How does the splitting in the Künneth theorem work?
In the algebraic Künneth theorem, see
https://ncatlab.org/nlab/show/Künneth+theorem theorem 2.2
We assume we have two complexes, one of which is free over some PID.
Now this gives some short exact sequence whose left map is
$oplus{H_k(C) otimes H_{n-k}(C')} to
oplus {H_n(C otimes C')} $
And this injection is split noncanonically.
Now my question is about the proof of this splitting.
From all the sources I looked at, the first step is to notice the splitting of
$0 to Z to C to B to 0$
Now this induces a splitting after tensoring by $C'$ or $Z'$ however I cannot really finish the proof.
I would appreciate someone spelling out the details of the last bit, I looked at different sources but none are really detailed. Hatcher seems to assume C' are also free? Please do not assume any properties of C', the nLab statement doesn't impose any conditions on C'.
Edit: just realized nLab doesn't even talk about splitting... Rest assured that both Weibel and Rotman writes split for the conditions I gave
homological-algebra
edited Dec 9 at 17:58
Jean Marie
28.8k41949
asked Dec 9 at 17:39
davik
558317
add a comment |
In the algebraic Künneth theorem, see
https://ncatlab.org/nlab/show/Künneth+theorem theorem 2.2
We assume we have two complexes, one of which is free over some PID.
Now this gives some short exact sequence whose left map is
$oplus{H_k(C) otimes H_{n-k}(C')} to
oplus {H_n(C otimes C')} $
And this injection is split noncanonically.
Now my question is about the proof of this splitting.
From all the sources I looked at, the first step is to notice the splitting of
$0 to Z to C to B to 0$
Now this induces a splitting after tensoring by $C'$ or $Z'$ however I cannot really finish the proof.
I would appreciate someone spelling out the details of the last bit, I looked at different sources but none are really detailed. Hatcher seems to assume C' are also free? Please do not assume any properties of C', the nLab statement doesn't impose any conditions on C'.
Edit: just realized nLab doesn't even talk about splitting... Rest assured that both Weibel and Rotman writes split for the conditions I gave
homological-algebra
edited Dec 9 at 17:58
Jean Marie
28.8k41949
asked Dec 9 at 17:39
davik
558317
A detail that has its importance : I have transformed all the names you give in order they begin by a capital letter.
– Jean Marie
Dec 9 at 18:01
add a comment |
In the algebraic Künneth theorem, see
https://ncatlab.org/nlab/show/Künneth+theorem theorem 2.2
We assume we have two complexes, one of which is free over some PID.
Now this gives some short exact sequence whose left map is
$oplus{H_k(C) otimes H_{n-k}(C')} to
oplus {H_n(C otimes C')} $
And this injection is split noncanonically.
Now my question is about the proof of this splitting.
From all the sources I looked at, the first step is to notice the splitting of
$0 to Z to C to B to 0$
Now this induces a splitting after tensoring by $C'$ or $Z'$ however I cannot really finish the proof.
I would appreciate someone spelling out the details of the last bit, I looked at different sources but none are really detailed. Hatcher seems to assume C' are also free? Please do not assume any properties of C', the nLab statement doesn't impose any conditions on C'.
Edit: just realized nLab doesn't even talk about splitting... Rest assured that both Weibel and Rotman writes split for the conditions I gave
homological-algebra
edited Dec 9 at 17:58
Jean Marie
28.8k41949
asked Dec 9 at 17:39
davik
558317
In the algebraic Künneth theorem, see
https://ncatlab.org/nlab/show/Künneth+theorem theorem 2.2
We assume we have two complexes, one of which is free over some PID.
Now this gives some short exact sequence whose left map is
$oplus{H_k(C) otimes H_{n-k}(C')} to
oplus {H_n(C otimes C')} $
And this injection is split noncanonically.
Now my question is about the proof of this splitting.
From all the sources I looked at, the first step is to notice the splitting of
$0 to Z to C to B to 0$
Now this induces a splitting after tensoring by $C'$ or $Z'$ however I cannot really finish the proof.
I would appreciate someone spelling out the details of the last bit, I looked at different sources but none are really detailed. Hatcher seems to assume C' are also free? Please do not assume any properties of C', the nLab statement doesn't impose any conditions on C'.
Edit: just realized nLab doesn't even talk about splitting... Rest assured that both Weibel and Rotman writes split for the conditions I gave
homological-algebra
homological-algebra
edited Dec 9 at 17:58
Jean Marie
28.8k41949
asked Dec 9 at 17:39
davik
558317
edited Dec 9 at 17:58
Jean Marie
28.8k41949
asked Dec 9 at 17:39
davik
558317
edited Dec 9 at 17:58
Jean Marie
28.8k41949
edited Dec 9 at 17:58
Jean Marie
28.8k41949
edited Dec 9 at 17:58
Jean Marie
28.8k41949
28.8k41949
asked Dec 9 at 17:39
davik
558317
asked Dec 9 at 17:39
davik
558317
asked Dec 9 at 17:39
davik
558317
558317
A detail that has its importance : I have transformed all the names you give in order they begin by a capital letter.
– Jean Marie
Dec 9 at 18:01
add a comment |
A detail that has its importance : I have transformed all the names you give in order they begin by a capital letter.
– Jean Marie
Dec 9 at 18:01
A detail that has its importance : I have transformed all the names you give in order they begin by a capital letter.
– Jean Marie
Dec 9 at 18:01
A detail that has its importance : I have transformed all the names you give in order they begin by a capital letter.
– Jean Marie
Dec 9 at 18:01
add a comment |
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A detail that has its importance : I have transformed all the names you give in order they begin by a capital letter.
– Jean Marie
Dec 9 at 18:01