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m (tex done, typo: with an ad hoc definition of \Sha (for Tate-Shafarevich set))
 
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A [[Principal G-object|principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p0747201.png" />-object]] in the category of algebraic varieties or schemes. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p0747202.png" /> is a [[Scheme|scheme]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p0747203.png" /> is a [[Group scheme|group scheme]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p0747204.png" />, then a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p0747205.png" />-object in the category of schemes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p0747206.png" /> is said to be a principal homogeneous space. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p0747207.png" /> is the spectrum of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p0747208.png" /> (cf. [[Spectrum of a ring|Spectrum of a ring]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p0747209.png" /> is an algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472010.png" />-group (cf. [[Algebraic group|Algebraic group]]), then a principal homogeneous space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472011.png" /> is an algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472012.png" />-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472013.png" /> acted upon (from the left) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472014.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472015.png" /> is replaced by its separable algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472016.png" />, then each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472017.png" /> defines an isomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472018.png" /> of the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472020.png" />. A principal homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472021.png" /> is trivial if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472022.png" /> is non-empty. The set of classes of isomorphic principal homogeneous spaces over a smooth algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472023.png" /> can be identified with the set of [[Galois cohomology|Galois cohomology]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472024.png" />. In the general case the set of classes of principal homogeneous spaces over an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472025.png" />-group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472026.png" /> coincides with the set of one-dimensional non-Abelian cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472027.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472028.png" /> is some [[Grothendieck topology|Grothendieck topology]] on the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472029.png" /> [[#References|[2]]].
+
{{TEX|done}}
  
Principal homogeneous spaces have been computed in a number of cases. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472030.png" /> is a finite field, then each principal homogeneous space over a connected algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472031.png" />-group is trivial (Lang's theorem). This theorem also holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472032.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472033.png" />-adic number field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472034.png" /> is a simply-connected semi-simple group (Kneser's theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472035.png" /> is a multiplicative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472036.png" />-group scheme, then the set of classes of principal homogeneous spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472037.png" /> becomes identical with the [[Picard group|Picard group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472039.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472040.png" /> is the spectrum of a field, this group is trivial. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472041.png" /> is an additive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472042.png" />-group scheme, then the set of classes of principal homogeneous spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472043.png" /> becomes identical with the one-dimensional cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472044.png" /> of the structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472045.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472046.png" />. In particular, this set is trivial if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472047.png" /> is an affine scheme. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472048.png" /> is a global field (i.e. an algebraic number field or a field of algebraic functions in one variable), then the study of the set of classes of principal homogeneous spaces over an algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472049.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472050.png" /> is based on the study of the Tate–Shafarevich set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472051.png" />, which consists of the principal homogeneous spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472052.png" /> with rational points in all completions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472053.png" /> with respect to the valuations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472054.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472055.png" /> is an Abelian group over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472056.png" />, then the set of classes of principal homogeneous spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472057.png" /> forms a group (cf. [[Weil–Châtelet group|Weil–Châtelet group]]).
+
A [[Principal G-object|principal  $  G $-
 +
object]] in the category of algebraic varieties or schemes. If  $  S $
 +
is a [[Scheme|scheme]] and  $  \Gamma $
 +
is a [[Group scheme|group scheme]] over  $  S $,
 +
then a principal  $  G $-
 +
object in the category of schemes over  $  \Gamma $
 +
is said to be a principal homogeneous space. If  $  S $
 +
is the spectrum of a field  $  k $(
 +
cf. [[Spectrum of a ring|Spectrum of a ring]]) and  $  \Gamma $
 +
is an algebraic  $  k $-
 +
group (cf. [[Algebraic group|Algebraic group]]), then a principal homogeneous space over  $  \Gamma $
 +
is an algebraic  $  k $-
 +
variety  $  V $
 +
acted upon (from the left) by  $  \Gamma $
 +
such that if  $  k $
 +
is replaced by its separable algebraic closure  $  \overline{k}  $,
 +
then each point  $  v \in V ( \overline{k}  ) $
 +
defines an isomorphic mapping  $  g \rightarrow gv $
 +
of the varieties  $  V _{\overline{ {k}} } $
 +
and  $  \Gamma _{\overline{ {k}} } $.
 +
A principal homogeneous space  $  V $
 +
is trivial if and only if  $  V(k) $
 +
is non-empty. The set of classes of isomorphic principal homogeneous spaces over a smooth algebraic group  $  \Gamma $
 +
can be identified with the set of [[Galois cohomology|Galois cohomology]]  $  H ^{1} (k,\  \Gamma ) $.
 +
In the general case the set of classes of principal homogeneous spaces over an  $  S $-
 +
group scheme  $  \Gamma $
 +
coincides with the set of one-dimensional non-Abelian cohomology  $  H ^{1} ( S _{T} ,\  \Gamma ) $.
 +
Here  $  S _{T} $
 +
is some [[Grothendieck topology|Grothendieck topology]] on the scheme  $  S $[[#References|[2]]].
 +
 
