0
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ I don't get it: to which question answers this? $\endgroup$ – user26857 May 7 '15 at 21:42
  • $\begingroup$ I prove that the definitions are equivalent. $\endgroup$ – Martin Brandenburg May 7 '15 at 21:54
  • 1
    $\begingroup$ Which definitions: the wrong definition of finitely presented modules, and the definition of related modules? (I'm simply confused.) $\endgroup$ – user26857 May 7 '15 at 21:56
  • 2
    $\begingroup$ 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. $\endgroup$ – user26857 May 8 '15 at 16:13
  • 1
    $\begingroup$ Finitely generated finitely related modules are certainly finitely presented. This is what your answer is trying to say? $\endgroup$ – user26857 May 8 '15 at 23:24
3
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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 $K$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.