Timeline for Computing the induced homomorphisms of derived functors using acyclic resolutions
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 22 at 11:29 | comment | added | Z. M | I think that it suffices to assume that $\mathcal A$ has enough $F$-acyclic objects (e.g. sheaf categories usually do not have enough projectives). | |
| Mar 22 at 1:50 | comment | added | Benjamin Steinberg | @user509184, I’ll try to sort out what they are saying and if it fits my context. Thanks. | |
| Mar 22 at 0:48 | comment | added | user509184 | I think that's a very special case of Theorem XVII.4.3 in the original Cartan-Eilenberg homological algebra book. | |
| Mar 22 at 0:28 | comment | added | Benjamin Steinberg | @user509184 I was trying something along that line but my problem was to check that that the map from the projective resolution to Q is a quasi-isomorphism. The proof I’m used to that F-acyclic resolutions work is usually based on dimension shifts. | |
| Mar 21 at 23:20 | comment | added | user509184 | Here is how I faintly recall this working, but I haven't tried to write anything down to be careful about it, so caveat emptor! If your resolutions $Q$ and $Q'$ were projective resolutions, then it's standard that the map induced by $f$ in $L_*F$ is the one induced by $H_*(f)$ from $A$ to $A'$. So you just need to compare $Q$ and $Q'$ to projective resolutions. I think the standard approach is to take Cartan-Eilenberg resolutions of $Q$ and $Q'$, and use projectivity of the CE resolutions to give you a map between them which is induced by $f$ and yet also induces the right map in $L_*F$. | |
| Mar 21 at 23:05 | history | edited | Benjamin Steinberg | CC BY-SA 4.0 |
added 12 characters in body
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| Mar 21 at 23:04 | comment | added | Benjamin Steinberg | @user509184, Sorry I will fix | |
| Mar 21 at 22:23 | comment | added | user509184 | Looks like you forgot to apply $F$ to your resolutions before taking homology. | |
| Mar 21 at 22:12 | history | asked | Benjamin Steinberg | CC BY-SA 4.0 |