Principal homogeneous space

From Academic Kids

In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. That is, X is a homogeneous space for G such that the stabilizer of any point is trivial.

An analogous definition holds in other categories where

If G is nonabelian then one must distinguish between left and right torsors according to whether the action is on the left or right. For concreteness, we will use right actions. To state the definition more explicitly, X is a G-torsor if there is a map (in the appropriate category) X × GX such that

<math>x\cdot 1 = x<math>
<math>x\cdot(gh) = (x\cdot g)\cdot h<math>

for all xX and all g,hG and such that the map X × GX × X given by

<math>(x,g) \mapsto (x,x\cdot g)<math>

is an isomorphism. Note that this means X and G are isomorphic, however — and this is the essential point — there is no preferred 'identity' point in X. That is, X looks exactly like G but we have forgotten which point is the identity. This concept is often used in mathematics as a way of passing to a more intrinsic point of view, under the heading 'throw away the origin'.

Since X is not a group we cannot add elements; we can, however, take their 'difference'. That is, there is a map X × XG which sends (x,y) to the unique element gG such that y = xg.

Examples

Every group G can itself be thought of as a left or right G-torsor under the natural action of left or right multiplication.

Another example is the affine space concept: the idea of the affine space A underlying a vector space V can be said succinctly by saying that A is principal homogeneous space for V acting as the additive group of translations.

Given a vector space V we can take G to be the general linear group GL(V), and X to be the set of all (ordered) bases of V. Then G acts on X in the way that it acts on vectors of V; and it acts transitively since any basis can be transformed via G to any other. What is more, a linear transformation fixing each vector of a basis will fix all v in V, hence being the neutral element of the general linear group GL(V) : so that X is indeed a principal homogeneous space. One way to follow basis-dependence in a linear algebra argument is to track variables x in X.

Applications

The principal homogeneous space concept is a special case of that of principal bundle: it means a principal bundle with base a single point. In other words the local theory of principal bundles is that of a family of principal homogeneous spaces depending on some parameters in the base. The 'origin' can be supplied by a section of the bundle - such sections are usually assumed to exist locally on the base - the bundle being locally trivial, so that the local structure is that of a cartesian product. But sections will often not exist globally. For example a differential manifold M has a principal bundle of frames associated to its tangent bundle. A global section will exist, tautologically, only when M is parallelizable; which implies strong topological restrictions.

In number theory there is a (superficially different) reason to consider principal homogeneous spaces, for elliptic curves E defined over a field K (and more general abelian varieties). Once this was understood various other examples were collected under the heading, for other algebraic groups: quadratic forms for orthogonal groups, and Severi-Brauer varieties for projective linear groups being two.

The reason of the interest for Diophantine equations, in the elliptic curve case, is that K may not be algebraically closed. There can exist curves C that have no point defined over K, and which become isomorphic over a larger field to E, which by definition has a point over K to serve as identity element for its addition law. That is, for this case we should distinguish C that have genus 1, from elliptic curves E that have a K-point (or, in other words, provide a Diophantine equation that has a solution in K). The curves C turn out to be torsors over E, and form a set carrying a rich structure in the case that K is a number field (the theory of the Selmer group)). In fact a typical plane cubic curve C over Q has no particular reason to have a rational point; the standard Weierstrass model always does, namely the point at infinity, but you need a point over K to put C into that form over K.

This theory has been developed with great attention to local analysis, leading to the definition of the Tate-Shafarevich group. In general the approach of taking the torsor theory, easy over an algebraically closed field, and trying to get back 'down' to a smaller field is an aspect of descent. It leads at once to questions of Galois cohomology, since the torsors represent classes in group cohomology H1.

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