Difference between revisions of "Form of an (algebraic) structure"
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− | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f0408001.png" /> be an extension of fields, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f0408002.png" /> be some | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f0408001.png" /> be an extension of fields, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f0408002.png" /> be some "object" defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f0408003.png" />. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f0408004.png" /> could be a vector space together with a quadratic form, a Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f0408005.png" />, an Azumaya algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f0408006.png" />, a variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f0408007.png" />, an algebraic group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f0408008.png" />, a representation of a finite group in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f0408009.png" />-vector space, etc. A form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080010.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080011.png" />, more precisely, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080013.png" />-form, is an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080014.png" /> of the same type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080017.png" /> become isomorphic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080018.png" />, i.e. after extending scalars from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080019.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080020.png" /> the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080022.png" /> become isomorphic. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080023.png" /> denote the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080024.png" />-isomorphism classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080025.png" /> forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080026.png" />. If now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080027.png" /> is a Galois extension, then under suitable circumstances one has a bijection between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080028.png" /> and the Galois cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080029.png" /> (cf. [[Galois cohomology|Galois cohomology]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080030.png" /> is the group of automorphisms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080032.png" />. Consider, for instance, the case where the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080033.png" /> is a finite-dimensional algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080034.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080035.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080036.png" /> is a form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080037.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080038.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080039.png" />-algebras. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080040.png" /> be an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080041.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080042.png" />, i.e. an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080043.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080044.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080045.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080046.png" /> is another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080047.png" />-automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080048.png" />. This defines the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080050.png" />. Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080051.png" /> be a form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080052.png" />. The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080053.png" />-isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080054.png" /> is naturally a principal homogeneous space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080055.png" /> and thus defines an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080056.png" />. This mapping is a bijection in this case. More generally one has such a bijection for the case that the structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080057.png" /> is a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080058.png" /> together with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080059.png" />-tensor (the previous case corresponds to the case of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080060.png" />-tensor). (To prove surjectivity one needs the generalization of Hilbert's theorem 90: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080061.png" />.) For the case of algebraic groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080062.png" /> cf. [[Form of an algebraic group|Form of an algebraic group]]. |
For the case of algebraic varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080063.png" /> one has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080064.png" /> is injective and that it is bijective if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080065.png" /> is quasi-projective. | For the case of algebraic varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080063.png" /> one has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080064.png" /> is injective and that it is bijective if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040800/f04080065.png" /> is quasi-projective. | ||
