Definition: An $R$-module $P$ is finitely-presented if there exists an exact sequence
$$0\to K\to F\to P\to 0$$
where $F$ is free and $F,K$ are finitely-generated.
I have to prove that if
$$0\to M\xrightarrow{f} N\xrightarrow{g} P\to 0$$
is an exact sequence of $R$-modules where $M$ and $N$ are finitely-presented then $P$ is finitely-presented.
By hypothesis we have two exact sequences $0\to K_m\to R^m\to M\to 0$ and $0\to K_n\to R^n\to N\to 0$, but I don't know how to continue from here.
Can you help me? Thank you
Best Answer
Since $P \cong N / \operatorname{Im}(f)$ and there are a free $R$-module $F$ and a finitely generated $R$-module $K$ such that $N \cong F / K$, you get the exact sequence $$ 0\to K' \hookrightarrow F\to P\to 0$$ where $K' \leqslant F$ is the preimage of $\operatorname{Im}(f)$ under the projection map $\pi : F \mapsto F / K$. All what remains to do is to show that $K'$ is finitely generated.
Let $m_1, \dots, m_r$ generate $M$, and $k_1, \dots, k_s$ generate $K$, then $\operatorname{Im}(f)$ is generated by $f(m_1), \dots, f(m_r)$. For each $1 \leqslant i \leqslant r$, pick any $g_i$ in $\pi^{-1}(f(m_i))$. In $F$, one gets $K' = \langle k_1, \dots, k_s, g_1, \dots, g_r \rangle$. $K'$ is hence finitely generated.