Principal homogeneous space |
In mathematics, a principal homogeneous space, or G -torsor, for a group (mathematics) G is a set X on which G group action 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 category (mathematics) 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 × G → X such that :xcdot 1 = x :xcdot(gh) = (xcdot g)cdot h for all x ∈ X and all g,h ∈ G and such that the map X × G → X × X given by :(x,g) mapsto (x,xcdot g) 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 × X → G which sends ( x , y ) to the unique element g ∈ G such that y = x · g .
=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 V can be said succinctly by saying that A is principal homogeneous space for V acting as the additive group of translations.
Given a argument is to track variables x in X .
=Applications=
The principal homogeneous space concept is a special case of that of ; which implies strong topological restrictions.
In s 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 (mathematics) 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 (category theory). It leads at once to questions of Galois cohomology, since the torsors represent classes in group cohomology H1.
=Other usage=
The term torsor is also used without the transitivity condition, especially in sheaf theory.
In that case, we talk about a (right) G -torsor E on a space X ( X a scheme (mathematics)/manifold/topological space etc.) being a space E with a free (right, say) G action such that the map
:E imes_X G ightarrow E imes_X E
given by
:(x,g) mapsto (x,xg)
is a bijection in the appropriate category. When we are in the smooth category, then a G -torsor (for G a Lie group) is then precisely a principal G principal bundle. Torsors in this sense correspond to classes in the cohomology H 1( X,G ).
=External links=
*[http://math.ucr.edu/home/baez/torsors.html Torsors made easy] by John Baez|
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