View source for Principal homogeneous space

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]]].

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]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1973) {{MR|0404227}} {{ZBL|0259.12011}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , Masson (1970) {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, J. Tate, "Principal homogeneous spaces over abelian varieties" ''Amer. J. Math.'' , '''80''' (1958) pp. 659–684 {{MR|0106226}} {{ZBL|0097.36203}} </TD></TR></table>



====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.