Timeline for Usage of étale cohomology in algebraic geometry
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jan 16, 2020 at 7:42 | answer | added | Angelo | timeline score: 4 | |
| Jan 10, 2020 at 15:55 | answer | added | user148212 | timeline score: 3 | |
| Jan 10, 2020 at 9:47 | answer | added | Alessio | timeline score: 4 | |
| Jan 10, 2020 at 0:41 | answer | added | anon | timeline score: 7 | |
| Jan 7, 2020 at 21:30 | comment | added | Sam Hopkins | It's not algebraic geometry, but an important application of $\ell$-adic cohomology which is not "number theory" is the representation theory of classical groups over finite fields, a.k.a., Deligne-Lusztig theory (en.wikipedia.org/wiki/Deligne%E2%80%93Lusztig_theory) | |
| S Jan 7, 2020 at 19:17 | history | suggested | Emily | CC BY-SA 4.0 |
Corrected typos and/or grammar, since question was already on front page. (I apologise in advance if these are unwelcome or just bad edits.)
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| Jan 7, 2020 at 19:07 | review | Suggested edits | |||
| S Jan 7, 2020 at 19:17 | |||||
| Jan 7, 2020 at 11:16 | comment | added | D.-C. Cisinski | Explicit instances of the principle I skteched above are Deligne's proof of the relative hard Lefschetz theorem (thm. 6.2.5 in Astérisque 100). The main tools are the version over finite fields (which uses the theory of weights) and Lemma 6.2.6 in loc. cit. which explains how to relate complex geometry and geometry over finite fields using suitable derived categories of étale l-adic sheaves. | |
| Jan 7, 2020 at 11:13 | comment | added | D.-C. Cisinski | Well, one can use étale cohomology to solve geometric problems using arithmetic methods. The idea is that a complex algebraic variety is in fact defined by polynomial equations with coefficients which are in a ring of finite type $R$ over $\mathbb Z$. Moereover, what is true for our variety is in fact true on a dense open subscheme of $R$. Playing with étale local systems, we see that many properties over $\mathbb C$ can be detected by pulling back over a closed point of $Spec (R)$. In other words, many theorems in algebraic geometry over finite fields produce theorems in complex geometry. | |
| Jan 7, 2020 at 9:40 | review | First posts | |||
| Jan 7, 2020 at 12:57 | |||||
| Jan 7, 2020 at 9:38 | history | asked | Daebeom Choi | CC BY-SA 4.0 |