Timeline for lists of computed cohomologies?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Jul 25, 2010 at 23:45 | comment | added | José Figueroa-O'Farrill | Thanks, Mariano. I have only ever worked over $\mathbb{R}$ or $\mathbb{C}$ and hence I've learnt to be careful here in MO, to be careful to state the ground field before I make a general statement. I'll remember that this particular criterion works in general! | |
| Jul 25, 2010 at 22:49 | comment | added | Mariano Suárez-Álvarez | (continued) and if you apply now the functor $\hom_{\mathfrak g}(k,\mathord-)$ and do the identification $\hom_{\mathfrak g}(k,\hom_k(\mathord -,\mathord-))=\hom_{\mathfrak g}(\mathord-,\mathord-)$, you see that $0\to\hom_{\mathfrak g}(D,A)\to\hom_{\mathfrak g}(D,B)\to\hom_{\mathfrak g}(D,C)\to0$ is exact: it follows that $D$ is projective, and, since $D$ was arbitrary, that $\mathfrak g$ is semisimple. | |
| Jul 25, 2010 at 22:49 | comment | added | Mariano Suárez-Álvarez | The criterion in your 2nd paragraph works for all fin.dim. Lie algebras over all fields: the condition that $H^1(\mathfrak g,\mathord-)=0$ identically means that the trivial $\mathfrak g$-module is projective in the category of finite dimensional modules; now pick a $\mathfrak g$-module $D$ and a short exact sequence $0\to A\to B\to C\to 0$ of $\mathfrak g$-modules: appling the functor $\hom_k(D,\mathord-)$ (this is the $\hom$ as vector spaces!), you get another short exact sequence $0\to\hom_k(D,A)\to\hom_k(D,B)\to\hom_k(D,C)\to0$ of $\mathfrak g$-modules, | |
| Jul 25, 2010 at 20:16 | comment | added | Emerton | I took question (b) to mean Lie alg. cohom. with trivial coeffs. | |
| Jul 25, 2010 at 20:00 | history | answered | José Figueroa-O'Farrill | CC BY-SA 2.5 |