  two isomorphisms $\sfiso$ and $\sfcan$.

we fix isomorphisms

\[
\sfc'_{\PPi_{1}} : \YYM_{1} \lotimes_{\Pa} \PPi_{1} \to \PPi_{1} \lotimes_{\Pa} \YYM_{1}, \ \
\sfe': \RHom_{\Pa^{\op}}(\YYM_{1}, \PPi_{1}) \cong \PPi_{1} \lotimes_{\Pa} \YYM_{1}^{\rvvee} \to \YYM_{1}
\]

in $\sfD(\Pa^{\mre})$ obtained in Corollary <ref> and Corollary <ref>.
We define an isomorphism
$\sfiso: \RHom_{\Pa^{\op}}(\YYM_{1} \lotimes_{\Pa} \YYM_{1}, \PPi_{1} ) \to \YYM_{1} \lotimes_{\Pa} \YYM_{1}$
to be the composition

\[
\begin{split}
\RHom_{\Pa^{\op}}(\YYM_{1} \lotimes_{\Pa} \YYM_{1}, \PPi_{1} )& \xrightarrow{\cong }
\PPi_{1} \lotimes_{\Pa}  \RHom_{\Pa^{\op}}( \YYM_{1}, \PPi_{1} ) \lotimes_{\Pa}  ( \YYM_{1})^{\rvvee} \\
&\xrightarrow{ {}_{\PPi_{1}}\sfe'_{ \YYM_{1}^{\rvvee} } }
\PPi_{1}\lotimes_{\Pa}  \YYM_{1} \lotimes_{\Pa} (\YYM_{1})^{\rvvee} \\
&\xrightarrow{(\sfc'_{\PPi_{1}} )^{-1}_{(\YYM_{1})^{\rvvee}  }}
\YYM_{1} \lotimes_{\Pa} \PPi_{1}\lotimes_{\Pa} (\YYM_{1})^{\rvvee} \\
&\xrightarrow{ {}_{  \YYM_{1}}  \sfe' }
\YYM_{1} \lotimes_{\Pa} \YYM_{1}.\end{split}
\]

Let $\sfcan: \RHom_{\Pa^{\op}}(\PPi_{1}, \PPi_{2}) \to \PPi_{1}$  be the canonical isomorphism.
We prove the composition  $\xxi_{2} := \lambda^{-1} \sfcan \RHom_{\Pa^{\op}}(\eeta^{*}, \PPi_{1}) \sfiso^{-1}: \YYM_{1} \lotimes_{\Pa} \YYM_{1} \to \PPi_{1}$
has the desired properties.

\[
\xxi_{2} : \YYM_{1} \lotimes_{\Pa} \YYM_{1} \xrightarrow{ \sfiso^{-1}} \RHom_{\Pa^{\op}}(\YYM_{1} \lotimes_{\Pa} \YYM_{1}, \PPi_{2})
\xrightarrow{ \RHom_{\Pa^{\op}}(\eeta^{*}, \PPi_{1}) }
\RHom_{\Pa^{\op}}(\PPi_{1}, \PPi_{2}) \xrightarrow{ \lambda^{-1}  \sfcan} \PPi_{1}.
\]

Applying  $\RHom_{\Pa^{\op}}(-, \PPi_{2})$ to the morphism
${}_{ \YYM_{1}}\ppi_{1}:  \YYM_{1} \lotimes_{\Pa} \YYM_{1} \to \YYM_{1} \lotimes_{\Pa} \PPi_{1}$, we obtain the following commutative diagram

\begin{equation}\label{2021022216151}
\begin{xymatrix}@C=60pt{
(   \YYM_{1} \lotimes_{\Pa} \PPi_{1}, \PPi_{2} )
\ar[r]^{({}_{  \YYM_{1}} \ppi_{1}, \PPi_{2} )} \ar[d] &
(   \YYM_{1} \lotimes_{\Pa} \YYM_{1}, \PPi_{2} ) \ar[d] \\
\PPi_{1} \lotimes_{\Pa}  ( \PPi_{1}, \PPi_{1} )   \lotimes_{\Pa} ( \YYM_{1})^{\rvvee}
\ar[r]^{ {}_{\PPi_{1}} (\ppi_{1}, \PPi_{2} )_{ ( \YYM_{1})^{\rvvee} } } \ar[d] &
\PPi_{1} \lotimes_{\Pa}  ( \YYM_{1}, \PPi_{1} ) \lotimes_{\Pa}  ( \YYM_{1})^{\rvvee}
\ar[d]^{{}_{\PPi_{1}}\sfe'_{ \YYM_{1}^{\rvvee} } }
\\
\PPi_{1}\lotimes_{\Pa} (\YYM_{1})^{\rvvee}\ar@{=}[d]
\ar[r]^{ {}_{\PPi_{1} }  \rrho_{ (\YYM_{1})^{\rvvee}  } }
\PPi_{1}\lotimes_{\Pa}  \YYM_{1} \lotimes_{\Pa} (\YYM_{1})^{\rvvee}
\ar[d]^{(\sfc'_{\PPi_{1}} )^{-1}_{(\YYM_{1})^{\rvvee}  }} \\
\PPi_{1}\lotimes_{\Pa} (\YYM_{1})^{\rvvee}\ar[d]_{\sfe' }
\ar[r]^{   \rrho_{ \PPi_{1} \lotimes (\YYM_{1})^{\rvvee} } }
\YYM_{1} \lotimes_{\Pa} \PPi_{1}\lotimes_{\Pa} (\YYM_{1})^{\rvvee}
\ar[d]^{ {}_{  \YYM_{1}}  \sfe'} \\
\YYM_{1}\ar[r]_{ \rrho_{\YYM_{1}}} &
\YYM_{1} \lotimes_{\Pa}  \YYM_{1}
\end{equation}

we use the abbreviation  $(-, +) := \RHom_{\Pa^{\op}}(-, +)$.
We note that the right column is $\sfiso$.

On the other hand, applying $\RHom_{\Pa^{\op}}(- , \PPi_{2})$
to the morphism $\rrho_{\PPi_{1}} :  \PPi_{1} \to  \YYM_{1} \lotimes_{\Pa} \PPi_{1}$,
we obtain the following commutative diagram.

\[
\begin{xymatrix}@C=60pt{
\RHom_{\Pa^{\op}}( \YYM_{1} \lotimes_{\Pa} \PPi_{1}, \PPi_{2} )
\ar[r]^{\RHom_{\Pa^{\op}}(\rrho_{\PPi_{1}} , \PPi_{2})}\ar[d]
\RHom_{\Pa^{\op}}(  \PPi_{1}, \PPi_{2} ) \ar[d] \\
\RHom_{\Pa^{\op}}(  \YYM_{1} , \PPi_{1} ) \ar[r]^{\RHom_{\Pa^{\op}}(\rrho , \PPi_{1})}\ar[d]^{\sfe'} &
\RHom_{\Pa^{\op}}(\Pa, \PPi_{1} ) \ar[d] \\
\YYM_{1}\ar[r]_{ - \lambda\ppi_{ 1 }} &
\PPi_{1}.
\]

Observe that the right column coincides with that of the diagram (<ref>).

