Proof. Let e be an idempotent of A and $\varepsilon :=1 -e$ . It suffices to show that e is a neat idempotent of A if and only if $A\varepsilon A$ is a strong idempotent ideal and $A/A\varepsilon A$ is semisimple. Indeed, one can apply this claim to each subalgebra $\varepsilon _{i}A \varepsilon _{i}$ . By [Reference Ágoston, Dlab and Wakamatsu1, Proposition 1.1], e is a neat idempotent of A if and only if it satisfies that (a) the multiplication map: $A \varepsilon \otimes _{\varepsilon A \varepsilon } \varepsilon A \to A \varepsilon A$ is an isomorphism, (b) $\operatorname {Tor}\nolimits _{j}^{\varepsilon A \varepsilon }(A \varepsilon , \varepsilon A)=0$ for each $j \ge 1$ and (c) $e A \varepsilon A e=eJe$ . Moreover, by [Reference Conde6, Remark 2.1.2(a)], the conditions (a) and (b) are equivalent to the condition that $A \varepsilon A$ is a strong idempotent ideal of A. Thus we show that the condition (c) holds if and only if $A/A \varepsilon A$ is semisimple. The “only if” part holds since $A/ A\varepsilon A \cong eAe / eA \varepsilon A e = eAe/ J(eAe)$ . We show the “if” part. By $\varepsilon =1-e$ , we obtain that $eA\varepsilon A e \subseteq eJe$ . Since $A/ A \varepsilon A $ is semisimple, so is $eAe/e A \varepsilon A e$ . Thus $0=J(eAe/eA\varepsilon Ae) =J(eAe)/eA\varepsilon Ae$ , and hence the assertion holds.▪

Proof. We assume that A is a quasi-hereditary algebra. Then there exists a heredity chain

$$ \begin{align*} A> A\varepsilon_{2} A > \cdots > A \varepsilon_{n} A > A \varepsilon_{n+1} A=0. \end{align*} $$

We show that this chain is a neat chain. By definition, $\varepsilon _{i} A\varepsilon _{i}/ \varepsilon _{i} A \varepsilon _{i+1} A \varepsilon _{i}$ is semisimple. Thus it suffices from Proposition 3.3 to prove that $\varepsilon _{i}A \varepsilon _{i+1}A \varepsilon _{i}$ is a strong idempotent ideal of $\varepsilon _{i}A \varepsilon _{i}$ for each $1 \le i\le n$ . By $A \varepsilon _{i}A/A\varepsilon _{i+1}A \in \mathsf {proj}\, (A/ A \varepsilon _{i+1}A)$ , it follows from [Reference Auslander, Platzeck and Todorov3, Proposition 5.2] that $A \varepsilon _{i}A/A\varepsilon _{i+1}A$ is a strong idempotent ideal of $A/A\varepsilon _{i+1}A$ . Since

$$ \begin{align*} \frac{A/ A \varepsilon_{i+1}A}{A\varepsilon_{i}A/A\varepsilon_{i+1}A }\cong A/A\varepsilon_{i}A, \end{align*} $$

we obtain that $\operatorname {Ext}\nolimits _{A/A \varepsilon _{i}A}^{j}(X, Y) \cong \operatorname {Ext}\nolimits _{A/A\varepsilon _{i+1}A}^{j}(X, Y)$ for each $X, Y \in \mathsf {mod}\, (A/A \varepsilon _{i}A)$ and $j \ge 1$ . Thus, $A \varepsilon _{i}A$ is inductively a strong idempotent ideal of A for each $2 \le i\le n$ . Since $A \varepsilon _{i+1}A$ is a strong idempotent ideal of A, it follows from [Reference Auslander, Platzeck and Todorov3, Theorem 2.1] that $\varepsilon _{i}A \varepsilon _{i+1}A \varepsilon _{i}$ is also a strong idempotent ideal of $\varepsilon _{i}A \varepsilon _{i}$ . Hence the assertion holds.▪

Proof. By an equivalence $\operatorname {Hom}\nolimits _{A}(\varepsilon _{i}A, -) : \mathsf {add}\, \varepsilon _{i}A \xrightarrow {\sim } \mathsf {proj}\, \varepsilon _{i}A \varepsilon _{i}$ , we obtain that $\mathsf {add}\, \varepsilon _{i+1}A$ is a cosemisimple right rejective (respectively, coreflective) subcategory of $\mathsf {add}\, \varepsilon _{i}A$ if and only if $\mathsf {add}\, \varepsilon _{i+1}A\varepsilon _{i}$ is a cosemisimple right rejective (respectively, coreflective) subcategory of $\mathsf {proj}\, \varepsilon _{i}A\varepsilon _{i}$ .

