Timeline for How to memorise (understand) Nakayama's lemma and its corollaries?

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May 16, 2013 at 19:00	comment	added	Ryan Reich		@Toink: Because by flatness we know that the sequence you wrote is actually of the form $0 \to K \otimes k \to R^n \otimes k \to M \otimes k \to 0$. Therefore $K \otimes k = 0$, so by Nakayama $K = 0$. Without flatness we'd only have $K \otimes k \to R^n \otimes k \to M \otimes k \to 0$, and thus $K \otimes k \twoheadrightarrow \operatorname{ker}(R^n \otimes k \to M \otimes k) = 0$, which is exactly zero information about $K$ :)
May 16, 2013 at 18:53	comment	added	Toink		I don't understand the last example. if you reduce that s.e.s. to $k$, you get an exact sequence $0\to 0\to k^n\to M\otimes k\to 0$. But we knew that already and you don't need flatness of M for that. How does it follow that in the original sequence $K$ vanishes?
Apr 12, 2011 at 19:51	history	edited	Ryan Reich	CC BY-SA 3.0	small fixes
Apr 12, 2011 at 19:41	history	answered	Ryan Reich	CC BY-SA 3.0