Thus applying $\RHom_{\Pa^{\op}}(- , \PPi_{2})$ to the commutative diagram

\[
\begin{xymatrix}@C=60pt{\PPi_{1} \ar[d]_{ \eeta^{*}} \ar[dr]^{-\rrho_{\PPi_{1}}} & \\
\YYM_{1} \lotimes_{\Pa} \YYM_{1} \ar[r]_{  {}_{\YYM_{1} } \ppi_{1} } &
\YYM_{1} \lotimes_{\Pa} \PPi_{1},
\]

we obtain the following commutative diagram

\[
\begin{xymatrix}@C=60pt{\YYM_{1} \ar[r]^{\rrho_{\YYM_{1} }} \ar[dr]_{\ppi_{1} }&
\YYM_{1} \lotimes_{\Pa} \YYM_{1}  \ar[d]^{\xxi_{2}}.\\
& \PPi_{1}.
\]

This proves the first equality of (1).

Applying  $\RHom_{\Pa^{\op}}(-, \PPi_{2})$ to the morphism
$\ppi_{1,\YYM_{1} }:  \YYM_{1} \lotimes_{\Pa} \YYM_{1} \to \PPi_{1} \lotimes_{\Pa} \YYM_{1}$,
we obtain the following commutative diagram

\begin{equation}\label{2021022216152}
\begin{xymatrix}@C=60pt{
(   \PPi_{1} \lotimes_{\Pa} \YYM_{1}, \PPi_{2} )
\ar[r]^{(\ppi_{1,\YYM_{1} }, \PPi_{2} )} \ar[d] &
(   \YYM_{1} \lotimes_{\Pa} \YYM_{1}, \PPi_{2} ) \ar[d]
\\
\PPi_{1} \lotimes_{\Pa}  ( \YYM_{1}, \PPi_{1} )   \lotimes_{\Pa} ( \PPi_{1})^{\rvvee}
\ar[r]^{ {}_{\PPi_{1} \lotimes  (\YYM_{1}, \PPi_{2} )} \ppi_{1}^{\rvvee} } \ar[d]_{{}_{\PPi_{1}} \sfe'_{\PPi_{1}^{\rvvee} } } &
\PPi_{1} \lotimes_{\Pa}  ( \YYM_{1}, \PPi_{1} ) \lotimes_{\Pa}  ( \YYM_{1})^{\rvvee}
\ar[d]^{{}_{\PPi_{1}}\sfe'_{ \YYM_{1}^{\rvvee} } }
\\
\PPi_{1}\lotimes_{\Pa} \YYM_{1}  \lotimes_{\Pa} \PPi_{1}^{\rvvee} \ar[d]_{ (\sfc'_{\PPi_{1}})^{-1}_{\PPi_{1}^{\rvvee} }}
\ar[r]^{ {}_{\PPi_{1}  \lotimes \YYM_{1} }  \ppi^{\rvvee}  }
\PPi_{1}\lotimes_{\Pa}  \YYM_{1} \lotimes_{\Pa} (\YYM_{1})^{\rvvee}
\ar[d]^{(\sfc'_{\PPi_{1}} )^{-1}_{(\YYM_{1})^{\rvvee}  }}
\\
\YYM_{1} \lotimes_{\Pa} \PPi_{1}\lotimes_{\Pa} (\PPi_{1})^{\rvvee}\ar[d]
\ar[r]^{ {}_{\YYM_{1}  \lotimes \PPi_{1} }  \ppi^{\rvvee}  }
\YYM_{1} \lotimes_{\Pa} \PPi_{1}\lotimes_{\Pa} (\YYM_{1})^{\rvvee}
\ar[d]^{ {}_{  \YYM_{1}}  \sfe'} \\
\YYM_{1}\ar[r]_{ {}_{\YYM_{1}} \rrho} &
\YYM_{1} \lotimes_{\Pa}  \YYM_{1}
\end{equation}

we use the abbreviation  $(-, +) := \RHom_{\Pa^{\op}}(-, +)$.
We note that the right column is $\sfiso$.

On the other hand, applying $\RHom_{\Pa^{\op}}(- , \PPi_{2})$
to the morphism ${}_{\PPi_{1}} \rrho :  \PPi_{1} \to  \PPi_{1} \lotimes_{\Pa} \YYM_{1}$,
we obtain the following commutative diagram.

\[
\begin{xymatrix}@C=60pt{
(   \PPi_{1} \lotimes_{\Pa} \YYM_{1}, \PPi_{2} )
\ar[r]^{( {}_{\PPi_{1}} \rrho, \PPi_{2} )} \ar[d] &
(   \PPi_{1}, \PPi_{2} ) \ar[d]
\\
\PPi_{1} \lotimes_{\Pa}  ( \YYM_{1}, \PPi_{1} )   \lotimes_{\Pa} ( \PPi_{1})^{\rvvee}
\ar[r]^{ {}_{\PPi_{1}} (\rrho, \PPi_{1})_{\PPi_{1}^{\rvvee}}  }
\ar[d]_{{}_{\PPi_{1}} \sfe'_{\PPi_{1}^{\rvvee} } } &
\PPi_{1} \lotimes_{\Pa}  ( \Pa, \PPi_{1} ) \lotimes_{\Pa}  ( \PPi_{1})^{\rvvee}
\ar[d]
\\
\PPi_{1}\lotimes_{\Pa} \YYM_{1}  \lotimes_{\Pa} \PPi_{1}^{\rvvee} \ar[d]_{ (\sfc'_{\PPi_{1}})^{-1}_{\PPi_{1}^{\rvvee} }}
\ar[r]^{ - \lambda {}_{\PPi_{1}}\ppi_{1, \PPi_{1}^{\rvvee} } }
\PPi_{1}\lotimes_{\Pa}  \PPi_{1} \lotimes_{\Pa} (\PPi_{1})^{\rvvee}
\ar@{=}[d]
\\
\YYM_{1} \lotimes_{\Pa} \PPi_{1}\lotimes_{\Pa} (\PPi_{1})^{\rvvee}\ar[d]
\ar[r]^{ -  \lambda^{2} \ppi_{1, \PPi_{1} \lotimes \PPi_{1}^{\rvvee} } }
\PPi_{1} \lotimes_{\Pa} \PPi_{1}\lotimes_{\Pa} (\PPi_{1})^{\rvvee}
\ar[d] \\
\YYM_{1}\ar[r]_{ - \lambda^{2} \ppi_{1}  } &
\PPi_{1}.
\]

The commutativity of third square is proved in Corollary <ref>.
Therefore, the right column is $\sfcan$.

