Talk:Principal homogeneous space

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The page says "Note that this means X and G are isomorphic"; however this is only the case if X is non-empty.

Actually the definition of principal homogeneous space given here is incorrect: the assumption that X is non-empty should be added. This is a common mistake. Without this assumption we obtain what is called a "formally principal homogeneous space".

Also note that until the definition is corrected, the statement that the cohomology group H^1(X,G) parametrizes isomorphism classes of torsors is erroneous (take G=1 and X=point).

[edit] Axiomatic Characterization of a Torsor

A torsor can also be defined as an algebra in its own right through the ternary "torsor" operation (a,b,c) |-> a/b.c, subject to the axioms a/a.b = b, a/b.b = a, a/b.(c/d.e) = (a/b.c)/d.e. The additional axiom a/b.c = c/b.a characterizes torsors that correspond to Abelian groups.

The group G can be recovered from X intrinsically through the operator a\b, defined as the equivalence class [(a,b)] of the equivalence relation generated from (a,b/c.d) ~ (c/b.a,d), thus effecting the axiom a\(b/c.d) = (c/b.a)\d. In turn, this is identified as the corresponding group product (a\b)(c\d). One can then prove that a\a = b\b and that (a\b)(b\c) = a\c, thus providing the structure of a group, with the inverse of a\b being b\a.

The group can also be recovered as the fibre X_e associated with an element e by identifying e as the identity, the product as (a,b) |-> a/e.b and the inverse as a |-> e/a.e. Each fibre X_e is isomorphic to the group G via the map f_e: X_e -> G given by f_e(a) = e\a, or the inverse map g_e: G -> X_e given by g_e((a\b)) = e/a.b.

This characterization provides the structure where the group acts on the right, via a (b\c) = a/b.c. The corresponding structure with left actions is obtained by reversing all the slashes and order of operations in the foregoing (a kind of duality principle).

I'll leave it to others to relate this definition to the formal definition provided in the article (i.e., to find the isomorphism mentioned in the article); incorporating these observations in the article, proper.

[edit] Principal and Generalized Affine Bundles as Torsor Bundles

Besides the Principal Bundle, one has a more generalized notion of a "generalized affine bundle" which is just a torsor bundle. A Principal Bundle, itself, is just a trivial torsor bundle. Both of these can also be directly characterized algebraically in a similar fashion as the foregoing. -- Mark, 13 October 2006