MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This question might be trivial but I cann't see.

Let $A$ and $B$ be two modules. is it always possible to have an exact sequence which begins with $A$, ends with $B$ with all modules in the sequence (other than $A$ and $B$) projective !?

share|cite|improve this question
Do you mean a long exact sequence? Should it begin with $0 \to A$ or just $A$? – Martin Brandenburg Jan 2 '13 at 12:05
@Martin Brandenburg: It must mean just $A$. Otherwise, it would require embedding $A$ into a projective module (or directly into $B$), and any non-free $\mathbb Z$-module would provide a counterexample. – Andreas Blass Jan 2 '13 at 14:35
Although with that interpretation, the sequence $A\rightarrow 0\rightarrow B$ provides an example. – Jeremy Rickard Jan 2 '13 at 15:01
@Jeremy: Right, I missed that. So we don't seem to have any non-trivial interpretation of the question. What if we ask for just $A$ at the beginning but $B\to0$ at the end? Your idea would handle any $B$ of finite projective dimension, but there seems to be no such exact sequence if $A$ has finite projective dimension and $B$ doesn't. (Maybe I should stop trying to de-trivialize the question and let the OP tell us what (s)he actually wants.) – Andreas Blass Jan 2 '13 at 15:23
I give for granted that the question is about extensions, thus my answer. – Fernando Muro Jan 2 '13 at 20:15
up vote 1 down vote accepted

Not in general. The keyword is stable module category, the quotient of the module category by the ideal of morphisms which factor through a projective. The leftmost term is functorial on the rightmost term in this category if all intermediate modules are projective. This imposes some restrictions. If you take a hereditary ring, eg the integers, you get easy counterexamples as any submodule of a projective module is projective.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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