Thus applying $\RHom_{\Pa^{\op}}(- , \PPi_{2})$ to the commutative diagram

\[
\begin{xymatrix}@C=60pt{\PPi_{1} \ar[d]_{ \eeta^{*}} \ar[dr]^{{}_{\PPi_{1}} \rrho } & \\
\YYM_{1} \lotimes_{\Pa} \YYM_{1} \ar[r]_{   \ppi_{1, \YYM_{1}} } &
\PPi_{1} \lotimes_{\Pa} \YYM_{1},
\]

we obtain the following commutative diagram

\[
\begin{xymatrix}@C=60pt{\YYM_{1} \ar[r]^{{}_{\YYM_{1}} \rrho } \ar[dr]_{-\lambda \ppi_{1} }&
\YYM_{1} \lotimes_{\Pa} \YYM_{1}  \ar[d]^{\xxi_{2}}.\\
& \PPi_{1}.
\]

This proves the second equality of (1).

Let $i \in Q_{0}$.
By (1), we have $\xxi_{2,\Pa e_{i}} \rrho_{\YYM_{1} e_{i}} = \ppi_{1, \Pa e_{i}}$.
It follows from
Corollary <ref> and Corollary <ref> that
the endomorphism
$( \xxi_{2} \eeta^{*}_{2} + (1 +\lambda) \id_{\PPi_{1}}) \lotimes \Pa e_{i} =
\xxi_{2, \Pa e_{i}} \eeta^{*}_{2, \Pa e_{i}} + (1 +\lambda) \id_{\PPi_{1}e_{i}}$ of $\PPi_{1}e_{i}$ belongs to  in $\rad \End_{\Pa}(\PPi_{1}e_{i})$.
the endomorphism
$\xxi_{2} \eeta^{*}_{2} + (1 +\lambda) \id_{\PPi_{1}}$ of $\PPi_{1}$ in $\sfD(\Pa)$ belongs to  in $\rad \End_{\Pa}(\PPi_{1})$.

Since $\xxi_{2} \eeta^{*}_{2} + (1 +\lambda) \id_{\PPi_{1}}$ is an endomorphism of $\PPi_{1}$ in $\sfD(\Pa^{\mre})$,
it follows from by Lemma <ref> that
the endomorphism $\xxi_{2} \eeta^{*}_{2} + (1 +\lambda) \id_{\PPi_{1}}$ belongs to $\rad \End_{\Pa^{\mre}}(\PPi_{1})$.

Observe that $\End_{\Pa^{\mre}}(\PPi_{1}) \cong \End_{\Pa^{\mre}}(\Pa) = (\textup{the center of } A) = \kk$ and $\rad \End_{\Pa^{\mre}}(\PPi_{1}) = 0$.
Therefore we conclude that $\xxi_{2} \eeta^{*}_{2} = - (1 +\lambda) \id_{\PPi_{1}}$ as desired.

(3) is a consequence of (2).

Assume that a sincere weight $v \in \kk^{\times} Q_{0}$
is a regular (resp. semi-regular) eigenvector of $\Psi$ with the eigenvalue $\lambda$.

Assume moreover that there is   a natural number  $n \geq 2$
satisfies the condition:

\[
V_{m}(\lambda )  \neq 0, \textup{ for } 1 \leq m \leq n.
\]

there exists a morphism of
$\oomega_{n -1}: \YYM_{n -1}  \to \YYM_{1} \lotimes_{\Pa} \YYM_{n -2} $ in $\Dbmod{\Pa^{\mre}}$
that has the following properties

* $\zzeta_{n -1} \oomega_{n -1} = \id_{\YYM_{n -1}}$.

* If we set $\xxi_{n, M} := (\xxi_{2, \YYM_{ n-2}})({}_{\YYM_{1}} \oomega_{n -1})$

\[
\xxi_{n}
: \YYM_{1} \lotimes_{\Pa} \YYM_{n -1}
\xrightarrow{{}_{\YYM_{1}} \oomega_{n -1} }
\YYM_{1} \lotimes_{\Pa} \YYM_{1} \lotimes_{\Pa} \YYM_{n -2}
\xrightarrow{\xxi_{2, \YYM_{n -2}  } }
\PPi_{1} \lotimes_{\Pa} \YYM_{n -2},
\]

then the following equality holds  in $\ResEnd_{\Pa^{\mre} } (\PPi_{1} \lotimes_{\Pa} \YYM_{n -2})$

\[
\xxi_{n} \eeta^{*}_{n} =-  \frac{V_{n}(\lambda)}{V_{n-1}(\lambda) } \neq 0.
\]

Therefore it is an automorphism of $\PPi_{1} \lotimes_{\Pa} \YYM_{n -2}$ in $\Dbmod{\Pa^{\mre}}$.

* We have $
(\xxi_{n})( \rrho_{\YYM_{ n-1}}) =(\ppi_{1, \YYM_{n-2}})(\oomega_{n-1})$.

* Setting $\eepsilon_{m} := (\xxi_{m} \eeta^{*}_{m} )^{-1}, \  \varppi_{m}:= ( -1)^{n} (\ppi_{1, \YYM_{m-2}})(\oomega_{m-1})$ for $m = 2,3, \ldots, n $,
we obtain  the commutative diagram below for $m=2,3,\ldots, n$

\[
\begin{xymatrix}@C=60pt{
& \PPi_{1} \lotimes_{\Pa} \YYM_{m -2}
\ar@<5pt>[d]^{ ( \eeta^{*}_{m})(\eepsilon_{m}) } \ar@/^5pt/[dr]^{({}_{\PPi_{1}} \rrho_{\YYM_{m -1} })(\eepsilon_{m})} &  &\\
\YYM_{m-1}  \ar[r]^{\rrho_{\YYM_{m-1} } }
\ar@/^5pt/[ur]^{(-1)^{n -1} \varppi_{m-1} }  \ar@/_5pt/[dr]_{\brho_{m}}
\YYM_{1} \lotimes_{\Pa} \YYM_{m-1}
\ar[r]^{ \ppi_{1, \YYM_{n-1}  } } \ar@<5pt>[u]^{\xxi_{m}}
\ar@<-5pt>[d]_{\zzeta_{m} } & \PPi_{1} \lotimes_{\Pa} \YYM_{m-1} \\
&  \YYM_{m}   \ar@<-5pt>[u]_{\oomega_{m}} \ar@/_5pt/[ur]_{( -1)^{m}\varppi_{m} } &&
\]

Hence we obtain the following homotopy Cartesian diagram that is folded to a direct sum of AR-triangles

\[
\begin{xymatrix}@C=60pt{
\YYM_{m-1}   \ar[d]_{ \varppi_{m-1}  } \ar[r]^{\brho_{m}}
& \YYM_{m} \ar[d]^{\varppi_{m}} \\
\PPi_{1} \lotimes_{\Pa} \YYM_{m -2}
\ar[r]_{ ( {}_{\PPi_{1}} \brho_{m -1} )(\eepsilon_{m })  } & \PPi_{1} \lotimes_{\Pa} \YYM_{m-1}.
\]

Applying $- \lotimes_{\Pa} M$ where $M$ is an object of $\Dbmod{\Pa}$ (resp. $\cU_{\Pa}[\ZZ]$)
to the above diagrams,
we obtain the diagram
(<ref>) and (<ref>).