(1) $\Leftrightarrow $ (3): We consider an exact sequence in $\mathsf {mod}\, \varepsilon _{i}A\varepsilon _{i}$

$$ \begin{align*} 0 \to e_{i}J\varepsilon_{i} \to e_{i}A\varepsilon_{i} \to S(e_{i})\varepsilon_{i} \to 0. \end{align*} $$

If $\mathsf {add}\, \varepsilon _{i+1} A$ is a cosemisimple right rejective subcategory of $\mathsf {add}\, \varepsilon _{i}A$ , then $e_{i}J\varepsilon _{i} \in \mathsf {add}\, \varepsilon _{i+1} A\varepsilon _{i}$ by Proposition 2.3(3). Thus $\operatorname {pd}\nolimits _{\varepsilon _{i}A\varepsilon _{i}} S(e_{i})\varepsilon _{i} \le 1$ . Conversely, we assume that $\operatorname {pd}\nolimits _{\varepsilon _{i}A\varepsilon _{i}} S(e_{i})\varepsilon _{i} \le 1$ . Then $e_{i}J\varepsilon _{i} \in \mathsf {proj}\, \varepsilon _{i}A\varepsilon _{i}$ . Since $e_{i}A\varepsilon _{i}$ is not a direct summand of $e_{i}J\varepsilon _{i}$ , we have $e_{i}J\varepsilon _{i} \in \mathsf {add}\, \varepsilon _{i+1} A\varepsilon _{i}$ . Hence $\mathsf {add}\, \varepsilon _{i+1} A$ is a cosemisimple right rejective subcategory of $\mathsf {add}\, \varepsilon _{i}A$ by Proposition 2.3(3). Thus the assertion holds.

(1) $\Rightarrow{}$ (2): Since right rejective chains are coreflective chains, it suffices to show that the chain $A>A \varepsilon _{2}A > \cdots > A \varepsilon _{n}A >0$ is a neat chain by Proposition 3.3. Due to our assumption, $\mathsf {add}\, \varepsilon _{i+1}A$ is a right rejective subcategory of $\mathsf {add}\, \varepsilon _{i}A$ . By Proposition 2.3(2), we obtain that $\varepsilon _{i} A\varepsilon _{i+1}A \varepsilon _{i} \in \mathsf {proj}\, \varepsilon _{i}A\varepsilon _{i}$ for each ${1 \le i \le n}$ . Thus $\varepsilon _{i} A\varepsilon _{i+1}A \varepsilon _{i}$ is a strong idempotent ideal of $\varepsilon _{i}A\varepsilon _{i}$ by [Reference Auslander, Platzeck and Todorov3, Proposition 5.2]. Let $\mathcal {C}:=\mathsf {add}\, \varepsilon _{i}A/ [\mathsf {add}\, \varepsilon _{i+1}A]$ . Then $\mathcal {J}_{\mathcal {C}}=0$ holds by cosemisimplicity. Since $J(\varepsilon _{i}A \varepsilon _{i}/ \varepsilon _{i}A \varepsilon _{i+1} A \varepsilon _{i})=J(\operatorname {End}\nolimits _{\mathcal {C}}(\varepsilon _{i}A)) = \mathcal {J}_{\mathcal {C}}(\varepsilon _{i}A, \varepsilon _{i}A)=0$ , we have the assertion.

