Timeline for $H_n(Z \otimes C) = (Z \otimes C)_n$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Aug 18, 2020 at 8:33 | comment | added | Jeremy Rickard | Crossposted on MSE | |
| Aug 18, 2020 at 5:12 | comment | added | Aaron | Since $Z$ has $0$ differential, the differential of $Z\otimes C$ is $(-1)^a_Z\otimes \Delta_C$, and since everything is flat, you should get $H(Z\otimes C)\cong Z\otimes H(C)$. I cannot figure out what else might be going on. | |
| Aug 18, 2020 at 3:44 | history | edited | user163897 | CC BY-SA 4.0 |
added 206 characters in body
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| Aug 18, 2020 at 3:26 | comment | added | user163897 | OK. I really appreciate it! | |
| Aug 18, 2020 at 3:25 | history | edited | user163897 | CC BY-SA 4.0 |
added 304 characters in body
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| Aug 18, 2020 at 3:20 | comment | added | Yemon Choi | Just to let you know it is 4am here so I am unlikely to respond for some time ... | |
| Aug 18, 2020 at 3:16 | comment | added | user163897 | yes! it is the proof of his theorem! I will add all from the beginning up to this point! There's not much! Thx again! | |
| Aug 18, 2020 at 3:12 | comment | added | Yemon Choi | Something doesn't look right here: if I start with A already having zero differential, so that A=Z, then the claim you mention seems to assert that the homology of the total complex $A\otimes C$ doesn't depend on the differential of $C$. Since I don't have a copy of Rotman's book at hand, could you add some of the surrounding context? Is this something to do with a K\"unneth formula? | |
| Aug 18, 2020 at 3:05 | comment | added | user163897 | Yes! It is from page 679 of Rotman's introduction to homological algebra. Thx! | |
| Aug 18, 2020 at 3:03 | comment | added | Yemon Choi | Could you tell us which source you are using for this notation/terminology? I am not sure what a "cycle subcomplex" of $A$ is supposed to be: do you mean that you are taking $Z_n$ to be the kernel of $A_n \to A_{n-1}$, and then viewing $Z_{\bullet}$ as a complex with zero differential? | |
| Aug 18, 2020 at 2:50 | review | First posts | |||
| Aug 18, 2020 at 9:41 | |||||
| Aug 18, 2020 at 2:50 | history | asked | user163897 | CC BY-SA 4.0 |