Principal Homogenous Spaces
Jose Capco
Whenever I try to read works of Tate or Milne (who was obviously
Tate's student), I very often see something about Principal Homogenous
Spaces. It's been a while since I have been staring at the definitions
with a blank head, I most definitely have a motivational problem here.
Therefore, I thought maybe someone out there who knows a little bit
about PHS could motivate me with some nontrivial examples (which are
lacking in the references I have been looking at) and into why they
could be so interesting. I certainly don't have a lot of confidence in
dealing with Group Schemes, but at least I see how beautiful they are
in an abstract sense of groups (in fact Group Objects in category
theory), but their application in Principal Homogenous Spaces totally
baffles me. Why and how was PHS made?
Sincerely,
Jose Capco
David Madore
The typical intuition of a p.h.s. under G is "a copy of G with the
origin forgotten".
Simplest nontrivial example: Spec(C)->Spec(R) is a principal
homogeneous space under the group Z/2Z (seen as a constant group
scheme over Spec(R): so as a scheme it is Spec(R\times R) =
Spec(R)+Spec(R) if you will). Here, C is the field of complex numbers
and R is the field of reals; and Z/2Z acts on Spec(C) above Spec(R) by
complex conjugation.
More generally, if L/K is a Galois field extension, then
Spec(L)->Spec(K) is a p.h.s. under the Galois group Gal(L/K) seen as a
constant group scheme over Spec(K).
A simple nontrivial example over a nonconstant group scheme: consider
the group scheme \mu_n = \{ x | x^n=1 \} of n-th roots of unity (over
Spec(Q), say). Then for any non-zero a, the scheme \{ x | x^n = a \}
is a p.h.s. under \mu_n. For example, in the case of cube roots of 2
over the rationals, this means Spec(Q(2^{1/3})) is a p.h.s. under
\mu_3 where the latter, as a scheme, is Spec(Q) +
Spec(Q((1+\sqrt{-3})/2)).
For an example over a nonfinite group, let E be the elliptic curve
over the reals, or the rationals, with affine equation y^2 = x^3 - x
(the origin being, as usual, the point at infinity). Then the
(projective completion of the) curve C with affine equation y^2 = - 1
- 4x^4 over the same field can be shown to be a p.h.s. under E: this
is a genus 1 curve, obviously having no points over the field in
question, but as soon as you add a point on C (by adding a square root
of -1 in the base field which has none), choosing this point as origin
makes C into an elliptic curve isomorphic to E.
Hope this helps,
--
David A. Madore
( http://www.madore.org/~david/ )
(Do _NOT_ remove the "+news" extension to email me.)
wellsoberlin
wrote:
Try posting this question on Math Overflow. Recently some "how do I
think about..." questions on MO have drawn some wonderfully helpful
answers.
Charles Wells