(2) $\Rightarrow{}$ (1): Since (3.2) is a coreflective chain, $\mathsf {add}\, \varepsilon _{i+1}A$ is a cosemisimple subcategory of $\mathsf {add}\, \varepsilon _{i}A$ . Hence, it is enough to show that $\mathsf {add}\, \varepsilon _{i+1}A$ is a right rejective subcategory of $\mathsf {add}\, \varepsilon _{i}A$ for each $1 \le i \le n-1$ . By Proposition 2.3(2), we prove that $\varepsilon _{i}A \varepsilon _{i+1}A\varepsilon _{i} \in \mathsf {proj}\, \varepsilon _{i}A \varepsilon _{i}$ . Since A is a neat algebra with a neat sequence $(e_{1}, e_{2}, \ldots , e_{n})$ , it follows from Proposition 3.3 that $\varepsilon _{i}A\varepsilon _{i+1}A\varepsilon _{i}$ is a strong idempotent ideal of $\varepsilon _{i}A\varepsilon _{i}$ . Due to our assumption, $\mathsf {add}\, \varepsilon _{i+1}A$ is a coreflective subcategory of $\mathsf {add}\, \varepsilon _{i}A$ . Thus, $\varepsilon _{i}A \varepsilon _{i+1} \in \mathsf {proj}\, \varepsilon _{i+1}A \varepsilon _{i+1}$ by Proposition 2.3(1). Since $\varepsilon _{i}A\varepsilon _{i+1}A\varepsilon _{i}$ is a strong idempotent ideal of $\varepsilon _{i}A\varepsilon _{i}$ and $\varepsilon _{i}A \varepsilon _{i+1} \in \mathsf {proj}\, \varepsilon _{i+1}A\varepsilon _{i+1}$ , it follows from [Reference Auslander, Platzeck and Todorov3, Corollary 3.8(b), Proposition 5.2] that $\varepsilon _{i}A \varepsilon _{i+1}A\varepsilon _{i} \in \mathsf {proj}\, \varepsilon _{i}A \varepsilon _{i}$ . Thus the proof is complete.▪

Proof. Let $f \in \operatorname {Hom}\nolimits _{A}(eA, X \otimes _{\varepsilon A \varepsilon } \varepsilon A)$ . Then $f \otimes _{A}A\varepsilon : eA \otimes _{A}A\varepsilon \to X \otimes _{\varepsilon A \varepsilon } \varepsilon A\otimes _{A}A \varepsilon $ . Since $\varphi $ is a right $\mathsf {add}\, eA\varepsilon $ -approximation, there exists $\varphi ' : eA \otimes _{A} A \varepsilon \to Y$ such that $\alpha (f \otimes _{A} A \varepsilon ) = \varphi \varphi '$ , where $\alpha : X \otimes _{\varepsilon A \varepsilon } \varepsilon A \otimes _{A} A\varepsilon \to X$ is an isomorphism. Let $\beta : eA \otimes _{A} A \varepsilon \otimes _{\varepsilon A \varepsilon } \varepsilon A \to eA$ be an isomorphism via $ea \otimes b\varepsilon \otimes \varepsilon c \mapsto eab \varepsilon c$ . Then $(\alpha \otimes _{\varepsilon A \varepsilon } \varepsilon A) (f \otimes _{A} A \varepsilon \otimes _{\varepsilon A \varepsilon } \varepsilon A)= f \beta $ holds. Since $(\varphi \otimes _{\varepsilon A \varepsilon } \varepsilon A) (\varphi ' \otimes _{\varepsilon A \varepsilon } \varepsilon A) \beta ^{-1}=(\varphi \varphi ' \otimes _{\varepsilon A \varepsilon } \varepsilon A) \beta ^{-1}=(\alpha \otimes _{\varepsilon A \varepsilon } \varepsilon A ) (f \otimes _{A} A \varepsilon \otimes _{\varepsilon A \varepsilon } \varepsilon A) \beta ^{-1}=f$ , we have the assertion.▪

Proof. Let $\varphi : P \to Q$ be a monomorphism in $\mathsf {add}\, \varepsilon A$ . We show that $\operatorname {Hom}\nolimits _{A}(\varepsilon A,\operatorname {Ker}\nolimits \varphi )=0$ . Let $\psi \in \operatorname {Hom}\nolimits _{A}(\varepsilon A, \operatorname {Ker}\nolimits \varphi )$ . Then, we have a composition map

$$ \begin{align*} \varepsilon A \xrightarrow{\psi} \operatorname{Ker}\nolimits \varphi \xrightarrow{i} P \xrightarrow{\varphi} Q \end{align*} $$

such that $\varphi i \psi =0$ holds. Since $\varphi : P \to Q$ is a monomorphism in $\mathsf {add}\, \varepsilon A$ , we have ${i \psi =0}$ . Hence $\psi =0$ holds.▪

