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).

Contents 1 Axiomatic Characterization of a Torsor 2 Principal and Generalized Affine Bundles as Torsor Bundles 3 “Torsor” 4 Section "Other usage" 5 Local Triviality

[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

[edit] “Torsor”

I'm curious about the name “torsor”. Is this statement accurate?:

This “torque”-related word comes from the action of rotation? That is, if I apply a 20° rotation, it twists everything by 20°—there is a zero rotation, but there needn't be a zero orientation.

Is that about right? Also, does this relate to other mathematical uses of the word “torsion”? —Ben FrantzDale 18:53, 4 January 2007 (UTC)

As far as I understand Torsors it's right what you have written. But I think there is no direct relation to "torsion". Or at least I don't see it. Florianhe 21:28, 11 January 2007 (UTC)

[edit] Section "Other usage"

As far as I understand, the definition given here, i.e. a free action of G on E, so that the given map von E x G to E x E is bijective, implies transitivity. It also implies that the action is free. So I don't see the difference to the definition given above. Florianhe 21:28, 11 January 2007 (UTC)

You're correct, assuming that G is an ordinary group, and E is an object whose isomorphisms are determined by bijections on an underlying set. But when G and E are sheaves on some topological space, for instance, with G being group-valued and E being set-valued, the requirement that the map E x G to E x E be an isomorphism makes sense, whereas requiring that G act freely and transitively on E no longer has any meaning beyond that. The point is, since E x G --> E x E being an isomorphism is a category-theoretic requirement, that definition will work in a much larger range of categories. QBobWatson 04:40, 25 September 2007 (UTC)

[edit] Local Triviality

This is partially in response to the anonymous comment at the top of the page -- I agree that a principal homogeneous space should be nonempty; more generally (as in the "Other Usage" section), a torsor should be locally trivial on the base. This is certainly required if one is going to classify torsors using cohomology groups. I'm going to go ahead and update the page to reflect this. QBobWatson 23:42, 23 October 2007 (UTC)