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In my research I confronted with the following question. I do not have a counterexample, so I ask it here if anyone knows. I would like a positive answer, however a negative answer seems more likely to me.

A complex $C$ is said to be a minimal complex provided every morphism $\beta:C\rightarrow C$ that is homotopic to id$_C$ is an isomorphism.

Original question: Suppose that $(R,\mathfrak{m})$ is a commutative local ring and $\{F_j\}_{j\in\mathbb{N}}$ is a direct system of minimal complexes of finite free modules, each of which a minimal free resolution of $H_0(F_j)$. Then is the direct limit $F=\lim\limits_{\longrightarrow} F_j$ also a minimal complex?

The original question has a couneterexample by the answer below. Later on I modified the question as below.

Modified Question: Suppose that $(R,\mathfrak{m})$ is a commutative local ring and $\{F_j\}_{j\in\mathbb{N}}$ is a direct system of minimal complexes of finite free modules, each of which a minimal free resolution of $H_0(F_j)$, such that $$\lim\limits_{\longrightarrow}H_0(F_j)\neq 0.$$ Then is the direct limit $F=\lim\limits_{\longrightarrow} F_j$ also a minimal complex?

In my research I confronted with the following question. I do not have a counterexample, so I ask it here if anyone knows. I would like a positive answer, however a negative answer seems more likely to me.

A complex $C$ is said to be a minimal complex provided every morphism $\beta:C\rightarrow C$ that is homotopic to id$_C$ is an isomorphism.

Original question: Suppose that $(R,\mathfrak{m})$ is a commutative local ring and $\{F_j\}_{j\in\mathbb{N}}$ is a direct system of minimal complexes of finite free modules, each of which a minimal free resolution of $H_0(F_j)$. Then is the direct limit $F=\lim\limits_{\longrightarrow} F_j$ also a minimal complex?

The original question has a couneterexample by the answer below. Later on I modified the question as below.

Modified Question: Suppose that $(R,\mathfrak{m})$ is a commutative local ring and $\{F_j\}_{j\in\mathbb{N}}$ is a direct system of minimal complexes of finite free modules, each of which a minimal free resolution of $H_0(F_j)$, such that $$\lim\limits_{\longrightarrow}H_0(F_j)\neq 0.$$ Then is the direct limit $F=\lim\limits_{\longrightarrow} F_j$ also a minimal complex?

P.S.: The counterexample below definitely can not be changed simply to answer negatively the second modified question, because the main point of the counter-example below is that the resulted direct limit complex is split exact.

In my research I confronted with the following question. I do not have a counterexample, so I ask it here if anyone knows. I would like a positive answer, however a negative answer seems more likely to me.

A complex $C$ is said to be a minimal complex provided every morphism $\beta:C\rightarrow C$ that is homotopic to id$_C$ is an isomorphism.

Suppose that $(R,\mathfrak{m})$ is a commutative local ring and $\{F_j\}_{j\in\mathbb{N}}$ is a direct system of minimal complexes of finite free modules, each of which a minimal free resolution of $H_0(F_j)$, such that $$\lim\limits_{\longrightarrow}H_0(F_j)\neq 0.$$ Then is the direct limit $F=\lim\limits_{\longrightarrow} F_j$ also a minimal complex?

In my research I confronted with the following question. I do not have a counterexample, so I ask it here if anyone knows. I would like a positive answer, however a negative answer seems more likely to me.

A complex $C$ is said to be a minimal complex provided every morphism $\beta:C\rightarrow C$ that is homotopic to id$_C$ is an isomorphism.

Original question: Suppose that $(R,\mathfrak{m})$ is a commutative local ring and $\{F_j\}_{j\in\mathbb{N}}$ is a direct system of minimal complexes of finite free modules, each of which a minimal free resolution of $H_0(F_j)$. Then is the direct limit $F=\lim\limits_{\longrightarrow} F_j$ also a minimal complex?

The original question has a couneterexample by the answer below. Later on I modified the question as below.

Modified Question: Suppose that $(R,\mathfrak{m})$ is a commutative local ring and $\{F_j\}_{j\in\mathbb{N}}$ is a direct system of minimal complexes of finite free modules, each of which a minimal free resolution of $H_0(F_j)$, such that $$\lim\limits_{\longrightarrow}H_0(F_j)\neq 0.$$ Then is the direct limit $F=\lim\limits_{\longrightarrow} F_j$ also a minimal complex?

In my research I confronted with the following question. I do not have a counterexample, so I ask it here if anyone knows. I would like a positive answer, however a negative answer seems more likely to me.

A complex $C$ is said to be a minimal complex provided every morphism $\beta:C\rightarrow C$ that is homotopic to id$_C$ is an isomorphism.

Suppose that $(R,\mathfrak{m})$ is a commutative local ring and $\{F_j\}_{j\in\mathbb{N}}$ is a direct system of minimal complexes of finite free modules (e.g. each of which a minimal free resolution) such that each direct limit map induces a non-zero non-isomorphism homomorphism on the $0$-th homologies. Then is the direct limit $F=\lim\limits_{\longrightarrow} F_j$ also a minimal complex?

In my research I confronted with the following question. I do not have a counterexample, so I ask it here if anyone knows. I would like a positive answer, however a negative answer seems more likely to me.

A complex $C$ is said to be a minimal complex provided every morphism $\beta:C\rightarrow C$ that is homotopic to id$_C$ is an isomorphism.

Suppose that $(R,\mathfrak{m})$ is a commutative local ring and $\{F_j\}_{j\in\mathbb{N}}$ is a direct system of minimal complexes of finite free modules, each of which a minimal free resolution of $H_0(F_j)$, such that $$\lim\limits_{\longrightarrow}H_0(F_j)\neq 0.$$ Then is the direct limit $F=\lim\limits_{\longrightarrow} F_j$ also a minimal complex?