Proof. First, we assume (1). Then $e_{i}J\varepsilon _{i} \in \mathsf {add}\, \varepsilon _{n} A \varepsilon _{i}$ implies that $\operatorname {pd}\nolimits _{\varepsilon _{i}A \varepsilon _{i}} S(e_{i})\varepsilon _{i} \leq 1$ holds. Thus, by Theorem 3.5, (3.3) is a right rejective chain. Since $\varphi _{j}: e_{j}J \varepsilon _{j} \to e_{j}A \varepsilon _{j}$ is a right $\mathsf {add}\, \varepsilon _{j+1}A \varepsilon _{j}$ -approximation by Proposition 2.3(3), it follows from Lemma 3.7 that a composition map of $\varphi _{j} \otimes _{\varepsilon _{j}A \varepsilon _{j}} \varepsilon _{j}A: e_{j}J \varepsilon _{j}\otimes _{\varepsilon _{j}A \varepsilon _{j}} \varepsilon _{j}A \to e_{j}A \varepsilon _{j}\otimes _{\varepsilon _{j}A \varepsilon _{j}} \varepsilon _{j}A$ and an isomorphism $e_{j}A \varepsilon _{j} \otimes _{\varepsilon _{j}A \varepsilon _{j}}\varepsilon _{j}A \to e_{j}A$ is a right $\mathsf {add}\, \varepsilon _{j+1}A$ -approximation of $e_{j}A$ . By $e_{j}J\varepsilon _{j} \in \mathsf {add}\, \varepsilon _{n} A \varepsilon _{j}$ , we obtain that $e_{j}J \varepsilon _{j}\otimes _{\varepsilon _{j}A \varepsilon _{j}} \varepsilon _{j}A \in \mathsf {add}\, \varepsilon _{n}A$ . Take a minimal right $\mathsf {add}\, \varepsilon _{j+1}A$ -approximation $\varphi ^{\prime }_{j}: P' \to e_{j}A$ of $e_{j}A$ . Then $P' \in \mathsf {add}\, \varepsilon _{n}A$ . On the other hand, we have that $\varepsilon _{n} J \varepsilon _{n} =0$ since $\varepsilon _{n} J \varepsilon _{n} \in \mathsf {add}\, \varepsilon _{n} A \varepsilon _{n}$ and $\varepsilon _{n} A \varepsilon _{n}$ is an indecomposable $\varepsilon _{n} A \varepsilon _{n}$ -module. Thus, we obtain that $A\varepsilon _{n}A$ is a heredity ideal of A since $A \varepsilon _{n}A \in \mathsf {proj}\, A$ . By [Reference Burgess and Fuller4, Lemma 1.7], $\varphi ^{\prime }_{j}$ is a monomorphism, and hence the assertion holds.

Next, we assume (2). We show that (3.3) is a total right rejective chain by induction on n. If $n=1$ , then this is clear. Let $n \ge 2$ . If $j=1$ , then $\varepsilon _{2}A\varepsilon _{2}$ satisfies (1). Thus, we obtain a total right rejective chain of $\mathsf {proj}\, \varepsilon _{2}A\varepsilon _{2}$ . If $j>1$ , then we also have a total right rejective chain of $\mathsf {proj}\, \varepsilon _{2}A\varepsilon _{2}$ by induction hypothesis. Since $\operatorname {pd}\nolimits S(e_{1})=1$ , it follows from Proposition 2.3(3) that $\mathsf {add}\, \varepsilon _{2}A$ is a cosemisimple right rejective subcategory of $\mathsf {proj}\, A$ . Since an equivalence $\mathsf {add}\, \varepsilon _{i}A \varepsilon _{2} \simeq \mathsf {add}\, \varepsilon _{i}A$ holds for each $2 \le i \le n$ , the chain of subcategories

$$ \begin{align*} \mathsf{proj}\, A \supset \mathsf{add}\, \varepsilon_{2} A \supset \cdots \supset \mathsf{add}\, \varepsilon_{n} A \supset 0. \end{align*} $$

satisfies that $\mathsf {add}\, \varepsilon _{2}A$ is a cosemisimple right rejective subcategory of $\mathsf {proj}\, A$ and $\mathsf {add}\, \varepsilon _{2} A \supset \cdots \supset \mathsf {add}\, \varepsilon _{n} A \supset 0$ is a total right rejective chain. Thus, each minimal right $\mathsf {add}\, \varepsilon _{i}A$ -approximation $\varphi : P \to Q$ of $Q \in \mathsf {add}\, \varepsilon _{2}A$ is a monomorphism in $\mathsf {proj}\, \varepsilon _{2}A\varepsilon _{2}$ . Since $\mathsf {add}\, \varepsilon _{2} A$ is a cosemisimple subcategory of $\mathsf {proj}\, A$ , there exists $l \ge 0$ such that $\operatorname {Ker}\nolimits \varphi \cong S(e_{1})^{\oplus l}$ by Lemma 3.8. Suppose to the contrary that $l>0$ . Due to our assumption, we have that $\operatorname {Ker}\nolimits \varphi \in \mathsf {proj}\, A$ . Thus $S(e_{1}) \in \mathsf {proj}\, A$ , a contradiction. Hence, the proof is complete.▪

