Exercise 12. 8. 7, page 510 0f Grillet's Abstract Algebra

Question

In the exercise:

For every $R$-module $A$, show that $pd(A)=n$ implies $Ext_R^n(A, R) \neq 0.$ It is true for every $R$-module $A$ ? I think that $A$ should be finitely generated.

Answer 1

Napo, I agree: it looks as though a hypothesis is missing. You are certainly right that the claim is provable if we add the condition of finite generation.

There is a problem with the exercise even in a very classical context: if we take $R = \mathbb{Z}$ and $n=1$, then the condition $\text{pd}(A) = 1$ is equivalent to $A$'s being nonprojective or nonfree. So even in this special case the exercise asks us to prove that $A$ nonfree (as a $\mathbb{Z}$-module) implies $\text{Ext}^1(A, \mathbb{Z}) \neq 0$. This is equivalent to the http://en.wikipedia.org/wiki/Whitehead_problem which is famously undecidable in ZFC.

If this is an exercise in a course and there are indeed no missing hypotheses, then you should bring this to the attention of your instructor, and perhaps consider also contacting the author of the textbook.