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I'd like to give a comment for fpqc localness(not affine case) after the answer of BCnrd and Hovey. (I'm so sorry that I don't have enough reputation to add a comment). The example of Hovey can't disprove the Zariski localness of projectiveness because that can be reduced to affine case. But this example contradicts the fpqc localness. Famous Wedderburn's theorem tell us that a commutative semisimple ring is a finite product of fields. Let $R$ be an infinite product of fields, which is a $0-$dimensional reduced ring. So, localization of $R$ at primes are fields. If we assume fpqc decent of local projectiveness, we conclude that every $R-$module is projective because it's free after localization at primes. So, $R$ is semisimple then $R$ is a finite product of fields, which is obviously impossible. So when we want to use the decent of local projectiveness, we had better to assure we can reduce to the case of ring map or we just need the fppf decent.

It's curious that the only finitely generated maximal ideals of $R$ are which corresponds to principal ultrafilters. So we also get examples of modules which are finite flat but not projective.

I'd like to give a comment for fpqc localness(not affine case) after the answer of BCnrd and Hovey. (I'm so sorry that I don't have enough reputation to add a comment). The example of Hovey can't disprove the Zariski localness of projectiveness because that can be reduced to affine case. But this example contradicts the fpqc localness. Famous Wedderburn's theorem tell us that a commutative semisimple ring is a finite product of fields. Let $R$ be an infinite product of fields, which is a $0-$dimensional reduced ring. So, localization of $R$ at primes are fields. If we assume fpqc decent of local projectiveness, we conclude that every $R-$module is projective because it's free after localization at primes. So, $R$ is semisimple then $R$ is a finite product of fields, which is obviously impossible. So when we want to use the decent of local projectiveness, we had better assure we can reduce to the case of ring map or we just need the fppf decent.

It's curious that the only finitely generated maximal ideals of $R$ are which corresponds to principal ultrafilters. So we also get examples of modules which are finite flat but not projective.

I'd like to give a comment for fpqc localness(not affine case) after the answer of BCnrd and Hovey. (I'm so sorry that I don't have enough reputation to add a comment). The example of Hovey can't disprove the Zariski localness of projectiveness because that can be reduced to affine case. But this example contradicts the fpqc localness. Famous Wedderburn's theorem tell us that a commutative semisimple ring is a finite product of fields. Let $R$ be an infinite product of fields, which is a $0-$dimensional reduced ring. So, localization of $R$ at primes are fields. If we assume fpqc decent of projectiveness, we conclude that every $R-$module is projective because it's free after localization at primes. So, $R$ is semisimple then $R$ is a finite product of fields, which is obviously impossible. So when we want to use the decent of local projectiveness, we had better to assure we can reduce to the case of ring map or we just need the fppf decent.

It's curious that the only finitely generated maximal ideals of $R$ are which corresponds to principal ultrafilters. So we also get examples of modules which are finite flat but not projective.

I'd like to give a comment for fpqc localness(not affine case) after the answer of BCnrd and Hovey. (I'm so sorry that I don't have enough reputation to add a comment). The example of Hovey can't disprove the Zariski localness of projectiveness because that can be reduced to affine case. But this example contradicts the fpqc localness. Famous Wedderburn's theorem tell us that a commutative semisimple ring is a finite product of fields. Let $R$ be an infinite product of fields, which is a $0-$dimensional reduced ring. So, localization of $R$ at primes are fields. If we assume fpqc decent of local projectiveness, we conclude that every $R-$module is projective because it's free after localization at primes. So, $R$ is semisimple then $R$ is a finite product of fields, which is obviously impossible. So when we want to use the decent of local projectiveness, we had better to assure we can reduce to the case of ring map or we just need the fppf decent.

It's curious that the only finitely generated maximal ideals of $R$ are which corresponds to principal ultrafilters. So we also get examples of modules which are finite flat but not projective.

I'd like to give a comment for fpqc localness(not affine case) after the answer of BCnrd and Hovey. (I'm so sorry that I don't have enough reputation to add a comment). The example of Hovey can't disprove the Zariski localness of projectiveness because that can be reduced to affine case. But this example contradicts the fpqc localness. Famous Wedderburn's theorem tell us that a commutative semisimple ring is a finite product of fields. Let $R$ be an infinite product of fields, which is a $0-$dimensional reduced ring. So, localization of $R$ at primes are fields. If we assume fpqc decent of projectiveness, we conclude that every $R-$module is projective because it's free after localization at primes. So, $R$ is semisimple then $R$ is a finite product of fields, which is obviously impossible. So we can use fppf decent of projectiveness at most.

It's curious that the only finitely generated maximal ideals of $R$ are which corresponds to principal ultrafilters. So we also get examples of modules which are finite flat but not projective.

I'd like to give a comment for fpqc localness(not affine case) after the answer of BCnrd and Hovey. (I'm so sorry that I don't have enough reputation to add a comment). The example of Hovey can't disprove the Zariski localness of projectiveness because that can be reduced to affine case. But this example contradicts the fpqc localness. Famous Wedderburn's theorem tell us that a commutative semisimple ring is a finite product of fields. Let $R$ be an infinite product of fields, which is a $0-$dimensional reduced ring. So, localization of $R$ at primes are fields. If we assume fpqc decent of projectiveness, we conclude that every $R-$module is projective because it's free after localization at primes. So, $R$ is semisimple then $R$ is a finite product of fields, which is obviously impossible. So when we want to use the decent of local projectiveness, we had better to assure we can reduce to the case of ring map or we just need the fppf decent.

It's curious that the only finitely generated maximal ideals of $R$ are which corresponds to principal ultrafilters. So we also get examples of modules which are finite flat but not projective.