Proof. Since $e_{i}J\varepsilon _{i} =0 \in \mathsf {add}\, \varepsilon _{n} A \varepsilon _{i}$ , it is enough to prove $A \varepsilon _{n} A \in \mathsf {proj}\, A$ . By $\operatorname {rad}\nolimits _{A}(\varepsilon _{i}A, e_{i}A) =e_{i}J\varepsilon _{i}=0$ , we obtain that $\operatorname {Hom}\nolimits _{A}(P(e_{n}), e_{i}J)=0$ holds for each $1 \le i \le n$ . Hence $A \varepsilon _{n}A =(e_{1} + \cdots +e_{n})A \varepsilon _{n}A =\varepsilon _{n}A \in \mathsf {add}\, \varepsilon _{n}A$ .▪

Proof. Let A be a Nakayama algebra and $\varepsilon $ an idempotent of A. Note that $\varepsilon A \varepsilon $ is also a Nakayama algebra since $\operatorname {Hom}\nolimits _{A}(\varepsilon A, -)$ is an exact and dense functor. We may assume that A is connected and fix a complete set $(e_{1},e_{2},\ldots , e_{n})$ of primitive orthogonal idempotents. By [Reference Anderson and Fuller2, Theorem 32.4], we can order the primitive idempotents such that there are projective covers

(3.4) $$ \begin{align} e_{i} A \to e_{i+1}J \to 0 \end{align} $$

for each $1 \le i \le n-1$ and

(3.5) $$ \begin{align} e_{n} A \to e_{1}J \to 0 \end{align} $$

if $e_{1}J \neq 0$ .

(1) If A has a simple projective module $S(e)$ , then $(1-e)A(1-e)$ is also a Nakayama algebra with a simple projective module by [Reference Anderson and Fuller2, Theorem 32.4]. It follows from Corollary 3.10 that $\mathsf {proj}\, A$ admits a total right rejective chain.

We assume that there exists a heredity ideal of A. We prove that A satisfies the condition (1) in Theorem 3.9 by induction on n. If $n=1$ , then this is clear. Let $n \geq 2$ . We assume that A has no simple projective modules. Then $e_{i}J \neq 0$ holds for each $1 \le i \le n$ . By (3.4) and (3.5), we obtain an exact sequence $e_{i} A \to e_{i+1}J \to 0$ for each $1 \le i \le n$ , where $e_{n+1}:=e_{1}$ . Since A has a heredity ideal, we may assume that $Ae_{n}A$ is a heredity ideal of A. Then we have a composition map $e_{n}A \to e_{1} J \to e_{1}A$ . Since $Ae_{n}A$ is a heredity ideal of A, this composition map is a monomorphism by [Reference Burgess and Fuller4, Lemma 1.7]. Thus, we obtain that $e_{n}A \cong e_{1} J \neq 0$ , and hence $\operatorname {pd}\nolimits S(e_{1})=1$ . Let $\varepsilon _{2}:=1-e_{1}$ . Then $\varepsilon _{2} A e_{n} A\varepsilon _{2} \in \mathsf {proj}\, \varepsilon _{2} A \varepsilon _{2}$ and $e_{n} J(\varepsilon _{2} A\varepsilon _{2})e_{n}=e_{n}J(A)e_{n}=0$ hold. Thus, $\varepsilon _{2} A\varepsilon _{2}$ is a Nakayama algebra with a heredity ideal $\varepsilon _{2} A e_{n} A\varepsilon _{2}$ . By induction hypothesis, $\varepsilon _{2} A \varepsilon _{2}$ satisfies the condition (1) in Theorem 3.9. Since $A e_{n} A \in \mathsf {proj}\, A$ and $e_{1} J \in \mathsf {add}\, e_{n} A$ , we have the assertion.

