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Construction of $(p)$-local spectrum.
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I have a question regarding the construction of the $(p)$-local spectrum of a given spectrum $X$. I have seen in many papers people mentioning it, but never anyone giving the precise definition. So I was wondering if anyone is able to give the definition and construction of the $p$-local spectrum associated to a spectrum $X$, or even better to provide a reference of it.
algebraic-topology homotopy-theory spectra
asked Jul 25 at 17:52
user430191
1557
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up vote
2
down vote
favorite
I have a question regarding the construction of the $(p)$-local spectrum of a given spectrum $X$. I have seen in many papers people mentioning it, but never anyone giving the precise definition. So I was wondering if anyone is able to give the definition and construction of the $p$-local spectrum associated to a spectrum $X$, or even better to provide a reference of it.
algebraic-topology homotopy-theory spectra
asked Jul 25 at 17:52
user430191
1557
A. Bousfield, The localization of spectra with respect to homology, Topology vol. 18, 1979
– xsnl
Jul 25 at 18:20
I have checked that paper (which is the starting point of this idea); so is $X_(p)$ just defined the Bousfield localization with respect a prime number $p$?
– user430191
Jul 25 at 18:27
1
If you want some explicit construction, then you can take your favourite spectrum $M$ and take homotopy colimit of diagram looking like $M to M to M dots$, where arrows are multiplication by all $p$-free natural numbers in some order. Informally, localization with respect to some good class of morphisms (e. g. giving isomorphism on some homology theory) is obtained by taking homotopy colimit by all endomorphisms in that class. (It's not true as stated, but IMO gives good sense of how it looks).
– xsnl
Jul 25 at 18:29
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I have a question regarding the construction of the $(p)$-local spectrum of a given spectrum $X$. I have seen in many papers people mentioning it, but never anyone giving the precise definition. So I was wondering if anyone is able to give the definition and construction of the $p$-local spectrum associated to a spectrum $X$, or even better to provide a reference of it.
algebraic-topology homotopy-theory spectra
asked Jul 25 at 17:52
user430191
1557
I have a question regarding the construction of the $(p)$-local spectrum of a given spectrum $X$. I have seen in many papers people mentioning it, but never anyone giving the precise definition. So I was wondering if anyone is able to give the definition and construction of the $p$-local spectrum associated to a spectrum $X$, or even better to provide a reference of it.
algebraic-topology homotopy-theory spectra
asked Jul 25 at 17:52
user430191
1557
asked Jul 25 at 17:52
user430191
1557
asked Jul 25 at 17:52
user430191
1557
asked Jul 25 at 17:52
user430191
1557
1557
A. Bousfield, The localization of spectra with respect to homology, Topology vol. 18, 1979
– xsnl
Jul 25 at 18:20
I have checked that paper (which is the starting point of this idea); so is $X_(p)$ just defined the Bousfield localization with respect a prime number $p$?
– user430191
Jul 25 at 18:27
1
If you want some explicit construction, then you can take your favourite spectrum $M$ and take homotopy colimit of diagram looking like $M to M to M dots$, where arrows are multiplication by all $p$-free natural numbers in some order. Informally, localization with respect to some good class of morphisms (e. g. giving isomorphism on some homology theory) is obtained by taking homotopy colimit by all endomorphisms in that class. (It's not true as stated, but IMO gives good sense of how it looks).
– xsnl
Jul 25 at 18:29
add a comment |Â
A. Bousfield, The localization of spectra with respect to homology, Topology vol. 18, 1979
– xsnl
Jul 25 at 18:20
I have checked that paper (which is the starting point of this idea); so is $X_(p)$ just defined the Bousfield localization with respect a prime number $p$?
– user430191
Jul 25 at 18:27
1
If you want some explicit construction, then you can take your favourite spectrum $M$ and take homotopy colimit of diagram looking like $M to M to M dots$, where arrows are multiplication by all $p$-free natural numbers in some order. Informally, localization with respect to some good class of morphisms (e. g. giving isomorphism on some homology theory) is obtained by taking homotopy colimit by all endomorphisms in that class. (It's not true as stated, but IMO gives good sense of how it looks).
– xsnl
Jul 25 at 18:29
A. Bousfield, The localization of spectra with respect to homology, Topology vol. 18, 1979
– xsnl
Jul 25 at 18:20
A. Bousfield, The localization of spectra with respect to homology, Topology vol. 18, 1979
– xsnl
Jul 25 at 18:20
I have checked that paper (which is the starting point of this idea); so is $X_(p)$ just defined the Bousfield localization with respect a prime number $p$?
– user430191
Jul 25 at 18:27
I have checked that paper (which is the starting point of this idea); so is $X_(p)$ just defined the Bousfield localization with respect a prime number $p$?
– user430191
Jul 25 at 18:27
1
1
If you want some explicit construction, then you can take your favourite spectrum $M$ and take homotopy colimit of diagram looking like $M to M to M dots$, where arrows are multiplication by all $p$-free natural numbers in some order. Informally, localization with respect to some good class of morphisms (e. g. giving isomorphism on some homology theory) is obtained by taking homotopy colimit by all endomorphisms in that class. (It's not true as stated, but IMO gives good sense of how it looks).
– xsnl
Jul 25 at 18:29
If you want some explicit construction, then you can take your favourite spectrum $M$ and take homotopy colimit of diagram looking like $M to M to M dots$, where arrows are multiplication by all $p$-free natural numbers in some order. Informally, localization with respect to some good class of morphisms (e. g. giving isomorphism on some homology theory) is obtained by taking homotopy colimit by all endomorphisms in that class. (It's not true as stated, but IMO gives good sense of how it looks).
– xsnl
Jul 25 at 18:29
add a comment |Â
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A. Bousfield, The localization of spectra with respect to homology, Topology vol. 18, 1979
– xsnl
Jul 25 at 18:20
I have checked that paper (which is the starting point of this idea); so is $X_(p)$ just defined the Bousfield localization with respect a prime number $p$?
– user430191
Jul 25 at 18:27
1
If you want some explicit construction, then you can take your favourite spectrum $M$ and take homotopy colimit of diagram looking like $M to M to M dots$, where arrows are multiplication by all $p$-free natural numbers in some order. Informally, localization with respect to some good class of morphisms (e. g. giving isomorphism on some homology theory) is obtained by taking homotopy colimit by all endomorphisms in that class. (It's not true as stated, but IMO gives good sense of how it looks).
– xsnl
Jul 25 at 18:29