Exercise 10, § 3

Let $E$ be an $A$-module. We say that $E$ is pseudo-coherent if every finitely generated submodule of $E$ is of finite presentation; every submodule of a pseudo-coherent module is pseudo-coherent. We say that $E$ is coherent if it is pseudo-coherent and of finite type (hence of finite presentation). a) Let $0 \to E' \to E \to E'' \to 0$ be an exact sequence of $A$-modules. Prove that, if $E$ is pseudo-coherent (resp. coherent) and $E'$ is of finite type, $E''$ is pseudo-coherent (resp. coherent). Prove that, if $E'$ and $E''$ are pseudo-coherent (resp. coherent), the same is true of $E$. Prove that, if $E$ and $E''$ are coherent, the same is true of $E'$ (use $X$, p. 11, prop. 6). b) Let $E$ be a coherent $A$-module, and let $E'$ be a pseudo-coherent (resp. coherent) $A$-module. Prove that, for every homomorphism $u : E \to E'$, $\mathrm{Im}\,(u)$ and $\mathrm{Ker}\,(u)$ are coherent and $\mathrm{Coker}\,(u)$ is pseudo-coherent (resp. coherent) (use a). c) Prove 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 if $M, N$ are coherent submodules of $E$, prove that $M + N$ and $M \cap N$ are coherent (use a and c). e) Suppose that $A$ is commutative. Prove that, if $E$ is a coherent $A$-module and $F$ is a coherent (resp. pseudo-coherent) $A$-module, $\mathrm{Hom}_A(E, F)$ is a coherent (resp. pseudo-coherent) $A$-module. (Reduce to the case where $F$ is coherent; consider a finite presentation of $E$, then use b.)