# Exact sequences

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 !?

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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

## 1 Answer

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.

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