This is exercise 5.29 from A Term of Commutative Algebra by A. Altman & S. Kleiman. A digital version may be found here. The statement of the exercise is the following.
Let $R$ be a ring, $X_1,X_2,\cdots$ infinitely many variables. Set $P:=R[X_1,X_2,\cdots]$ and $M:=P/\langle X_1,X_2,\cdots\rangle$. Is $M$ finitely presented? Explain.
The solution attached at the end of the book is the following.
No, otherwise by (5.18), the ideal $\langle X_1, X2, \cdots\rangle$ would be generated by some $f_1,\cdots,f_n\in P$, so also by $X_1,\cdots,X_m$ for some $m$, but plainly it isn’t.
The (5.18) in the proof above refers to the following.
Let $R$ be a ring, and $0 \rightarrow L \rightarrow R^n \rightarrow M \rightarrow 0$ an exact sequence. Prove $M$ is finitely presented if and only if $L$ is finitely generated.
However, I have the following questions:
- By $M$ finitely presented, does the author mean finitely presented as a module over $R$?
- By referring to (5.18), the author seems to be considering the following exact sequence: $$0\rightarrow \langle X_1,X_2,\cdots\rangle \rightarrow R[X_1,X_2,\cdots] \rightarrow M \rightarrow 0.$$ However, $R[X_1,X_2,\cdots]$ does not have the form $R^n$ required by (5.18).
- Isn't $M\cong R$ and therefore finitely presented?