* Consequently,
we have an exact triangle

\[
\Pa \xrightarrow{ \brho^{n} } \YYM_{n} \xrightarrow{ \varppi_{n}} \PPi_{1} \lotimes_{\Pa} \YYM_{n-1} \to
\]

in $\Dbmod{\Pa^{\mre}}$
such that
applying  $- \lotimes_{\Pa} M$ where  $M$ is an object of $\Dbmod{\Pa}$ (resp. $\cU_{\Pa}[\ZZ]$),
we obtain an exact triangle

\[
M \xrightarrow{ \brho^{n}_{M} } \YYM_{n} \lotimes_{\Pa} M \xrightarrow{ \varppi_{n, M}} \PPi_{1} \lotimes_{\Pa} \YYM_{n-1} \lotimes_{\Pa} M \to
\]

in $\Dbmod{\Pa}$
the first arrow $\brho^{n}_{M}$  of which is a minimal left $\rad^{n}$-approximation of $M$.

We can prove Theorem <ref> in the same way of Theorem <ref> by using
Proposition <ref> and Lemma <ref>.
Hence we leave details to the readers.

We note we have a left ladder in $\Dbmod{\Pa^{\mre}}$ of the following form
such that if we apply $ -\lotimes M$ then we obtain the diagram (<ref>).

\[
\begin{xymatrix}@C=30pt{
\Pa \ar[r]^{\brho } & \YYM_{1} \ar[r]^{\brho} \ar[d]_{\ppi} &
\YYM_{2}  \ar[r]^{\brho } \ar[d]_{\varpi} &  \hspace{20pt} \cdots \hspace{-20pt}  &\ar[r] &
\YYM_{n-1}   \ar[d]_{\varpi  } \ar[r]^{\brho}
& \YYM_{n}  \ar[d]^{ \varpi} \\
& \PPi_{1}  \ar[r]_-{{}_{\PPi_{1}} \brho \epsilon } & \PPi_{1} \lotimes_{\Pa} \YYM_{1}  \ar[r]_-{{}_{\PPi_{1}} \brho \epsilon }  &
\hspace{20pt} \cdots \hspace{-20pt} &\ar[r]
\PPi_{1} \lotimes_{\Pa} \YYM_{n -2}
\ar[r]_{  {}_{\PPi_{1}} \brho \epsilon } & \PPi_{1} \lotimes_{\Pa} \YYM_{n-1} .
\]

§ HOMOTOPY CARTESIAN SQUARE

Let $\sfD$ be a triangulated category.
We recall from <cit.> that
a commutative diagram

\[
\begin{xymatrix}{
W \ar[r]^{k} \ar[d]_{w} & V \ar[d]^{v} \\
X \ar[r]_{f} & Y

is called homotopy cartesian if
there exists a morphism $c: Y \to W[1]$ that makes the following exact triangle

\[
W \xrightarrow{(-w, k)^{t} } X \oplus V \xrightarrow{ (f, v) } Y \xrightarrow{ c  } \Sigma W.
\]

We remark that the definition is  modified from that given in [41].
In op cit., the morphism $(w, -k)^{t}$ is used in place of $(-w, k)^{t}$.
But it is easy to check that these two definitions  are equivalent.

In other words, if we have a diagram

\[
\begin{xymatrix}@C=60pt{
& X\ar@<-5pt>[d]_{i} &  &\\
W \ar[r]^{a} & U \ar[r]^{b} \ar@<-5pt>[u]_{p} \ar@<-5pt>[d]_{q} & Y \ar[r]^{c} & \Sigma W \\
&  V \ar@<-5pt>[u]_{j} &&
\]

whose row is an exact triangle and the column gives a splitting of
of $U$, i.e., we have $pi= \id_{X}, qj  = \id_{V}, ip + j q = \id_{U}$,
then the following diagram is homotopy Cartesian

\[
\begin{xymatrix}{
W \ar[r]^{qa} \ar[d]_{-pa} & V \ar[d]^{bj} \\
X \ar[r]_{bi} & Y.

We use the following lemma that compare two homotopy Cartesian square.

Let $Z[-1] \xrightarrow{h} X \xrightarrow{f} Y \xrightarrow{ g} Z $ be an exact triangle
and $k: W \to V$ a morphism in $\sfD$.
Assume two commutative squares are given

\[
\begin{xymatrix}{
W \ar[r]^{k} \ar[d]_{w_{i}} & V \ar[d]^{v_{i}} \\
X \ar[r]_{f} & Y
}\end{xymatrix} \ \ ( i =1,2).
\]

Assume moreover that $gv_{1} =g v_{2}$ and
$\Hom_{\sfD}(W, Z[-1]) = 0$.
Then there exists
a morphism $s: V \to X$ such that
the automorphism $\begin{pmatrix} \id  & s \\ 0 & \id \end{pmatrix}$ of $V \oplus X$ completes the following commutative diagram.

\[
\begin{xymatrix}@C=40pt{
W \ar[r]^{\begin{pmatrix} -w_{1} \\ k  \end{pmatrix} }\ar@{=}[d] & X \oplus V  \ar[r]^{(f , v_{1})} \ar[d]_{\cong}^{\tiny \begin{pmatrix} \id  & s \\ 0 & \id \end{pmatrix}} & Y \ar@{=}[d] \\
W \ar[r]_{\begin{pmatrix}-w_{2} \\ k  \end{pmatrix}  }& X \oplus V  \ar[r]_{(f, v_{2})} & Y
\]

(We note that if the squares are homotopy Cartesian then, the both rows are exact triangles.)

We have $g(v_{1} -v_{2} ) = 0$.
Thus there exists a morphism $s: V \to X$ such that $v_{1} - v_{2} = fs$ or equivalently $v_{1} =v_{2} + fs$.

\[
\begin{xymatrix}@C=50pt{
& W \ar[d]_{w_{i}} \ar[r]^{k} & V \ar@{-->}[dl]_{s}  \ar[d]^{v_{i}} \ar[r]^{gv_{1} = gv_{2} } & Z \ar@{=}[d] \\
Z[-1] \ar[r]_{h} & X \ar[r]_{f} & Y \ar[r]_{g} & Z
\end{xymatrix}
\]

We have $fw_{1} = v_{1} k = ( v_{2} + fs)k=v_{2}k + fsk$ and hence $v_{2} k = fw_{1} - fsk =f(w_{1} -sk)$.
On the other hand, we have $v_{2}k = fw_{2}$.
Consequently we see  $f(w_{1} -sk -w_{2} ) = 0$.
It follows from the assumption $\Hom_{\sfD}(W, Z[-1]) =0$ that
$w_{1} -sk -w_{2}=0$
and hence $-w_{2} = -w_{1} +sk$.
Now it is straightforward to check that
the automorphism $ \begin{pmatrix} \id & s \\ 0 & \id \end{pmatrix}$ of $X \oplus V$ satisfies the desired property.

§ INVERSE OF SERRE FUNCTORS AND HAPPEL'S CRITERION

In this section, we fix notations for Serre duality and recall Happel's criterion for AR-triangles.

Let $R$ be a finite dimensional algebra of finite global dimension.
We set $\sfD := \Dbmod{R}$.

