Timeline for T^i functors are isomorphic for analytically isomorphic isolated singular points

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Sep 20, 2012 at 2:05	comment	added	anon		I mean the derived generalisation of the defining property of the limit: $\mathrm{Hom}(M,\lim_k N_k) \simeq \lim_k \mathrm{Hom}(M,N_k)$. Fixing an $M$ gives two isomorphic left-exact functors out of an abelian category of projective systems, so deriving the isomorphism gives an isomorphism $\mathrm{RHom}(M,\mathrm{R}\lim -) \simeq \mathrm{R}\lim \mathrm{RHom}(M,-)$ of derived functors on projective systems.
Sep 19, 2012 at 23:33	comment	added	Damian Rössler		Thank you very much for this comment. So the answer to Hnuer's question is positive in general (without any assumption on singularities). Could you expand on your parenthetical "... and commuting with RLim and RHom appropriately..." ?
Sep 19, 2012 at 0:22	comment	added	anon		For the last part, one has a much more general statement: $\mathrm{RHom}(L_\phi,N) = 0$ for any $m$-adically complete $\widehat{B}$-module $N$. Indeed, by completeness (and commuting $\mathrm{R}\lim$ with $\mathrm{RHom}$ appropriately), it suffices to show the claim for $N$ killed by some power $m^n$ of $m$. By adjunction, it suffices to show that $L_\phi \otimes_{\widehat{B}} \widehat{B}/m^n \simeq 0$, but this follows from the base change for the cotangent complex of $\phi$ together with fact that $\phi \mod m^n$ is an isomorphism. This also shows that excellence is not necessary.
Aug 13, 2012 at 9:43	history	answered	Damian Rössler	CC BY-SA 3.0