Thursday, 22 March 2007

gr.group theory - substitute for Serre's twisting when the "twisting" is outer

The short answer is that there is not much to say about the relationship between H1(G,B) and a twist H1(G,Bc) where c is a cocycle taking values in Aut(B). (I am going to write Bc for the twist instead of Serre's notation cB for the sake of easy typesetting.) You can get a good feel for what is possible by soaking in sections I.5.7 and III.1.4 of Serre's Galois Cohomology.



Section I.5.7



One thing you can do -- as exhibited in I.5.7 -- is twist all three terms in a short exact sequence of G-modules and get a new short exact sequence, assuming the obvious compatibility conditions hold. Serre starts with an exact sequence




1toAtoBtoCto1




where A is assumed central in B. Then he fixes a 1-cocycle c with values in C and twists to get an exact sequence




1toAtoBctoCcto1.




Note that this twist Bc is not an inner twist of B, because c need not be in the image of H1(G,B)toH1(G,C).



This may look like a lame example, in that the twist of B is "pretty close" to being inner. But already here you don't have any results regarding a connection between H1(G,B) and H1(G,Bc). That's a pretty fuzzy statement; Serre says as much as you can say with precision in Remark 1: "it is, in general, false that H1(G,Bc) is in bijective correspondence with H1(G,B)."



Section III.1.4



This section discusses your question for the specific case where G is the absolute Galois group of a field k and B is the group of n-by-n matrices of determinant 1 with entries in a separable closure of k. Serre explains what you get as Bc when you twist B by a cocycle with values in Aut(B). You can get, for example, a special unitary group.



You can find explicit descriptions of H1(G,Bc) for some Bc's in The Book of Involutions, pages 393 (Cor. 29.4) and 404 (box in middle of page). Note that for Bc as in Section I.5.7, H1(G,Bc) is a group (a nice coincidence) but in the case where you get a true special unitary group, H1(G,Bc) does not have a reasonable group structure--it is just a pointed set like you expect.

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