§.§ Inverse of  Serre functors

§.§.§ Exact functors

To fix a notation,
we recall that an exact functor $F: \sfD \to \sfD'$ is a pair $(F, \sigma_{F})$ of
a  functor $F: \sfD \to \sfD'$ between triangulated categories and a natural isomorphism
$\sigma_{F} : F \Sigma_{\sfD} \to \Sigma_{\sfD'} F $ that satisfies certain conditions.

We note that the pair $(\Sigma_{\sfD}, - \id_{\Sigma^{2}})$ is an exact autoequivalence of $\sfD$,
i.e., $\sigma_{\Sigma_{\sfD}} = - \id_{\Sigma^{2}}$.

§.§.§ Inverse of  Serre functors

Let $\sfS$ be the Serre functor of $\sfD$.
By $\isagl{-,+}$ we denote the pairing for Serre duality.
Namely it is a non-degenerate pairing

\[
\isagl{-, +} : \Hom_{\sfD}(Y, X) \otimes \Hom_{\sfD}(\sfS^{-1}(X), Y) \to \kk
\]

for $X, Y \in  \sfD$.
It follows from functoriality of the pairing that
for  morphisms given in the diagram below

\[
\sfS^{-1}(Z) \xrightarrow{ \sfS^{-1}(h)} \sfS^{-1}(X) \xrightarrow{ f} Y \xrightarrow{ g} Z \xrightarrow{h} X,
\]

we have the following equality

\begin{equation}\label{202012091955}
\isagl{h, gf} = \isagl{hg, f} = \isagl{g, f\sfS^{-1}(h)}.
\end{equation}

In particular,  we have

\begin{equation}\label{202102071308}
\isagl{f, g} = \isagl{\sfS^{-1}(f), \sfS^{-1}(g) }
\end{equation}

for all  $f: Y \to X, \ g : \sfS^{-1} (X) \to Y$.

Let $F$ be an exact autoequivalence of $\sfD$.
Then there exists a natural isomorphism $\gamma_{F}: \sfS^{-1} F \to F \sfS^{-1}$ that makes the following commutative diagram.

\[
\begin{xymatrix}@C=50pt{
\Hom_{\sfD}(Y, X) \otimes \Hom_{\sfD}(\sfS^{-1}(X), Y)  \ar[r]^-{\isagl{-,+}}  \ar[d]_{F_{Y, X} \otimes F_{\sfS^{-1}(X), Y} }
& \kk\\
\Hom_{\sfD}(F(Y)   F(X) ) \otimes \Hom_{\sfD}(F\sfS^{-1}(X), F(Y)) \ar[d]_{\id_{\Hom(F(Y) , F(X) )} \otimes \Hom(\gamma_{F,X}, F(Y))} &\\
\Hom_{\sfD}(F(Y), F(X)) \otimes \Hom_{\sfD}(\sfS^{-1}F(X), F(Y) )  \ar@/^-30pt/[ruu]_-{\isagl{-,+}}
\]

In other words, we have

\[
\isagl{f, g} = \isagl{F(f), F(g)\gamma_{F, X} }
\]

for all  $f: Y \to X, \ g : \sfS^{-1} (X) \to Y$.
Comparing it with (<ref>),
we see that

\begin{equation}\label{202102071309}
\gamma_{\sfS^{-1} } = \id_{\sfS^{-1}}.
\end{equation}

It is also easy to check that
for exact autoequivalences $F, G$ we have

\[
\gamma_{FG}= F(\gamma_{G}) (\gamma_{F})_{G}: \sfS^{-1}FG \xrightarrow{ (\gamma_{F})_{G} } F\sfS^{-1}G \xrightarrow{ F(\gamma_{G}) } FG\sfS^{-1}.
\]

It is shown by van den Bergh <cit.> that
the pair $(\sfS^{-1}, - \gamma_{\Sigma})$ is an exact autoequivalence of $\sfD$.
Therefore we set

\[
\sigma_{\sfS^{-1} }:  = - \gamma_{\Sigma}: \sfS^{-1} \Sigma \to \Sigma \sfS^{-1}.
\]

§.§.§ A lemma

We provide a technical lemma.

For simplicity we set $\sfP := \nu_{1}^{-1} := \Sigma \sfS^{-1}$.
Since there are two natural isomorphism $\gamma_{-}$ and $\sigma_{-}$,
it is necessary to care about the way that we exchange, for example,
$\sfP^{n}$ with $\Sigma^{-1} $.

For an exact endofunctor  $F$  of $\sfD$,
we denote by $\sigma'_{ F} :    \Sigma^{-1}F \to F \Sigma^{-1}  $
the natural isomorphism induced from $\sigma_{F} : F \Sigma \to \Sigma F$.

\[
\sigma'_{F}:  \Sigma^{-1}F \xrightarrow{\Sigma^{-1} F(\beta)}
\Sigma^{-1} F \Sigma \Sigma^{-1}
\xrightarrow{ \Sigma^{-1} ((\sigma_{F})_{\Sigma^{-1}})}
\Sigma^{-1}\Sigma   F\Sigma^{-1} \xrightarrow{\alpha_{F\Sigma^{-1}} } F\Sigma^{-1}
\]

where  we denote the canonical natural isomorphisms by  $\alpha: \Sigma^{-1} \Sigma \to \id_{\sfD},
\ \beta: \id_{\sfD} \to \Sigma \Sigma^{-1} $.

We note that $\sigma'_{\Sigma} $ may be  identified with $- 1: \id_{\sfD} \to \id_{\sfD}$.

* Let $n \geq 1$ be a positive integer. Then the following composition equals to $\id_{ \sfP^{n}}$.

\[
\mathsf{comp}^{(n)}:\sfP^{n} = \Sigma \sfS^{-1} \sfP^{n -1} \xrightarrow{ \Sigma (\gamma_{\sfP^{n -1} }) } \Sigma \sfP^{n -1} \sfS^{-1}
\xrightarrow{ (\sigma^{-1}_{\sfP^{n -1}})_{\sfS^{-1}} } \sfP^{ n-1} \Sigma \sfS^{-1} =\sfP^{n}.
\]

* Let $n \geq 1$ be a positive integer.
Then the following diagram is commutative

\[
\begin{xymatrix}{
\Sigma^{-1}\sfP^{n} \ar[r]^{ (\sigma'_{\sfP^{n -1} } )_{\sfP} }
\ar@{=}[d]
\sfP^{n -1} \Sigma^{-1} \sfP \ar@{=}[r] &
\sfP^{n -1}\Sigma^{-1} \Sigma \sfS^{-1} \ar[d]^{ \sfP^{n -1} (\alpha_{\sfS^{-1} }) }
\\
\Sigma^{-1}\Sigma \sfS^{-1} \sfP^{n -1} \ar[r]_{ \alpha_{\sfS^{-1} \sfP^{n -1} } }
\sfS^{-1} \sfP^{n -1} \ar[r]_{\gamma_{\sfP^{n -1}}}
\sfP^{n -1} \sfS^{-1}
\]

(1) The case $n =1$ is clear.
The case $ n= 2$ is proved in the following computation:

\[
\begin{split}
\mathsf{comp}^{(2)}
&= \Sigma(\sigma^{-1}_{\sfS^{-1}}) \circ  (\sigma^{-1}_{\Sigma})_{\sfS^{-2}} \circ  \Sigma^{2}(\gamma_{\sfS^{-1}}) \circ \Sigma(\gamma_{\Sigma})_{\sfS^{-1}}\\
&=- \Sigma(\sigma^{-1}_{\sfS^{-1}}) \circ \Sigma(\gamma_{\Sigma})_{\sfS^{-1}}\\
& = \id_{P^{2}}
\end{split}
\]

where for the second equality we use $\sigma_{\Sigma}= -\id, \ \gamma_{\sfS^{-1}} = \id$.