(2) Let A be an algebra of finite global dimension. By (1), we assume that A has no simple projective modules. Take an indecomposable projective module $e_{i}A$ such that its Loewy length $LL(e_{i}A)$ is maximal. By (3.4) and (3.5), we have $LL(e_{i-1}A)\ge LL(e_{i}A)-1$ . If $LL(e_{i-1}A)=LL(e_{i}A)-1$ , then $\operatorname {pd}\nolimits S(e_{i})=1$ . Thus, we assume that $LL(e_{i-1}A)>LL(e_{i}A)-1$ . Then $LL(e_{i-1}A)=LL(e_{i}A)$ by maximality of $LL(e_{i}A)$ . By $\operatorname {gldim}\nolimits A <\infty $ , A is non-self-injective. Therefore, we obtain a simple module with projective dimension exactly one by replacing $e_{i}A$ with $e_{i-1}A$ and repeating this argument. Let $S(e)$ be the simple module with projective dimension one. Since it follows from Proposition 2.3(2) that $\operatorname {gldim}\nolimits (1-e)A(1-e) \le \operatorname {gldim}\nolimits A<\infty $ , we inductively obtain that A satisfies the condition (3) in Theorem 3.5. Hence, $\mathsf {proj}\, A$ admits a right rejective chain.▪

Proof. (1) (c) $\Rightarrow{}$ (b) and (b) $\Rightarrow{}$ (a) follow from Theorem 3.5 and Proposition 3.2, respectively. (a) $\Rightarrow{}$ (c) follows from Proposition 3.13(2). Moreover, if A has n simple modules, then $\operatorname {gldim}\nolimits A \le 2n-2$ by Remark 2.6.

(2) (c) $\Rightarrow{}$ (b) and (b) $\Rightarrow{}$ (a) are clear. By Proposition 3.13(1), if A has a heredity ideal, then $\mathsf {proj}\, A$ has a total right rejective chain. Thus, (a) $\Rightarrow{}$ (c) holds by Proposition 2.8.▪

Proof. (1) Let $\varphi : P \to Q$ be a monomorphism in $\mathsf {add}\, \varepsilon A$ . Since $\mathsf {add}\, \varepsilon A$ is a cosemisimple subcategory of $\mathsf {proj}\, A$ , there exists $l \ge 0$ such that $\operatorname {Ker}\nolimits \varphi \cong S(e)^{\oplus l}$ by Lemma 3.8. Thus $\rho ^{\oplus l}: P(e)^{\oplus l} \to \operatorname {Ker}\nolimits \varphi $ is a projective cover of $\operatorname {Ker}\nolimits \varphi $ , where $\rho : P(e) \to S(e)$ is a projective cover of $S(e)$ . On the other hand, there exists an indecomposable direct summand $P'$ of P such that $S(e)$ is a direct summand of $\operatorname {soc}\nolimits P'$ . Let $\iota : S(e) \to P'$ be the inclusion. Then a composition map $\iota \rho : P(e) \to P'$ is a nonzero morphism between indecomposable projective modules. Since $\iota \rho $ is a monomorphism, $\rho $ is an isomorphism. Thus $\operatorname {Ker}\nolimits \varphi \in \mathsf {proj}\, A$ .

(2) Let $\operatorname {soc}\nolimits A =S(e_{i_{1}}) \oplus \cdots \oplus S(e_{i_{t}})$ , where $S(e_{i_{j}})$ is a simple module. For each $1 \le j \le t$ , there exists an indecomposable projective module $P_{j}$ such that $S(e_{i_{j}})$ is a direct summand of $\operatorname {soc}\nolimits P_{j}$ . Since a composition map of a projective cover $p : P(e_{i_{j}}) \to S(e_{i_{j}})$ and the inclusion $S(e_{i_{j}}) \to P_{j}$ is a monomorphism, p is an isomorphism. Thus $\operatorname {soc}\nolimits A \in \mathsf {proj}\, A$ .

(3) We show that there exists an indecomposable projective module P such that its Loewy length $LL (P)$ is two. If $LL(A)=2$ , then this is clear. We assume that $LL (A)\ge 3$ . Let Q be an indecomposable projective module with $LL (Q)=:l \ge 3$ . Then $QJ^{l-2}/QJ^{l-1}\cong S(e_{i_{1}}) \oplus \cdots \oplus S(e_{i_{s}}) \neq 0$ . Note that there exists $1 \le j \le s$ such that $S(e_{i_{j}}) \not \in \mathsf {proj}\, A$ . Thus, $LL(P(e_{i_{j}})) \ge 2$ . Since a composition map of $P(e_{i_{j}}) \to QJ^{l-2}$ and $QJ^{l-2} \to Q$ is a monomorphism, so is $P(e_{i_{j}}) \to QJ^{l-2}$ . Since $LL( P(e_{i_{j}}))\le 2$ , $P(e_{i_{j}})$ is a desired projective module. Thus, we obtain $\operatorname {pd}\nolimits S(e_{i_{j}})=1$ since it follows from (2) that $\operatorname {soc}\nolimits P(e_{i_{j}}) \in \mathsf {proj}\, A$ .▪

