Timeline for Can we characterise affine open subschemes of ${\rm Spec}(A)$?
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6 events
| when toggle format | what | by | license | comment | |
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| Sep 28, 2010 at 15:46 | answer | added | Hailong Dao | timeline score: 1 | |
| Sep 25, 2010 at 21:53 | comment | added | Karl Schwede | In fact, BCnrd's argument implies aniket's as well (at least in sufficiently geometric settings). This is because $H^{i+1}_{X\setminus U}(X, \widetilde{M})$ is non-zero at the generic points of $X \setminus U$ for appropriate $i$ (depending on the dimension of those generic points). | |
| Sep 25, 2010 at 10:12 | comment | added | pinaki | A necessary condition: $I$ has to be of pure codimension one (see e.g. jstor.org/pss/1970814) | |
| Sep 25, 2010 at 9:11 | comment | added | BCnrd | Assume $U$ is q-compact; equivalently, choose $I$ finitely generated. Inclusion $j:U \rightarrow X := {\rm{Spec}}(A)$ is a q-compact open immersion, so $j_{\ast}(F)$ is q-coh. on $X$ for all q-coh. $F$ on $U$. Restricting back to $U$ gives $F$, so $F = \widetilde{M}|_U$ for an $A$-module $M$. Then by excision ${\rm{H}}^i(U,F) = {\rm{H}}^i(U,\widetilde{M}) = {\rm{H}}^{i+1}_{X-U}(X,\widetilde{M}) = \injlim {\rm{Ext}}^{i+1}(A/I^n,M)$ (limit over $n \rightarrow \infty$), final equality since $I$ f. gen'td (univ. $\delta$-functor argument). Vanishing for $i > 0$ and all $M$ seems "impractical"... | |
| Sep 25, 2010 at 9:04 | comment | added | Martin Brandenburg | This was discussed here: mathoverflow.net/questions/20782/… | |
| Sep 25, 2010 at 8:53 | history | asked | unknown | CC BY-SA 2.5 |