The case $n \geq 3$ is shown by induction on $n$ using  the  equality
$\mathsf{comp}^{(n)} = \sfP^{n -2}(\mathsf{comp}^{(2)} ) \circ \mathsf{comp}^{(n-1)}.$

(2) can be checked by using  the following commutative diagram

\[
\begin{xymatrix}@C=60pt{
\Sigma^{-1} \sfP^{n} \ar[d]_{ \Sigma^{-1}\sfP^{ n-1}(\beta)_{\sfP} }
\ar[r]^{ (\sigma'_{\sfP^{n -1} })_{\sfP} }
\sfP^{n -1}\Sigma^{-1} \sfP \ar[d]^{ \alpha^{-1}_{\sfP^{n -1}\Sigma^{-1} \sfP }}
\\
\Sigma^{-1} \sfP^{n -1} \Sigma \Sigma^{-1} \Sigma \sfS^{-1}
\ar[r]_{ \Sigma^{-1}(\sigma_{\sfP^{n -1}})_{\Sigma^{-1}\Sigma\sfS^{-1}}}
\ar[d]_{ \Sigma^{-1}\sfP^{n-1} \Sigma (\alpha)_{\sfS^{-1}} }
\Sigma^{-1} \Sigma \sfP^{n -1} \Sigma^{-1} \Sigma \sfS^{-1}
\ar[d]^{\Sigma^{-1} \Sigma\sfP^{n-1}  (\alpha)_{\sfS^{-1}}}
\\
\Sigma^{-1}  \sfP^{n -1}  \Sigma \sfS^{-1} \ar@{=}[d]\ar[r]_{ \Sigma^{-1}(\sigma_{\sfP^{n -1}})_{\sfS^{-1}}} &
\Sigma^{-1} \Sigma \sfP^{n -1}\sfS^{-1} \ar@{=}[d] \\
\Sigma^{-1}\Sigma \sfS^{-1} \sfP^{n -1} \ar[r]_{ \Sigma^{-1}\Sigma( \gamma_{\sfP^{n-1}}) } &
\Sigma^{-1}\Sigma \sfP^{n -1} \sfS^{-1}
\]

where the bottom square is commutative by (1) and the left column is the identity morphism.

§.§ Happel's criterion

§.§.§ For indecomposable objects

Let  $X \in \ind  \sfD$.
A morphism $s: \sfS^{-1}(X) \to X$ is called Auslander-Reiten (AR)-coconnecting
if it is a coconnecting morphism of  an AR-triangle starting from $X$.
In other words, we have an AR-triangle of the following form:

\[
X \to Y \to \sfS^{-1}(X) [1]  \xrightarrow{ \ -s[1] \ } X[1].
\]

We note that if $s, t$ are AR-coconnecting to $X$,
then there exist automorphisms $\phi, \psi$ of $X$ such that
$t = \phi s, t= s\sfS^{-1}(\psi)$.

We recall Happel's criterion for AR-coconnecting morphisms.

Let $X \in \ind \sfD$.
Then a morphism $s: \sfS^{-1}(X) \to X$ is  an AR-coconnecting morphism
if the following equations hold

\[
\isagl{\id_{X}, s} \neq 0, \
\isagl{f, s} = 0
\]

where  $f \in \rad \End_{\sfD}(X)$.
The converse holds if $\dim \ResEnd_{\sfD}(X) = 1$.

Let $X$ be an indecomposable object of $\sfD$.
If $\dim \ResEnd_{\sfD}(X) = 1$, then the subspace of $\Hom_{\sfD}(\sfS^{-1}(X), X)$ formed by all AR-coconnecting morphisms is one dimensional over $\kk$.
Thus we have the following lemma.

Let $X$ be an  indecomposable object of $X$ such that $\dim \ResEnd_{\sfD}(X) = 1$.
Then, the following holds.

* Two AR-coconnecting morphisms $s,t$ to $X$ are proportional over $\kk$ to each other.
More precisely we have

\[
t = \frac{\isagl{\id_{X}, t}}{\isagl{\id_{X}, s}} s.
\]

* An element $s \in \Hom_{\sfD}(\sfS^{-1}(X), X)$ is $0$ if and only if it satisfies

\[
\isagl{\id_{X}, s } = 0, \ \ \isagl{f, s} = 0 \textup{ for } f \in \rad\End_{\sfD}(X).
\]

§.§ Happel's criterion for not necessarily indecomposable objects

The results given in Section <ref> and <ref> are not used in the main body of the paper, but may be of independent interest.

We deal with the case where $X$ is not necessarily indecomposable. From now on we assume that $\dim \ResEnd_{\sfD}(X) = 1$ for all indecomposable $X$. Notice that this is satisfied if the base field $\kk$ is algebraically closed.

Let  $X \in \Dbmod{R}$ an object.
A morphism $s: \sfS^{-1} (X) \to X$ is called AR-coconnecting
if there exists an indecomposable decomposition $\phi: X \xrightarrow{\cong} \bigoplus_{i =1}^{n} X_{i}$
and AR-coconnecting morphisms $s_{i}: \sfS^{-1}(X_{i}) \to X_{i}$ for all $i = 1,2, \ldots, n$
such that
the composition
$\phi s \sfS^{-1}(\phi^{-1}): \sfS^{-1}(\bigoplus_{i =1}^{n} X_{i}) \to \bigoplus_{i =1}^{n} X_{i}$
is equal to $\bigoplus_{i =1}^{n} s_{i}$.

In other words, for an AR-coconnecting morphism $s: \sfS^{-1} (X) \to X$ there is a direct sum of AR-triangles

\[
\sfS^{-1}(X) \xrightarrow{ s} X \to Y \to \sfS^{-1}(X) [1].
\]

We provide a Happel's criterion  for a not necessarily indecomposable object $X \in \Dbmod{R}$.

Let $X \in \Dbmod{R}$.
Then for  a morphism $s: \sfS^{-1} (X) \to X$ the following conditions are equivalent:

* The morphism $s$ AR-coconnecting.

* The following conditions hold.

* There exists a complete set $\{ e_{1}, e_{2}, \ldots, e_{n} \}$ of orthogonal primitive idempotent elements
of $\End_{R}(X)$
such that
$\isagl{e_{a},  s} \neq 0$ for all $a =1,2, \ldots, n$.

