Mixing
Every projective module is a submodule of a free module?
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I've seen this statement on the internet but I could not find a proof. Actually this is true for any module I think. Can a proof be given as follows?
Let $M$ be an $R$-module. Take a generating set $X$ of $M$ over $R$. Then consider the free $R$-module $F$ over the set $X$. Hence $M subset F$ and we are done. I hope this is not a nonsense idea.
free-modules
asked Jan 20 at 13:46
Philip JohnsonPhilip Johnson
193
$endgroup$
add a comment |
$begingroup$
I've seen this statement on the internet but I could not find a proof. Actually this is true for any module I think. Can a proof be given as follows?
Let $M$ be an $R$-module. Take a generating set $X$ of $M$ over $R$. Then consider the free $R$-module $F$ over the set $X$. Hence $M subset F$ and we are done. I hope this is not a nonsense idea.
free-modules
asked Jan 20 at 13:46
Philip JohnsonPhilip Johnson
193
$endgroup$
$begingroup$
Why $M subset F$?
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 13:51
$begingroup$
Every projective module is isomorphic to a direct factor of a free $R$-module.
$endgroup$
– Bernard
Jan 20 at 13:53
add a comment |
$begingroup$
I've seen this statement on the internet but I could not find a proof. Actually this is true for any module I think. Can a proof be given as follows?
Let $M$ be an $R$-module. Take a generating set $X$ of $M$ over $R$. Then consider the free $R$-module $F$ over the set $X$. Hence $M subset F$ and we are done. I hope this is not a nonsense idea.
free-modules
asked Jan 20 at 13:46
Philip JohnsonPhilip Johnson
193
$endgroup$
I've seen this statement on the internet but I could not find a proof. Actually this is true for any module I think. Can a proof be given as follows?
Let $M$ be an $R$-module. Take a generating set $X$ of $M$ over $R$. Then consider the free $R$-module $F$ over the set $X$. Hence $M subset F$ and we are done. I hope this is not a nonsense idea.
free-modules
free-modules
asked Jan 20 at 13:46
Philip JohnsonPhilip Johnson
193
asked Jan 20 at 13:46
Philip JohnsonPhilip Johnson
193
asked Jan 20 at 13:46
Philip JohnsonPhilip Johnson
193
asked Jan 20 at 13:46
Philip JohnsonPhilip Johnson
193
asked Jan 20 at 13:46
Philip JohnsonPhilip Johnson
193
193
$begingroup$
Why $M subset F$?
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 13:51
$begingroup$
Every projective module is isomorphic to a direct factor of a free $R$-module.
$endgroup$
– Bernard
Jan 20 at 13:53
add a comment |
$begingroup$
Why $M subset F$?
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 13:51
$begingroup$
Every projective module is isomorphic to a direct factor of a free $R$-module.
$endgroup$
– Bernard
Jan 20 at 13:53
$begingroup$
Why $M subset F$?
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 13:51
$begingroup$
Why $M subset F$?
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 13:51
$begingroup$
Every projective module is isomorphic to a direct factor of a free $R$-module.
$endgroup$
– Bernard
Jan 20 at 13:53
$begingroup$
Every projective module is isomorphic to a direct factor of a free $R$-module.
$endgroup$
– Bernard
Jan 20 at 13:53
add a comment |
1 Answer
1
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$begingroup$
What you get with your $X$ (a generating set of $M$) and your $F$ (free on $X$)
is a surjective homomorphism $pi:Fto M$, not an injection $Mto F$.
But if $M$ is projective, the surjection $pi$ splits, that is there's
an injective homomorphism $iota:Mto F$ with $picirciota=text{id}_M$.
answered Jan 20 at 13:52
Lord Shark the UnknownLord Shark the Unknown
105k1160133
$endgroup$
$begingroup$
Ok, that's the importance of projectivity! Thanks man.
$endgroup$
– Philip Johnson
Jan 20 at 13:59
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
What you get with your $X$ (a generating set of $M$) and your $F$ (free on $X$)
is a surjective homomorphism $pi:Fto M$, not an injection $Mto F$.
But if $M$ is projective, the surjection $pi$ splits, that is there's
an injective homomorphism $iota:Mto F$ with $picirciota=text{id}_M$.
answered Jan 20 at 13:52
Lord Shark the UnknownLord Shark the Unknown
105k1160133
$endgroup$
$begingroup$
Ok, that's the importance of projectivity! Thanks man.
$endgroup$
– Philip Johnson
Jan 20 at 13:59
add a comment |
$begingroup$
What you get with your $X$ (a generating set of $M$) and your $F$ (free on $X$)
is a surjective homomorphism $pi:Fto M$, not an injection $Mto F$.
But if $M$ is projective, the surjection $pi$ splits, that is there's
an injective homomorphism $iota:Mto F$ with $picirciota=text{id}_M$.
answered Jan 20 at 13:52
Lord Shark the UnknownLord Shark the Unknown
105k1160133
$endgroup$
$begingroup$
Ok, that's the importance of projectivity! Thanks man.
$endgroup$
– Philip Johnson
Jan 20 at 13:59
add a comment |
$begingroup$
What you get with your $X$ (a generating set of $M$) and your $F$ (free on $X$)
is a surjective homomorphism $pi:Fto M$, not an injection $Mto F$.
But if $M$ is projective, the surjection $pi$ splits, that is there's
an injective homomorphism $iota:Mto F$ with $picirciota=text{id}_M$.
answered Jan 20 at 13:52
Lord Shark the UnknownLord Shark the Unknown
105k1160133
$endgroup$
What you get with your $X$ (a generating set of $M$) and your $F$ (free on $X$)
is a surjective homomorphism $pi:Fto M$, not an injection $Mto F$.
But if $M$ is projective, the surjection $pi$ splits, that is there's
an injective homomorphism $iota:Mto F$ with $picirciota=text{id}_M$.
answered Jan 20 at 13:52
Lord Shark the UnknownLord Shark the Unknown
105k1160133
answered Jan 20 at 13:52
Lord Shark the UnknownLord Shark the Unknown
105k1160133
answered Jan 20 at 13:52
Lord Shark the UnknownLord Shark the Unknown
105k1160133
answered Jan 20 at 13:52
Lord Shark the UnknownLord Shark the Unknown
105k1160133
105k1160133
$begingroup$
Ok, that's the importance of projectivity! Thanks man.
$endgroup$
– Philip Johnson
Jan 20 at 13:59
add a comment |
$begingroup$
Ok, that's the importance of projectivity! Thanks man.
$endgroup$
– Philip Johnson
Jan 20 at 13:59
$begingroup$
Ok, that's the importance of projectivity! Thanks man.
$endgroup$
– Philip Johnson
Jan 20 at 13:59
$begingroup$
Ok, that's the importance of projectivity! Thanks man.
$endgroup$
– Philip Johnson
Jan 20 at 13:59
add a comment |
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$begingroup$
Why $M subset F$?
$endgroup$
– Tommaso Scognamiglio
Jan 20 at 13:51
$begingroup$
Every projective module is isomorphic to a direct factor of a free $R$-module.
$endgroup$
– Bernard
Jan 20 at 13:53