Timeline for Homology group of the étale homotopy type of a projective limit of schemes
Current License: CC BY-SA 4.0
7 events
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| Jun 15, 2020 at 19:55 | comment | added | D.-C. Cisinski | By "The Formula" I meant the fact that the homology of the limit is the limit of the homologies of the $X_j$'s (where the second instance of limit means in the sense of pro-objects in the derived category of abelian groups). The vanishing result you seem to want follows then immediately from there whatever you mean by limit. | |
| Jun 15, 2020 at 19:47 | comment | added | D.-C. Cisinski | By a Yoneda type argument, the Formula is equivalent to saying that for a locally constant sheaf A wih finite fibers, the $j$th cohomology of the limit is isomorphic to the filtered colimit of the $H^j(X_i,A)$ (with $j=0$ if $A$ is a sheaf of sets, $j=1$ if $A$ is a sheaf of groups, $j\geq 0$ if $A$ is a sheaf of abelian groups). This is established as Theorem 5.7 in Exposé VII of SGA4 in the case where $A$ is abelian. The case of non abelian coefficients is described in Remark 5.14 of loc. cit. | |
| Jun 15, 2020 at 14:32 | comment | added | Moutand Mohammed | @Denis-CharlesCisinski thank you for your comment ; pleas i need specific references | |
| Jun 15, 2020 at 8:58 | comment | added | D.-C. Cisinski | We need the limit to be op-filtered with affine transition maps, and the $X_i$'s quasi-compact and quasi-separated. Then this is a standard result from the 1960's. | |
| Jun 15, 2020 at 6:39 | comment | added | David Carchedi | Are they affine schemes? It so, you can see Corollary 4.4 here: arxiv.org/pdf/1905.06243.pdf | |
| Jun 15, 2020 at 4:34 | history | edited | Moutand Mohammed | CC BY-SA 4.0 |
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| Jun 15, 2020 at 4:16 | history | asked | Moutand Mohammed | CC BY-SA 4.0 |