Timeline for Why is the Tate local duality pairing compatible with the Cartier duality pairing?
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| Jul 30, 2014 at 2:31 | comment | added | user27920 | @QuestionMark: That is certainly the idea, but one has to be careful to ensure that the identifications made are functorial and not plagued by the ambiguity of some unknown isomorphism (as happens whenever using the TR3 axiom for triangulated categories to make a construction as opposed to verifying a property). So I grinded it out by working concretely with an injective resolution of $\mathbf{G}_m$, avoiding constructions based on abstract axioms; I don't think one can make the argument via abstract cone stuff (well, unless you're an $\infty$-categorical person, which I am not). | |
| Jul 30, 2014 at 1:25 | vote | accept | Question Mark | ||
| Jul 30, 2014 at 1:19 | comment | added | Question Mark | Thanks for such a nice answer! Regarding the homological algebra considerations left to the reader, is the idea in a nutshell that for an exact triangle $X \rightarrow Y \xrightarrow{y} Z \xrightarrow{z} X[1]$, if one builds the cone triangle $Z \xrightarrow{z} X[1] \rightarrow Cone(z) \xrightarrow{w} Z[1]$, then $w$ identifies with $y[1]$ (up to a universal sign?), whereas $\mathbf{R}\mathscr{Hom}$ and $\tau_{\le 1}$ are triangulated, so their intervention is harmless? | |
| Jul 29, 2014 at 19:20 | history | answered | user27920 | CC BY-SA 3.0 |