 +
Principal homogeneous spaces have been computed in a number of cases. If $  k $
 +
is a finite field, then each principal homogeneous space over a connected algebraic $  k $-
 +
group is trivial (Lang's theorem). This theorem also holds if $  k $
 +
is a p $-
 +
adic number field and $  \Gamma $
 +
is a simply-connected semi-simple group (Kneser's theorem). If $  \Gamma = \Gamma _{m,S} $
 +
is a multiplicative $  S $-
 +
group scheme, then the set of classes of principal homogeneous spaces over $  \Gamma $
 +
becomes identical with the [[Picard group|Picard group]] $  \mathop{\rm Pic}\nolimits (S) $
 +
of $  S $.  
 +
In particular, if $  S $
 +
is the spectrum of a field, this group is trivial. If $  \Gamma = \Gamma _{a,S} $
 +
is an additive $  S $-
 +
group scheme, then the set of classes of principal homogeneous spaces over $  \Gamma $
 +
becomes identical with the one-dimensional cohomology group $  H ^{1} (S,\  {\mathcal O} _{S} ) $
 +
of the structure sheaf $  {\mathcal O} _{S} $
 +
of $  S $.  
 +
In particular, this set is trivial if $  S $
 +
is an affine scheme. If $  k $
 +
is a global field (i.e. an algebraic number field or a field of algebraic functions in one variable), then the study of the set of classes of principal homogeneous spaces over an algebraic $  k $-
 +
group $  \Gamma $
 +
$\def\Sha{ {\mathop{\amalg\kern-0.30em\amalg}}}$
 +
is based on the study of the Tate–Shafarevich set $  \Sha ( \Gamma ) $,  
 +
which consists of the principal homogeneous spaces over $  \Gamma $
 +
with rational points in all completions $  k _{V} $
 +
with respect to the valuations of $  k $.  
 +
If $  \Gamma $
 +
is an Abelian group over the field $  k $,  
 +
then the set of classes of principal homogeneous spaces over $  \Gamma $
 +
forms a group (cf. [[Weil–Châtelet group|Weil–Châtelet group]]).
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
The notion of a principal homogeneous space is not restricted to algebraic geometry. For instance, it is defined in the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472058.png" />-sets, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472059.png" /> is a group. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472060.png" /> be a finite (profinite, etc.) group. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472061.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472063.png" />-set, i.e. a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472064.png" /> with an action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472065.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472066.png" /> on it. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472067.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472069.png" />-group, i.e. a group object in the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472070.png" />-sets, which means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472071.png" /> is a group and that the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472072.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472073.png" /> is by group automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472074.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472075.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472077.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472078.png" /> operates compatibly with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472079.png" />-action from the left on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472080.png" /> if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472081.png" />-action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472082.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472083.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472084.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472087.png" />. A principal homogeneous space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472088.png" /> in this setting is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472089.png" />-set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472090.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472091.png" /> acts compatibly with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472092.png" />-action and such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472093.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472094.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472095.png" />. (This is the property to which the word "principal" refers; one also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472096.png" /> is an affine space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472097.png" />.) In this case there is a natural bijective correspondence between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472098.png" /> and isomorphism classes of principal homogeneous spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p07472099.png" /> and, in fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p074720100.png" /> (for non-Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074720/p074720101.png" />) is sometimes defined this way.
+
The notion of a principal homogeneous space is not restricted to algebraic geometry. For instance, it is defined in the category of $  G $-
 +
sets, where $  G $
 +
is a group. Let $  G $
 +
be a finite (profinite, etc.) group. Let $  E $
 +
be a $  G $-
 +
set, i.e. a set $  E $
 +
with an action $  G \times E \rightarrow E $
 +
of $  G $
 +
on it. Let $  \Gamma $
 +
be a $  G $-
 +
group, i.e. a group object in the category of $  G $-
 +
sets, which means that $  \Gamma $
 +
is a group and that the action of $  G $
 +
on $  \Gamma $
 +
is by group automorphisms of $  \Gamma $:  
 +
$  (xy) ^ \gamma  = x ^ \gamma  y ^ \gamma  $
 +
for $  \gamma \in G $,
 +
$  x,\  y \in \Gamma $.  
 +
One says that $  \Gamma $
 +
operates compatibly with the $  G $-
 +
action from the left on $  E $
 +
if there is a $  \Gamma $-
 +
action $  \Gamma \times E \rightarrow E $
 +
on $  E $
 +
such that $  ( \gamma x ) ^{g} = ( \gamma ^{g} )(x ^{g} ) $
 +
for $  g \in G $,  
 +
$  \gamma \in \Gamma $,  
 +
$  x \in E $.  
 +
A principal homogeneous space over $  \Gamma $
 +
in this setting is a $  G $-
 +
set $  P $
 +
on which $  \Gamma $
 +
acts compatibly with the $  G $-
 +
action and such that for all $  x,\  y \in P $
 +
there is a $  \gamma \in \Gamma $
 +
such that $  y = \gamma x $.  
 +
(This is the property to which the word "principal" refers; one also says that $  P $
 +
is an affine space over $  \Gamma $.)  
 +
In this case there is a natural bijective correspondence between $  H ^{1} (G,\  \Gamma ) $
 +
and isomorphism classes of principal homogeneous spaces over $  \Gamma $
 +
and, in fact, $  H ^{1} (G,\  \Gamma ) $(
 +
for non-Abelian $  \Gamma $)  
 +
is sometimes defined this way.