* $\isagl{ f, s} = 0$ for all $f\in \rad\End(X)$.

The implication  (1) $\Rightarrow$ (2) is clear. We prove (2) $\Rightarrow$ (1).
For $a =1,2, \ldots, n$ we denote by $p_{a}: X \to X_{a}$ the projection induced from $e_{a}$ and
by $i_{a}: X_{a} \to X$   the injection induced from $e_{a}$.
The morphism $\phi:= (p_{1}, p_{2}, \ldots, p_{n})^{t}: X \to \bigoplus_{a=1}^{n}X_{a}$.
is an isomorphism and $\phi^{-1} = (i_{1}, i_{2}, \ldots, i_{n})$.
We set $s_{a} := p_{a} s \sfS^{-1}(i_{a})$ for $a =1,2, \ldots, n$.
Then it is immediate to check the equality $s = \phi^{-1} (\bigoplus_{a=1}^{n} s_{i}) \sfS^{-1}(\phi)$.
Thus it only remains to show that $s_{a}$ is AR-coconnecting to $X_{a}$.

We use Happel's criterion for indecomposable objects.
For $f \in \End_{R}(X_{a})$ we have

\[
\isagl{f, s_{a}} = \isagl{i_{a} f p_{a}, s} \neq 0.
\]

It follows from the assumption (a) that
$\isagl{\id_{X_{a}}, s_{a}} = \isagl{e_{a}, s} \neq 0$.
On the other hand,
if $f \in \rad\End_{R}(X_{a})$, then $i_{a} f p_{a} \in \rad\End_{R}(X)$.
Thus it follows from the assumption (b)
$\isagl{f, s_{a}} = \isagl{i_{a} f p_{a}, s}= 0$.

We point out the following property of AR-coconnecting morphisms.

Let $X \in \sfD$ and $s, t$ be AR-coconnecting to $X$.
Then there exist automorphisms $\phi, \psi$ of $X$ such that
$t= \phi s, \ t = s\sfS^{-1}(\psi)$.

§.§ Happel's criterion for left $\rad^{n}$-approximations

Let $n \geq 2$ be a positive integer, $M \in \sfD$
and $\lambda_{n -1}: M \to L_{n-1}$ a minimal left $\rad^{n -1}$-approximation, which fits an exact triangle
$M \xrightarrow{ \lambda_{n-1}} L_{n -1} \xrightarrow{ \lambda'_{n -1} } C \xrightarrow{\lambda''_{n -1}} M[1]$.
Assume that  $\lambda'_{n -1}: L_{n -1} \to C_{n -1}$ satisfies  the left $\rad$-fitting condition.
Then for a morphism $s: \sfS^{-1}(L_{n -1} ) \to M$  the following conditions are equivalent.

* A cone morphism $t: M \to C$ of $s$ is a minimal $\rad^{n}$-approximation.

* The following conditions are satisfied.

* There exists a complete set $\{e_{a}\}$ of orthogonal primitive idempotent elements of $\End(L_{n -1})$
such that $\isagl{ e_{a} \lambda_{n -1}, s} \neq 0$.

* We have $\isagl{ f \lambda_{ n-1}, s} = 0$ for all $ f \in \rad\End_{R}(L_{n -1})$.

The implication (1) $\Rightarrow$ (2) follows from Lemma <ref> and Proposition <ref>.
We prove the implication (2) $\Rightarrow$ (1).
By the octahedral axiom, we obtain the following diagram

\[
\begin{xymatrix}{
& C[-1] \ar@{=}[r] \ar[d] & C[-1] \ar[d]& \\
\sfS^{-1}(L_{n -1}) \ar@{=}[d] \ar[r]^{s} & M  \ar[d]_{\lambda_{n -1}} \ar[r] &\cone(s)  \ar[r] \ar[d] & \nu_{1}^{-1}(L_{n -1}) \ar@{=}[d] \\
\sfS^{-1}(L_{n -1})  \ar[r]_{\lambda_{n -1}s}  & L_{n -1} \ar[d]_{\lambda'_{n -1}} \ar[r] & N \ar[r] \ar[d] & \nu_{1}^{-1}(L_{n -1}) \\
& C \ar@{=}[r]  & C  &
\]

whose middle lows and columns are exact.
It follows from Proposition <ref> that
the condition  (a) and (b) implies that the composition $\lambda_{n -1} s$ is AR-coconnecting to $L_{n -1}$.
Thus the third row is a direct sum of AR-triangles.
Since we assume that the morphism $\lambda'_{n -1}$ satisfies the left $\rad$-fitting condition,
the morphism $N \to C_{n-1}$ is an split-epimorphism by  Lemma <ref>.
Thus, the morphism $\cone(s) \to N$ is a split-monomorphism.
Thus by Lemma <ref> we conclude that the morphism $M \to \cone(s)$ is a minimal left $\rad^{n}$-approximation of $M$.

§ NATURAL ISOMORPHISMS

In this section,
$\Pa$ denotes  a finite dimensional algebra such that $\gldim A^{\mre} < \infty$
(or more generally, a proper and homologically smooth dg-algebra).

First note that for $M, N \in \Dbmod{A}$, we have a natural isomorphism

\begin{equation}\label{202106091014}
\tuD( \tuD(N) \lotimes_{\Pa} M ) \cong \RHom_{\Pa}(M, \tuD\tuD(N) ) \cong \RHom_{\Pa}(M, N).
\end{equation}

Recall that
for $X \in \Dbmod{\Pa^{\mre}}$,
we set $X^{\vvee} := \RHom_{\Pa^{\mre}}(X, \Pa^{\mre})$.
Replacing $M$ with $X^{\vvee} \lotimes_{\Pa} M $ in (<ref>) and taking $\tuD$, we obtain the following natural isomorphism

\begin{equation}\label{202106091015}
\tuD(N) \lotimes_{\Pa}X^{\vvee} \lotimes_{\Pa} M
\cong \tuD\RHom_{\Pa}(X^{\vvee}\lotimes_{\Pa} M ,N).
\end{equation}

On the other hand,
we have the following natural  isomorphism

\begin{equation}\label{202106020808}
\begin{split}
\tuD(N) \lotimes_{\Pa}X^{\vvee} \lotimes_{\Pa} M
& \cong (\tuD(N) \otimes_{\kk} M) \lotimes_{\Pa^{\mre}} \RHom_{\Pa^{\mre}}(X, \Pa^{\mre}) \\
&\cong \RHom_{\Pa^{\mre}}(X, M \otimes_{\kk} \tuD(N)) \\
&\cong \RHom_{\Pa^{\mre}}(X, \Hom_{\kk}(N, M)) \\
&\cong \RHom_{\Pa }(X\lotimes_{\Pa}N , M). \\
\end{split}
\end{equation}

Combining (<ref>) and (<ref>) in the case $X = A$, we obtain
a natural isomorphism

\begin{equation}\label{202106020807}
\tuD\RHom_{\Pa}(\Pa^{\vvee} \lotimes_{\Pa} M ,N) \cong \RHom_{\Pa }(N , M).
\end{equation}

which shows that the functor $A^{\vvee} \lotimes_{\Pa} -$ is the inverse of a Serre functor.

