Difference between revisions of "Principal homogeneous space"
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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==== | ====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 | + | 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. |
Revision as of 14:51, 24 March 2012
A principal -object in the category of algebraic varieties or schemes. If
is a scheme and
is a group scheme over
, then a principal
-object in the category of schemes over
is said to be a principal homogeneous space. If
is the spectrum of a field
(cf. Spectrum of a ring) and
is an algebraic
-group (cf. Algebraic group), then a principal homogeneous space over
is an algebraic
-variety
acted upon (from the left) by
such that if
is replaced by its separable algebraic closure
, then each point
defines an isomorphic mapping
of the varieties
and
. A principal homogeneous space
is trivial if and only if
is non-empty. The set of classes of isomorphic principal homogeneous spaces over a smooth algebraic group
can be identified with the set of Galois cohomology
. In the general case the set of classes of principal homogeneous spaces over an
-group scheme
coincides with the set of one-dimensional non-Abelian cohomology
. Here
is some Grothendieck topology on the scheme
[2].
Principal homogeneous spaces have been computed in a number of cases. If is a finite field, then each principal homogeneous space over a connected algebraic
-group is trivial (Lang's theorem). This theorem also holds if
is a
-adic number field and
is a simply-connected semi-simple group (Kneser's theorem). If
is a multiplicative
-group scheme, then the set of classes of principal homogeneous spaces over
becomes identical with the Picard group
of
. In particular, if
is the spectrum of a field, this group is trivial. If
is an additive
-group scheme, then the set of classes of principal homogeneous spaces over
becomes identical with the one-dimensional cohomology group
of the structure sheaf
of
. In particular, this set is trivial if
is an affine scheme. If
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
-group
is based on the study of the Tate–Shafarevich set
, which consists of the principal homogeneous spaces over
with rational points in all completions
with respect to the valuations of
. If
is an Abelian group over the field
, then the set of classes of principal homogeneous spaces over
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 -sets, where
is a group. Let
be a finite (profinite, etc.) group. Let
be a
-set, i.e. a set
with an action
of
on it. Let
be a
-group, i.e. a group object in the category of
-sets, which means that
is a group and that the action of
on
is by group automorphisms of
:
for
,
. One says that
operates compatibly with the
-action from the left on
if there is a
-action
on
such that
for
,
,
. A principal homogeneous space over
in this setting is a
-set
on which
acts compatibly with the
-action and such that for all
there is a
such that
. (This is the property to which the word "principal" refers; one also says that
is an affine space over
.) In this case there is a natural bijective correspondence between
and isomorphism classes of principal homogeneous spaces over
and, in fact,
(for non-Abelian
) is sometimes defined this way.
Principal homogeneous space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_homogeneous_space&oldid=21910