Latest revision as of 11:51, 20 December 2019


A principal $ G $- object in the category of algebraic varieties or schemes. If $ S $ is a scheme and $ \Gamma $ is a group scheme over $ S $, then a principal $ G $- object in the category of schemes over $ \Gamma $ is said to be a principal homogeneous space. If $ S $ is the spectrum of a field $ k $( cf. Spectrum of a ring) and $ \Gamma $ is an algebraic $ k $- group (cf. Algebraic group), then a principal homogeneous space over $ \Gamma $ is an algebraic $ k $- variety $ V $ acted upon (from the left) by $ \Gamma $ such that if $ k $ is replaced by its separable algebraic closure $ \overline{k} $, then each point $ v \in V ( \overline{k} ) $ defines an isomorphic mapping $ g \rightarrow gv $ of the varieties $ V _{\overline{ {k}} } $ and $ \Gamma _{\overline{ {k}} } $. A principal homogeneous space $ V $ is trivial if and only if $ V(k) $ is non-empty. The set of classes of isomorphic principal homogeneous spaces over a smooth algebraic group $ \Gamma $ can be identified with the set of Galois cohomology $ H ^{1} (k,\ \Gamma ) $. In the general case the set of classes of principal homogeneous spaces over an $ S $- group scheme $ \Gamma $ coincides with the set of one-dimensional non-Abelian cohomology $ H ^{1} ( S _{T} ,\ \Gamma ) $. Here $ S _{T} $ is some Grothendieck topology on the scheme $ S $[2].

