Timeline for Definition of algebraic de Rham cohomology of non-smooth affine variety
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 30, 2018 at 15:30 | vote | accept | Jürgen Böhm | ||
| Jul 30, 2018 at 1:46 | answer | added | David Loeffler | timeline score: 13 | |
| Jul 29, 2018 at 21:14 | comment | added | David Loeffler | @PiotrAchinger It's not recent. The comparison between alg. de Rham coh and singular coh is in Hartshorne's work in the 1970's. See Thm 1.6 of Hartshorne's survey article in Manuscripta Math 7 (1972), or Chapter IV of his longer paper in PMIHES 45 (1975). | |
| Jul 29, 2018 at 20:45 | comment | added | Piotr Achinger | @JulianRosen Awesome, thanks! I'm surprised that the result is so recent. | |
| Jul 29, 2018 at 20:05 | comment | added | Julian Rosen | A reference for the isomorphism between algebraic de Rham cohomology and singular cohomology is the book Periods and Nori motives by Huber-Klawitter and Müller-Stach. The isomorphism is described in Definition 5.4.1. They use a different construction of algebraic de Rham cohomology, but Theorem 3.3.13 says their definition agrees with Hartshorne's. | |
| Jul 29, 2018 at 19:07 | comment | added | Piotr Achinger | I thought that the algebraic de Rham cohomology always agrees with singular cohomology over $\mathbf{C}$, by some form of $h$-descent of the usual isomorphism for smooth varieties. Unfortunately I cannot find a reference, so maybe it is false? Intuitively it should be true if we think of $\hat X$ as a tubular neighborhood of $Y$. | |
| Jul 29, 2018 at 18:26 | comment | added | Jürgen Böhm | The cited question asks for an example where the algebraic de Rham cohomology does not coincide with singular cohomology. My question is about two different definitions of algebraic de Rham cohomology in the singular case. | |
| Jul 29, 2018 at 18:21 | comment | added | Piotr Achinger | This question seems to be a duplicate of mathoverflow.net/questions/256379/… which in fact has an answer in the statement (a counterexample due to Arapura-Kang: $Y : x^5 + y^5 + x^2 y^2 = 0 $). | |
| Jul 29, 2018 at 18:13 | history | asked | Jürgen Böhm | CC BY-SA 4.0 |