# Finitely Presented Modules Definition

I am a little bit confused with the definition of finitely presented modules. In Lang's Algebra he defines a module $$M$$ to be finitely presented if and only if there is a exact sequence $$F'\to F\to M \to 0$$ such that both $$F', F$$ are free. However the standard definition I have seen elsewhere only demands $$F'$$ be finitely generated. Are these two definitions equivalent?

Looking at the situation of a non-principal ideal of a ring, say $$(x, y)$$ of $$\mathbb{R}[x, y]$$, it appears that this is finitely presented, by the usual definition, but I don't see any way of making it finitely presented by Lang's definition.

If $F' \to F \to M \to 0$ is exact and $F'$ is finitely generated, choose some finitely generated free module $F''$ which maps onto $F'$. Then $F'' \to F \to M \to 0$ is exact.

This shows: A finitely generated module is finitely related iff it is finitely presented.

Of course, this fails for modules which are not finitely generated.

• I don't get it: to which question answers this? – user26857 May 7 '15 at 21:42
• I prove that the definitions are equivalent. – Martin Brandenburg May 7 '15 at 21:54
• Which definitions: the wrong definition of finitely presented modules, and the definition of related modules? (I'm simply confused.) – user26857 May 7 '15 at 21:56
• So, the OP has made a "trivial typo" (???). Thus he has spoiled the definition of finitely presented modules. (The way he "defines" a finitely presented module is totally wrong, because for each module there is an exact sequence $F'\to F\to M\to 0$ with $F,F'$ free modules.) But then which definitions are you proving that are equivalent? (Finitely presented is equivalent to finitely related??? I don't think so!) Your answer has no meaning for me, and your comments didn't shed any light on the matter. – user26857 May 8 '15 at 16:13
• Finitely generated finitely related modules are certainly finitely presented. This is what your answer is trying to say? – user26857 May 8 '15 at 23:24

In Lang's Algebra he defines a module $$M$$ to be finitely presented if and only if there is an exact sequence $$F'\to F\to M \to 0$$ such that both $$F', F$$ are free of finite rank, and this is the definition of finitely presented modules. (Note that for each module $$M$$ there is an exact sequence $$F'\to F\to M\to 0$$ with $$F,F'$$ free modules.)

"However the standard definition I have seen elsewhere only demands $$F'$$ be finitely generated." This is the definition of finitely related modules.

"Are these two definitions equivalent?" In general they aren't: Let $$M$$ be a finitely presented module, and $$L$$ a free module which is not finitely generated. Then $$M\oplus L$$ is finitely related, but not finitely presented. However, if the module is finitely generated the two definitions coincide.

In Lang's definition, both $F$ and $F'$ must be finitely generated (and free). This definition is equivalent to: ‘There exists a short exact sequence: $$0\to K\to F\to M\to 0$$ such the $T$ is a finitely generated free module, and the module of relations $K$ is finitely generated. It is equivalent because there is exists a surjective homomorphism from a finitely generated free module onto $K$.