Recall that  $X^{\lvvee}:= \RHom_{\Pa}(X, \Pa), \ X^{\rvvee} := \RHom_{\Pa^{\op}}(X, A^{\op})$.
Setting  $M = N =A$ in  (<ref>), we obtain an isomorphism below in $\Dbmod{\Pa^{\mre}}$.

\[
\tuD(A) \lotimes_{\Pa} X^{\vvee} \cong X^{\lvvee}.
\]

Repeating the same argument with right modules,
we obtain an isomorphism $X^{\vvee} \lotimes_{\Pa} \tuD(A) \cong X^{\rvvee}$
in $\Dbmod{\Pa^{\mre}}$.

We remark that in the case where $M, N \in \Dbmod{\Pa^{\mre}}$,
the above isomorphisms are isomorphisms in $\Dbmod{\Pa^{\mre}}$.

We write $\sfS^{-1} := \Pa^{\vvee} \lotimes_{\Pa} -$, since, as is explained above,  it is the inverse of a Serre functor.

Let  $T \in \Dbmod{\Pa^{\mre}}$ be  a two-sided tilting complex over $A$.
We denote by  $F: = T \lotimes_{\Pa}-$ the associated autoequivalence  of $\Dbmod{\Pa}$.
Then, there exists a natural isomorphism $\gamma_{F}: \sfS^{-1} F \to F\sfS^{-1}$
induced from the defining property of a Serre functor (see Section <ref>).
Since $\sfS^{-1} F = \Pa^{\vvee} \lotimes_{\Pa} T \lotimes_{\Pa} -$ and
$F \sfS^{-1} = T\lotimes_{\Pa} \Pa^{\vvee} \lotimes_{\Pa} -$,
it is natural to expect that the natural isomorphism $\gamma_{F}$ is induced from
an isomorphism in $\Dbmod{\Pa^{\mre}}$.
In the following lemma, we prove that  it is the case.

There exists an isomorphism $\gamma_{T} : \Pa^{\vvee} \lotimes_{\Pa} T \to T \lotimes_{\Pa} \Pa^{\vvee}$
in $\Dbmod{\Pa^{\mre}}$
such that $\gamma_{F} = \gamma_{T}\lotimes_{\Pa} -$.

By Rickerd [44],
there exists an object  $T' \in \Dbmod{\Pa^{\mre}}$  such that the endofunctor $T'\lotimes -$ of $\Dbmod{\Pa}$ is a quasi-inverse of $T\lotimes -$

We define an isomorphism
\delta: T' \lotimes_{\Pa} \Pa^{\vvee} \lotimes_{\Pa} T \xrightarrow{ \cong } \Pa^{\vvee}
$ in $\Dbmod{\Pa^{\mre}}$ to be the following composition of isomorphisms

\begin{equation}\label{202106091125}
\begin{split}
T' \lotimes_{\Pa} \Pa^{\vvee} \lotimes_{\Pa} T &\cong
\tuD\RHom_{\Pa}(T' \lotimes_{\Pa} \Pa^{\vvee} \lotimes_{\Pa} T, \tuD(\Pa) ) \\
& \cong \tuD\RHom_{\Pa}(\Pa^{\vvee} \lotimes_{\Pa} T,  T \lotimes_{\Pa} \tuD(\Pa) ) \\
& \cong  \RHom_{\Pa}( T \lotimes_{\Pa} \tuD(\Pa), T ) \\
& \cong  \RHom_{\Pa}( \tuD(\Pa), \Pa) \\
& \cong \tuD\RHom_{\Pa}( \Pa^{\vvee}, \tuD(\Pa)) \\ &
\cong \Pa^{\vvee}
\end{split}
\end{equation}

where  the third  isomorphism  and the fifth isomorphism are
Serre dualities.

There exists an isomorphism $\epsilon: A \to  T\lotimes_{\Pa} T'$ in $\Dbmod{\Pa^{\mre}}$.
We define an isomorphism $\gamma_{T}$ to  be $\gamma_{T} := ({}_{T}\delta)(\epsilon_{\Pa^{\vvee}\lotimes T})$.

\[
\gamma_{T}: \Pa^{\vvee}\lotimes_{\Pa} T  \xrightarrow{ \ \epsilon_{\Pa^{\vvee}\lotimes T} \ }
T \lotimes_{\Pa} T' \lotimes_{\Pa} \Pa^{\vvee} \lotimes_{\Pa}  T \xrightarrow{ {}_{T} \delta} T \lotimes_{\Pa} \Pa^{\vvee}
\]

Let $M, N \in \Dbmod{\Pa}$.
By Serre duality we have $\tuD(N) \cong (\Pa^{\vvee} \lotimes_{\Pa}N)^{\lvvee}$.
Applying $\tuD(N) \lotimes - \lotimes M = ( \Pa^{\vvee} \lotimes_{\Pa} N)^{\lvvee} \lotimes_{\Pa} - \lotimes_{\Pa} M$ to (<ref>),
we obtain the right square of the following commutative diagram

\[
\begin{xymatrix}{
\tuD(\Pa^{\vvee}\lotimes_{\Pa} T \lotimes_{\Pa} M, T\lotimes_{\Pa} N)
\ar@{=}[r]
\ar[d]_{\tuD(\epsilon_{\Pa^{\vvee} \lotimes T \lotimes M}^{*})} &
\tuD(\Pa^{\vvee}\lotimes_{\Pa} T \lotimes_{\Pa} M, T\lotimes_{\Pa} N)
\ar[r]^-{\cong} \ar[d]_{\tuD(\mathsf{adj})}^-{\cong} &
(T\lotimes_{\Pa} N, T\lotimes_{\Pa} M) \\
\tuD(T \lotimes_{\Pa} T' \lotimes_{\Pa} \Pa^{\vvee}\lotimes_{\Pa} T \lotimes_{\Pa} M, T\lotimes_{\Pa} N)
\ar[r]^-{T} \ar[d]_{\tuD({}_{T}\delta_{ M}^{*})}
\tuD(T' \lotimes_{\Pa} \Pa^{\vvee}\lotimes_{\Pa} T \lotimes_{\Pa} M,  N)
\ar[d]_{\tuD(\delta_{ M}^{*})} & \\
\tuD(T \lotimes_{\Pa} \Pa^{\vvee}\lotimes_{\Pa} M, T\lotimes_{\Pa} N) \ar[r]_-{T} &
\tuD(\Pa^{\vvee}\lotimes_{\Pa} M,  N)  \ar[r] &
(N, M) \ar[uu]_{T}
\]

where we use the abbreviation $(-,+) := \RHom_{\Pa}(-, +)$,
the arrows labeled by $T$ are induced from the functor $T\lotimes_{\Pa}-$
and $\mathsf{adj}$ denotes the adjoint isomorphism.

Since the left column is induced from $\gamma_{T}$,
this proves that $\gamma_{T}$ has the desired property.

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