I am looking for an example of an exact sequence of $R$-modules $$ 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0 $$ and a $R$-module $N$, such that $$ 0 \rightarrow M' \otimes N \rightarrow M \otimes N \rightarrow M'' \otimes N \rightarrow 0 $$ fails to be exact.
What is a typical example of the tensor product of modules failing to be left exact?
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commutative-algebra
modules
examples-counterexamples
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5Have you tried coming up with one yourself? – 2011-05-16
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0@Qiaochu yes, I did. I didn't have the right idea, but maybe I should have tried harder. Many thanks to Henri in any case. – 2011-05-16
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0In fact, I'd say it's a bit harder to produce examples where the sequence remains exact (provided you start with non-split sequences) – 2011-05-16
1 Answers
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The typical example is made up with $M=M'=\mathbb Z$, and $N= \mathbb Z/ n \mathbb Z$, with the injection $\mathbb Z \to \mathbb Z, x \mapsto n x$. When tensorized by $Id_{\mathbb Z/ n \mathbb Z}$, this map becomes zero.