Let be an integral domain, ideals such that . If is principal, then and are projective.

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# Wish I had known this in 2012

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2 thoughts on “Wish I had known this in 2012”

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Let be an integral domain, ideals such that . If is principal, then and are projective.

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The sequence , where the last arrow is , is exact by hypothesis. Now is principal and therefore isomorphic to , since is a domain, and the sequence splits, since the rightmost term is projective. Then and so both ideals are projective.

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Consider the canonical exact sequence involving and .

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