Timeline for cohomology theory for algebraic groups
Current License: CC BY-SA 3.0
12 events
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| Jul 28, 2021 at 16:41 | comment | added | LSpice | Name of the important paper 'here': Cline, Parshall, Scott, and van der Kallen - Rational and generic cohomology. @ChristopherDrupieski's reference: Cline, Parshall, and Scott - Cohomology, hyperalgebras, and representations. | |
| Sep 12, 2017 at 22:41 | comment | added | Jim Humphreys | @Chris: Thanks for the clarifications. The notion of "algebraic group" has become rather slipppery in view of group scheme notions. So what I've said certainly has to be qualified. On the other hand, most of the serious questions so far have risen either in the context of reduced group schemes (old-fashioned affine algebraic groups) or in the context of finite group schemes. | |
| Sep 12, 2017 at 17:49 | comment | added | Christopher Drupieski | A comment about the fifth paragraph of Jim's answer: If $G$ is a finite group scheme, then the category of rational $G$-modules is equivalent to the category of $Dist(G)$-modules, where $Dist(G) = k[G]^*$ is the cocommutative Hopf algebra dual to the coordinate algebra $k[G]$. But in general, if $G$ is a non-finite affine group scheme, then rational $G$-modules correspond to $Dist(G)$-modules that have some local finiteness assumption (cf. Theorems 6.8 and 9.4 of the CPS paper Cohomology, Hyperalgebras, and Representations). | |
| Sep 22, 2015 at 16:20 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| Jul 13, 2011 at 7:33 | vote | accept | sim | ||
| Jan 15, 2011 at 20:46 | comment | added | Ralph | Jim, thanks for this information. It follows in particular that properties of hopf algebra cohomology (like cup products, Steenrod operations, Tate cohomology, etc.) carry over to the cohomology of group schemes. | |
| Jan 15, 2011 at 13:31 | comment | added | Jim Humphreys | @Dmitri: This can be useful for the immediate purpose, since the foundational Part I in the original 1987 Academic Press edition is essentially unchanged in the newer edition (though the longer Part II has been greatly expanded and partly rewritten). | |
| Jan 15, 2011 at 13:22 | history | edited | Jim Humphreys | CC BY-SA 2.5 |
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| Jan 15, 2011 at 4:41 | comment | added | Dmitri Pavlov | The first edition of Jantzen's book is available online: gen.lib.rus.ec/… | |
| Jan 14, 2011 at 23:27 | comment | added | Ralph | @Jim: Let $G$ be a finite group scheme (defined over the field $k$) and let $A$ be the corresponding cocommutative hopf algebra (i.e. A is the dual hopf algebra of the coordinate ring $k[G]$ of $G$). Is there a connection between the rational cohomology of $G$ and the cohomology of $A$ defined as $Ext_A(k,-)$ ? | |
| Jan 14, 2011 at 21:51 | history | edited | Jim Humphreys | CC BY-SA 2.5 |
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| Jan 14, 2011 at 21:44 | history | answered | Jim Humphreys | CC BY-SA 2.5 |