In [4], Lü and Zhao proved the following theorem, which is one of the main results of that paper.

Theorem 1.1   Let $\matrix{ 0 \to & K\to & M\to & N \to & 0 }$ be an exact sequence in the category of quasi-$d$-Koszul modules such that $JK=K\cap JM$. Then the minimal Horseshoe lemma holds.

One of the aims of this note is to prove that the above result is a necessary and sufficient condition. More precisely, we obtain

Theorem 1.2   Let $\matrix{\xi: 0 \to & K \to & M\to & N \to & 0}$ be a short exact sequence in the category of quasi-$d$-Koszul modules. Then $JK=K\cap JM$ if and only if the "minimal Horseshoe lemma" holds with respect to $\xi$.

Moreover, we provide some applications of minimal Horseshoe lemma.

Theorem 1.3   Let $\matrix{\xi: 0 \to & K \to & M\to & N \to & 0}$ be a short exact sequence of finitely generated $R$-modules, where $R$ denotes a Noetherian semiperfect augmented algebra and $J$ denotes the Jacobson radical of $R$. Denote $\mathcal{Q}^d(R)$ the category of quasi-$d$-Koszul modules. Then we have the following statements:

(1) if minimal Horseshoe lemma is true with respect to $\xi$, then $K\in\mathcal{Q}^d(R)$ provided that $M,\;N\in\mathcal{Q}^d(R)$;

(2) if minimal Horseshoe lemma is true with respect to $\xi$ and we have $J^{d-1}\Omega^i(K)=\Omega^i(K)\cap J^{d-1}\Omega^{i}(M)$ and $J^d\Omega^i(K)=\Omega^i(K)\cap J^d\Omega^i(M)$ for all positive odd integers $i$, and $J^2\Omega^j(K)=\Omega^j(K)\cap J^2\Omega^j(M)$ for all nonnegative even integers $j$, then $N\in\mathcal{Q}^d(R)$ provided that $K,\;M\in\mathcal{Q}^d(R)$.

We end this section with the following definition:

Definition 1.4 Let $R$ be a Noetherian semiperfect augmented algebra and $J$ be the Jacobson radical of $R$. A finitely generated $R$-module $M$ is called a quasi-$d$-Koszul module provided that $M$ has a minimal graded projective resolution

such that $J\ker d_i=\ker d_i\cap J^2P_i$ for $i$ being even, and $J\ker d_i=\ker d_i\cap J^dP_i$ for $i$ being odd, where $d\geq 2$ is a fixed integer.

Lemma 2.1  Let $\matrix{\xi: 0 \to & K \to & M\to & N \to & 0}$ be a short exact sequence of finitely generated $R$-modules, where $R$ denotes a Noetherian semiperfect augmented algebra and $J$ denotes its Jacobson radical. Then $J\Omega^i(K)=\Omega^i(K)\cap J\Omega^i(M)$ for all $i\geq 0$ if and only if for any given commutative diagram

with $\mathcal{P}_*$ and $\mathcal{Q}_*$ being minimal projective resolutions of $K$ and $N$, respectively. Then we can complement the above diagram into the following commutative diagram with exact rows and columns

such that $\matrix{ \mathcal{L}_* \to & M \to & 0 }$ is also a minimal projective resolution and for all $n\geq 0$, $L_n\cong P_n\oplus Q_n$. That is, the minimal Horseshoe lemma holds.

Proof First we claim that, for $\xi$, $JK=K\cap JM$ if and only if we have the following commutative diagram with exact rows and columns

such that $P_0\rightarrow K\rightarrow 0$, $L_0\rightarrow M\rightarrow 0$ and $Q_0\rightarrow N\rightarrow 0$ are projective covers. In fact, we obtain the exact sequence

since $JK=K\cap JM$. Note that for any finitely generated $R$-module $M$, $R\otimes_{R/J}M/JM\longrightarrow M\longrightarrow 0$ is a projective cover. Now put $P_0:=R\otimes_{R/J}K/JK$, $L_0:=R\otimes_{R/J}M/JM$ and $Q_0:=R\otimes_{R/J}N/JN$. We have the following exact sequence

since $R/J$ is semisimple. Now by Snake lemma, we get the exact sequence

which implies the desired diagram. Conversely, suppose that we have the above diagram. Note that the projective cover of a module is unique up to isomorphisms. We may assume that $P_0:=R\otimes_{R/J}K/JK$, $L_0:=R\otimes_{R/J}M/JM$ and $Q_0:=R\otimes_{R/J}N/JN$. From the middle row of the diagram, we have the following exact sequence

Note that $R/J$ is semisimple, we have $JK=K\cap JM$ since we have the short exact sequence as $R/J$-modules

Now we prove the claim.

$\Rightarrow$ By the claim, $J\Omega^i(K)=\Omega^i(K)\cap J\Omega^i(M)$ for all $i\geq 0$ if and only if, for all $i\geq 0$, we have the following commutative diagram with exact rows and columns

such that $P_i$, $L_i$ and $Q_i$ are projective covers of $\Omega^i(K)$, $\Omega^i(M)$ and $\Omega^i(N)$, respectively. Now putting these commutative diagrams together, we finish the proof of necessity.

