The minimal complexes and the direct limit

Question

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.

Answer 1

Here's a "minimal" counterexample. Let $R$ be the localization of $k[x]$ at $(x)$, and let $F_j := R \to^x R$ for every $j$, where $\to^x$ denotes multiplication by $x$. Consider the direct system $$\dots \to F_{j-1} \to^x F_j \to^x F_{j+1}\to \dots$$ Then the colimit is $k(x) \to^x k(x)$ and the $0$ map is homotopic to the identity via $x^{-1}$.