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Bitorsors and non abelian cohomology. (Bitorseurs et cohomologie non abélienne.) (French) Zbl 0743.14034
The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. I, Prog. Math. 86, 401-476 (1990).
[For the entire collection see Zbl 0717.00008.]
Let \(G,H\) be a pair of groups (or group schemes, or topological groups). A \((G,H)\)-biprincipal bundle (or bitorsor) over a space (or scheme) \(X\) consists in an \(X\)-object \(P\to X\), together with a left action of \(G\) (respectively a right action of \(H\)) on \(P\) which commute with each other, and for which \(P\) is both a left principal \(G\)-bundle and a right principal \(H\)-bundle. When \(G=H\), \(P\) is simply said to be a \(G\)-bitorsor on \(X\). Isomorphism classes of such \(G\)-bitorsors form a group, which may be interpreted as the group opposite to the group \(H^ 0(X,G\to Aut^ 0(G))\), with values in the crossed module \(G\to Aut^ 0(G)\) defined by the inner conjugation map of \(G\). It follows that the group of \(G\)- bitorsors is functorial in the crossed module in question, rather than merely in the group \(G\). Dévissage of this crossed module yields two exact sequences involving the group \(H^ 0(X,G\to Aut^ 0(G))\), and one of them terminates in the pointed set \(H^ 1(X,G\to Aut^ 0(G))\). The latter is interpreted as the set of classes of \(G\)-gerbes on \(X\), a slightly more restrictive notion than that of all gerbes on \(X\) in the sense of Giraud.
The classical description, due to Schreier, of the set \(Ext(H,G)\) of extension of a group \(H\) by \(G\), and its generalization to topological groups, or group schemes due to Grothendieck, may be viewed as a particular case of the above theory of \(G\)-gerbes, in which the role of the space \(X\) is played by the classifying space \(BH\) of the group \(H\). It is well known that the set \(Ext(H,G)\) is not in general functorial in the group \(G\). It follows from the previous discussion that a morphism of crossed modules \((G\to Aut^ 0(G))\to(G'\to Aut^ 0(G'))\) induces a map of pointed sets \(Ext(H,G)\to Ext(H,G')\).
Reviewer: L.Breen

14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
18G50 Nonabelian homological algebra (category-theoretic aspects)
Biographic References:
Grothendieck, Alexander