When does localisation behave badly?
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Localisation seems to be a very useful tool in commutative algebra/number theory, and it seems like in every case I've come across, it behaves incredibly well.
By behaves well, I mean that it is exact, preserves integral closure, preserves noetherian-ness, and so on (at least in the natural prime ideal complement case).
Furthermore, it seems like the "data" that is lost by localising is often very clear, eg, ideals remain essentially the same, or are killed completely.
What are some instances where an important property is not preserved by localisation? Note there are easy examples of global properties, such as the class group which are trivial locally, but ideally I am after a property that is not (obviously) global in nature.
I am particularly interested in cases when the "bad behaviour" arises naturally, eg, where the localisation is with respect to the complement of a prime ideal, since one can probably cook up strange multiplicative sets to get some kind of strange behaviour.
abstract-algebra commutative-algebra algebraic-number-theory localization
asked Dec 3 '18 at 11:42
user277182user277182
431212
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add a comment |
$begingroup$
Localisation seems to be a very useful tool in commutative algebra/number theory, and it seems like in every case I've come across, it behaves incredibly well.
By behaves well, I mean that it is exact, preserves integral closure, preserves noetherian-ness, and so on (at least in the natural prime ideal complement case).
Furthermore, it seems like the "data" that is lost by localising is often very clear, eg, ideals remain essentially the same, or are killed completely.
What are some instances where an important property is not preserved by localisation? Note there are easy examples of global properties, such as the class group which are trivial locally, but ideally I am after a property that is not (obviously) global in nature.
I am particularly interested in cases when the "bad behaviour" arises naturally, eg, where the localisation is with respect to the complement of a prime ideal, since one can probably cook up strange multiplicative sets to get some kind of strange behaviour.
abstract-algebra commutative-algebra algebraic-number-theory localization
asked Dec 3 '18 at 11:42
user277182user277182
431212
$endgroup$
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I would have said that the class group is obviously global, since you can always kill it with a localization.
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– Lubin
Dec 3 '18 at 21:35
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Yea, thats the kind of global thing I had in mind, ill edit so its clearer.
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– user277182
Dec 4 '18 at 1:56
add a comment |
$begingroup$
Localisation seems to be a very useful tool in commutative algebra/number theory, and it seems like in every case I've come across, it behaves incredibly well.
By behaves well, I mean that it is exact, preserves integral closure, preserves noetherian-ness, and so on (at least in the natural prime ideal complement case).
Furthermore, it seems like the "data" that is lost by localising is often very clear, eg, ideals remain essentially the same, or are killed completely.
What are some instances where an important property is not preserved by localisation? Note there are easy examples of global properties, such as the class group which are trivial locally, but ideally I am after a property that is not (obviously) global in nature.
I am particularly interested in cases when the "bad behaviour" arises naturally, eg, where the localisation is with respect to the complement of a prime ideal, since one can probably cook up strange multiplicative sets to get some kind of strange behaviour.
abstract-algebra commutative-algebra algebraic-number-theory localization
asked Dec 3 '18 at 11:42
user277182user277182
431212
$endgroup$
Localisation seems to be a very useful tool in commutative algebra/number theory, and it seems like in every case I've come across, it behaves incredibly well.
By behaves well, I mean that it is exact, preserves integral closure, preserves noetherian-ness, and so on (at least in the natural prime ideal complement case).
Furthermore, it seems like the "data" that is lost by localising is often very clear, eg, ideals remain essentially the same, or are killed completely.
What are some instances where an important property is not preserved by localisation? Note there are easy examples of global properties, such as the class group which are trivial locally, but ideally I am after a property that is not (obviously) global in nature.
I am particularly interested in cases when the "bad behaviour" arises naturally, eg, where the localisation is with respect to the complement of a prime ideal, since one can probably cook up strange multiplicative sets to get some kind of strange behaviour.
abstract-algebra commutative-algebra algebraic-number-theory localization
abstract-algebra commutative-algebra algebraic-number-theory localization
asked Dec 3 '18 at 11:42
user277182user277182
431212
asked Dec 3 '18 at 11:42
user277182user277182
431212
edited Dec 4 '18 at 1:59
user277182
asked Dec 3 '18 at 11:42
user277182user277182
431212
asked Dec 3 '18 at 11:42
user277182user277182
431212
asked Dec 3 '18 at 11:42
user277182user277182
431212
431212
$begingroup$
I would have said that the class group is obviously global, since you can always kill it with a localization.
$endgroup$
– Lubin
Dec 3 '18 at 21:35
$begingroup$
Yea, thats the kind of global thing I had in mind, ill edit so its clearer.
$endgroup$
– user277182
Dec 4 '18 at 1:56
add a comment |
$begingroup$
I would have said that the class group is obviously global, since you can always kill it with a localization.
$endgroup$
– Lubin
Dec 3 '18 at 21:35
$begingroup$
Yea, thats the kind of global thing I had in mind, ill edit so its clearer.
$endgroup$
– user277182
Dec 4 '18 at 1:56
$begingroup$
I would have said that the class group is obviously global, since you can always kill it with a localization.
$endgroup$
– Lubin
Dec 3 '18 at 21:35
$begingroup$
I would have said that the class group is obviously global, since you can always kill it with a localization.
$endgroup$
– Lubin
Dec 3 '18 at 21:35
$begingroup$
Yea, thats the kind of global thing I had in mind, ill edit so its clearer.
$endgroup$
– user277182
Dec 4 '18 at 1:56
$begingroup$
Yea, thats the kind of global thing I had in mind, ill edit so its clearer.
$endgroup$
– user277182
Dec 4 '18 at 1:56
add a comment |
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$begingroup$
I would have said that the class group is obviously global, since you can always kill it with a localization.
$endgroup$
– Lubin
Dec 3 '18 at 21:35
$begingroup$
Yea, thats the kind of global thing I had in mind, ill edit so its clearer.
$endgroup$
– user277182
Dec 4 '18 at 1:56