Principal homogeneous spaces have been computed in a number of cases. If $ k $ is a finite field, then each principal homogeneous space over a connected algebraic $ k $- group is trivial (Lang's theorem). This theorem also holds if $ k $ is a $ p $- adic number field and $ \Gamma $ is a simply-connected semi-simple group (Kneser's theorem). If $ \Gamma = \Gamma _{m,S} $ is a multiplicative $ S $- group scheme, then the set of classes of principal homogeneous spaces over $ \Gamma $ becomes identical with the Picard group $ \mathop{\rm Pic}\nolimits (S) $ of $ S $. In particular, if $ S $ is the spectrum of a field, this group is trivial. If $ \Gamma = \Gamma _{a,S} $ is an additive $ S $- group scheme, then the set of classes of principal homogeneous spaces over $ \Gamma $ becomes identical with the one-dimensional cohomology group $ H ^{1} (S,\ {\mathcal O} _{S} ) $ of the structure sheaf $ {\mathcal O} _{S} $ of $ S $. In particular, this set is trivial if $ S $ is an affine scheme. If $ k $ is a global field (i.e. an algebraic number field or a field of algebraic functions in one variable), then the study of the set of classes of principal homogeneous spaces over an algebraic $ k $- group $ \Gamma $ $\def\Sha{ {\mathop{\amalg\kern-0.30em\amalg}}}$ is based on the study of the Tate–Shafarevich set $ \Sha ( \Gamma ) $, which consists of the principal homogeneous spaces over $ \Gamma $ with rational points in all completions $ k _{V} $ with respect to the valuations of $ k $. If $ \Gamma $ is an Abelian group over the field $ k $, then the set of classes of principal homogeneous spaces over $ \Gamma $ forms a group (cf. Weil–Châtelet group).

References

[1] J.-P. Serre, "Cohomologie Galoisienne" , Springer (1973) MR0404227 Zbl 0259.12011
[2] M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970) MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401
[3] S. Lang, J. Tate, "Principal homogeneous spaces over abelian varieties" Amer. J. Math. , 80 (1958) pp. 659–684 MR0106226 Zbl 0097.36203


Comments

The notion of a principal homogeneous space is not restricted to algebraic geometry. For instance, it is defined in the category of $ G $- sets, where $ G $ is a group. Let $ G $ be a finite (profinite, etc.) group. Let $ E $ be a $ G $- set, i.e. a set $ E $ with an action $ G \times E \rightarrow E $ of $ G $ on it. Let $ \Gamma $ be a $ G $- group, i.e. a group object in the category of $ G $- sets, which means that $ \Gamma $ is a group and that the action of $ G $ on $ \Gamma $ is by group automorphisms of $ \Gamma $: $ (xy) ^ \gamma = x ^ \gamma y ^ \gamma $ for $ \gamma \in G $, $ x,\ y \in \Gamma $. One says that $ \Gamma $ operates compatibly with the $ G $- action from the left on $ E $ if there is a $ \Gamma $- action $ \Gamma \times E \rightarrow E $ on $ E $ such that $ ( \gamma x ) ^{g} = ( \gamma ^{g} )(x ^{g} ) $ for $ g \in G $, $ \gamma \in \Gamma $, $ x \in E $. A principal homogeneous space over $ \Gamma $ in this setting is a $ G $- set $ P $ on which $ \Gamma $ acts compatibly with the $ G $- action and such that for all $ x,\ y \in P $ there is a $ \gamma \in \Gamma $ such that $ y = \gamma x $. (This is the property to which the word "principal" refers; one also says that $ P $ is an affine space over $ \Gamma $.) In this case there is a natural bijective correspondence between $ H ^{1} (G,\ \Gamma ) $ and isomorphism classes of principal homogeneous spaces over $ \Gamma $ and, in fact, $ H ^{1} (G,\ \Gamma ) $( for non-Abelian $ \Gamma $) is sometimes defined this way.

How to Cite This Entry:
Principal homogeneous space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Principal_homogeneous_space&oldid=44311
This article was adapted from an original article by V.E. VoskresenskiiI.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article