Schanuel’s lemma fails for flats

Recall that Schanuel’s lemma asserts that if 0\to K_i \to P_i\to M\to 0 are short exact sequences i=1,2 and P_i is projective, then P_1\oplus K_2\simeq P_2\oplus K_1. Show this fails if the P_i are only assumed to be flat.

 

 

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Schanuel’s lemma fails for flats

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