Proof. Let A be a locally hereditary algebra. If A is semisimple, then it clearly satisfies the conditions (1) and (2) in Theorem 3.9. Thus, we assume that A is not semisimple. First, we show that A satisfies the condition (1). By Lemma 3.15(2), there exists a simple module $S(e)$ such that $\operatorname {pd}\nolimits S(e)=0$ . Let $\varepsilon :=1-e$ . Then $\varepsilon A \varepsilon $ is also a locally hereditary algebra by an equivalence of categories $\mathsf {add}\, \varepsilon A \simeq \mathsf {proj}\, \varepsilon A\varepsilon $ . Due to Corollary 3.10, A inductively satisfies the condition (1) in Theorem 3.9.

Next, we show that A satisfies the condition (2). By Lemma 3.15(3), there exists a simple module $S(e)$ with $\operatorname {pd}\nolimits S(e)=1$ . Let $\varepsilon :=1-e$ . Then $\mathsf {add}\, \varepsilon A$ is a cosemisimple subcategory of $\mathsf {proj}\, A$ by Proposition 2.3(3). For each monomorphism $\varphi : P\to Q$ in $\mathsf {add}\, \varepsilon A$ , we obtain that $\operatorname {Ker}\nolimits \varphi \in \mathsf {proj}\, A$ by Lemma 3.15(1). Since $\varepsilon A \varepsilon $ is also a locally hereditary algebra, A inductively satisfies the condition (2) in Theorem 3.9.▪

Proof. If $\operatorname {gldim}\nolimits A \le 1$ , then A is a locally hereditary algebra. Thus, the assertion follows from Proposition 3.16. We assume that $\operatorname {gldim}\nolimits A =2$ . Let S be an A-module such that its length $\ell (S)$ is minimal among A-modules with the projective dimension one. We show that S is a simple module. Suppose to the contrary that S is not simple. Then there exists an exact sequence $0 \to S' \to S \to S/S' \to 0$ such that $S' \neq 0$ and $S/S' \neq 0$ . Since $\operatorname {pd}\nolimits S'\le \max \{\operatorname {pd}\nolimits S, \operatorname {pd}\nolimits S/S' -1\} = 1$ , it follows from the assumption on S that $\operatorname {pd}\nolimits S'=0$ . Thus, we have $\operatorname {pd}\nolimits S/S' \le \max \{\operatorname {pd}\nolimits S, \operatorname {pd}\nolimits S' +1\} = 1$ . By the assumption on S, we have $\operatorname {pd}\nolimits S/S'=0$ , a contradiction. Thus S is a simple module. We put $S(e):=S$ and $\varepsilon :=1-e$ . Then $\mathsf {add}\, \varepsilon A$ is a cosemisimple right rejective subcategory of $\mathsf {proj}\, A$ by Proposition 2.3(3). Let $\varphi : P \to Q$ be a monomorphism in $\mathsf {add}\, \varepsilon A$ . Then, we have an exact sequence in $\mathsf {mod}\, A$ .

$$ \begin{align*} \operatorname{Ker}\nolimits \varphi \to P \to Q \to \operatorname{Cok}\nolimits \varphi \to 0. \end{align*} $$

By $\operatorname {gldim}\nolimits A = 2$ , we have that $\operatorname {Ker}\nolimits \varphi \in \mathsf {proj}\, A$ . Since $\operatorname {gldim}\nolimits \varepsilon A \varepsilon \le \operatorname {gldim}\nolimits A=2$ by Proposition 2.3(2), we inductively obtain that A satisfies the condition (2) in Theorem 3.9. Hence, we have the assertion.▪

Proof of Theorem 3.12

Let A be an algebra in Theorem 3.12. By Proposition 2.8, it is enough to show that there exists a total right rejective chain of $\mathsf {proj}\, A$ . Thus, the assertion follows from Propositions 3.13, 3.16 and 3.17.▪