$\Leftarrow$ By the claim, it is easy to see that minimal Horseshoe lemma is true if and only if we have $J\Omega^i(K)=\Omega^i(K)\cap J\Omega^i(M)$ for all $i\geq 0$.

Proof of Theorem 1.2 By Theorem 3.1 of [4], it is enough to prove the sufficiency. In fact, by Lemma 2.1, the minimal Horseshoe lemma is true if and only if $J\Omega^i(K)=\Omega^i(K)\cap J\Omega^i(M)$ for all $i\geq 0$. In particular, we have $JK=K\cap JM$ for the case of $i=0$.

In fact, Theorem 2.8 (see [6]) is also a necessary and sufficient condition, which is immediate from Theorem 1.2. That is, we have

Corollary 2.2 Let $\matrix{ 0 \to & K\to & M\to & N \to & 0}$ be an exact sequence of nice modules. Then $JK=K\cap JM$ if and only if the minimal Horseshoe lemma holds with respect to such an exact sequence.

As an application of minimal Horseshoe lemma, we give some sufficient conditions such that the category of quasi-$d$-Koszul modules $\mathcal{Q}^d(R)$ preserves kernels of epimorphisms and cokernels of monomorphisms.

Lemma 3.1  Let $\matrix{\xi: 0 \to & K \to & M\to & N \to & 0}$ be a short exact sequence of finitely generated $R$-modules, where $R$ denotes a Noetherian semiperfect augmented algebra and $J$ denotes the Jacobson radical of $R$. If minimal Horseshoe lemma is true with respect to $\xi$, then $K\in\mathcal{Q}^d(R)$ provided that $M,\;N\in\mathcal{Q}^d(R)$.

Proof By Lemma 2.1, we have the commutative diagram

such that $P_i\rightarrow \Omega^{i}(K)\rightarrow 0$, $Q_i\rightarrow \Omega^{i}(M)\rightarrow 0$ and $L_i\rightarrow \Omega^{i}(N)\rightarrow 0$ are projective covers, respectively. Note that $M$ and $N$ are in $\mathcal{Q}^{d}(R)$, we have the following commutative diagram with exact rows and columns

for all positive odd integers $i$ and the following commutative diagram with exact rows and columns

for all nonnegative even integers $i$.

Apply the functor $R/J\otimes_R-$ to the above two diagrams, we have the following commutative diagram with exact rows and columns

for all positive odd integers $i$ and the following commutative diagram with exact rows and columns

for all nonnegative even integers $i$. Therefore, $\theta$ is a monomorphism. Thus we have $J\Omega^{i+1}(K)=\Omega^{i+1}(K)\cap J^{d}P_i$ for all positive odd integers $i$ and $J\Omega^{i+1}(K)=\Omega^{i+1}(K)\cap J^{2}P_i$ for all nonnegative even integers $i$, which implies that $K\in\mathcal{Q}^d(R)$, as desired.

Lemma 3.2  Let $\matrix{\xi: 0 \to & K \to & M\to & N \to & 0}$ be a short exact sequence of finitely generated $R$-modules, where $R$ denotes a Noetherian semiperfect augmented algebra and $J$ denotes the Jacobson radical of $R$. If minimal Horseshoe Lemma is true with respect to $\xi$ and we have $J^{d-1}\Omega^i(K)=\Omega^i(K)\cap J^{d-1}\Omega^{i}(M)$ and $J^d\Omega^i(K)=\Omega^i(K)\cap J^d\Omega^i(M)$ for all positive odd integers $i$, and $J^2\Omega^j(K)=\Omega^j(K)\cap J^2\Omega^j(M)$ for all nonnegative even integers $j$, then $N\in\mathcal{Q}^d(R)$ provided that $K,\;M\in\mathcal{Q}^d(R)$.

Proof Note that minimal Horseshoe lemma is true for $\xi$, which is equivalent to $J\Omega^i(K)=\Omega^i(K)\cap J\Omega^i(M)$ for all $i\geq 0$ by the claim in the proof of Lemma 2.1. By assumption, $J\Omega^i(K)=\Omega^i(K)\cap J\Omega^i(M)$ for all positive odd integers $i$, thus we have the exact sequences

for all positive odd integers $i$ and

for all $i\geq 0$.

Similar to the proof of (1), we have the following commutative diagrams with exact rows and columns

for all positive odd integers $i$ and

for all nonnegative even integers $i$.

Note that we have

for all positive odd integers $i$ and

for all nonnegative even integers $i$.

Now apply the functor $R/J\otimes_R-$ to the above two diagrams, we have the following commutative diagrams with exact rows and columns

for all positive odd integers $i$ and

for all nonnegative even integers $i$.

Now by "$3\times3$" lemma, we have $\zeta$ is a monomorphism, which implies that

for all positive odd integers $i$ and $J\Omega^{i+1}(N)=\Omega^{i+1}(N)\cap J^{2}L_i$ for all nonnegative even integers $i$, which implies that $N\in\mathcal{Q}^d(R)$, as desired.

Now Theorem 1.3 is immediate from Lemmas 3.1 and 3.2.