Exercise 11, § 2

Let $E$ be a right $A$-module. $E$ is called pseudo-coherent if every finitely generated submodule of $E$ is finitely presented; every submodule of a pseudo-coherent module is pseudo-coherent. $E$ is called coherent if it is pseudo-coherent and finitely generated (and therefore finitely presented).

(a) Let $0 \to E' \to E \to E'' \to 0$ be an exact sequence of right $A$-modules. Show that, if $E$ is pseudo-coherent (resp. coherent) and $E'$ is finitely generated $E''$ is pseudo-coherent (resp. coherent). Show that, if $E'$ and $E''$ are pseudo-coherent (resp. coherent), so is $E$. Show that, if $E$ and $E''$ are coherent, so is $E'$ (use Exercise 6 and Lemma 9 of no. 8).

(b) Let $E$ be a coherent $A$-module and $E'$ a pseudo-coherent (resp. coherent) $A$-module. Show that, for every homomorphism $u : E \to E'$, $\operatorname{Im}(u)$ and $\operatorname{Ker}(u)$ are coherent and that $\operatorname{Coker}(u)$ is pseudo-coherent (resp. coherent) (use (a)).

(c) Show that every direct sum (resp. every finite direct sum) of pseudo-coherent (resp. coherent) modules is a pseudo-coherent (resp. coherent) module.

(d) If $E$ is a pseudo-coherent module and $M, N$ are coherent submodules of $E$, show that $M + N$ and $M \cap N$ are coherent (use (a) and (c)).

(e) Suppose that $A$ is commutative. Show that, if $E$ is a coherent $A$-module and $F$ a coherent (resp. pseudo-coherent) $A$-module, $\operatorname{Hom}_A(E, F)$ is a coherent (resp. pseudo-coherent) $A$-module. (Reduce it to the case where $F$ is coherent and consider a finite presentation of $E$, then use (b).)