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.-A. Knus, M. Ojanguren, "Théorie de la descent et algèbres d'Azumaya" , Springer (1974) {{MR|417149}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Grothendieck, "Revêtements étales et groupe fondamental" , ''SGA 1960–1961'' , '''Exp. VI: Categories fibrées et descente''' , IHES (1961) {{MR|2017446}} {{MR|0354651}} {{MR|0217088}} {{MR|0217087}} {{ZBL|1039.14001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.P. Murre, "Lectures on an introduction to Grothendieck's theory of the fundamental group." , Tata Inst. Fund. Res. (1967) pp. Chapt. VII {{MR|302650}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1973) {{MR|0404227}} {{ZBL|0259.12011}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> G.B. Seligman, "Modular Lie algebras" , Springer (1967) pp. Chapt. IV {{MR|0245627}} {{ZBL|0189.03201}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) pp. Chapt. V, Sect. 20 {{MR|0103191}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> N. Jacobson, "Lie algebras" , Dover, reprint (1979) pp. Chapt. X ((also: Dover, reprint, 1979)) {{MR|0559927}} {{ZBL|0333.17009}} {{ZBL|0215.38701}} {{ZBL|0144.27103}} {{ZBL|0121.27601}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} {{ZBL|0198.05404}} {{ZBL|0064.27002}} {{ZBL|0064.03503}} {{ZBL|0046.03402}} {{ZBL|0043.26803}} {{ZBL|0039.02803}} {{ZBL|0063.03015}} {{ZBL|0025.30302}} {{ZBL|0025.30301}} {{ZBL|0022.19801}} {{ZBL|0019.19402}} {{ZBL|0018.10302}} {{ZBL|0017.29203}} {{ZBL|0016.20001}} </TD></TR></table> |
Revision as of 14:49, 24 March 2012
Let be an extension of fields, and let
be some "object" defined over
. For example,
could be a vector space together with a quadratic form, a Lie algebra over
, an Azumaya algebra over
, a variety over
, an algebraic group over
, a representation of a finite group in a
-vector space, etc. A form of
over
, more precisely, a
-form, is an object
of the same type over
such that
and
become isomorphic over
, i.e. after extending scalars from
to
the objects
and
become isomorphic. Let
denote the set of
-isomorphism classes of
forms of
. If now
is a Galois extension, then under suitable circumstances one has a bijection between
and the Galois cohomology group
(cf. Galois cohomology), where
is the group of automorphisms over
of
. Consider, for instance, the case where the object
is a finite-dimensional algebra
over
. Then
is a form of
if
as
-algebras. Let
be an automorphism of
over
, i.e. an isomorphism of
-algebras
, and let
. Then
is another
-automorphism of
. This defines the action of
on
. Now let
be a form of
. The set of
-isomorphisms
is naturally a principal homogeneous space over
and thus defines an element of
. This mapping is a bijection in this case. More generally one has such a bijection for the case that the structure
is a vector space
together with a
-tensor (the previous case corresponds to the case of a
-tensor). (To prove surjectivity one needs the generalization of Hilbert's theorem 90:
.) For the case of algebraic groups over
cf. Form of an algebraic group.
For the case of algebraic varieties over one has that
is injective and that it is bijective if
is quasi-projective.
The concept of forms makes sense in a far more general setting, e.g. in any category with base change, i.e. with fibre products. Indeed, let be such a category, and
an object in
. An object over
is a morphism in
,
. Let
be a morphism in
. Base change from
to
gives the pullback (fibre product)
defined by the Cartesian square
![]() |
(In case ,
and
is, for instance, the category of (affine) schemes this corresponds to extending scalars.)
An object is now an
-form of
if the objects
and
are isomorphic over
. For an even more general setting cf. [a2].
A related problem (to that of forms) is the subject of descent theory. In the setting of a category with base change as above this theory is concerned with the question: Given , does there exists an
over
such that
is isomorphic over
to
, and what properties must
satisfy for this to be the case.
Below this question is examined in the following setting: is a commutative algebra (with unit element) and
is a commutative
-algebra. Given a module
over
the question is whether there exists a module
over
such that
(as
-modules). Below all tensor products
are tensor products over
:
. If
is of the form
there is a natural isomorphism of
modules
given by
. Let
be an
-module. A descent datum on
is an isomorphism of
modules
such that
. Here
are the three natural
-module homomorphisms defined by
, where
is the identity on factor
and given by
on the other two components:
![]() |
![]() |
![]() |
The faithfully flat descent theorem now says that if is faithfully flat over
and
is a descent datum for
over
, then there exists an
-module
and an isomorphism
such that the following diagram commutes
![]() |
where the left vertical arrow is the descent datum on described above. Moreover, the pair
is uniquely defined by this property. One defines
by an invariance property:
(which is like invariance under the Galois group in the case of Galois descent).
There is a similar theorem for descent of algebras over .
In algebraic geometry one has for instance the following descent theorem (a globalization of the previous one for algebras). For a morphism of schemes , consider the fibre products
and
and let
be the projections
,
; and
the projections
,
. Let
be faithfully flat and compact. Then to give a scheme
affine over
is the same as to give a scheme
affine over
together with an isomorphism
such that
.
The theory of descent is quite general and includes such matters as specifying a section of a sheaf by local sections and the construction of locally trivial fibre bundles by glueing together trivial bundles over the elements of an open covering
of
. Indeed, let
be the disjoint union of the
and
the natural projection. Giving glueing data
is the same as giving an isomorphism
, where
is the trivial vector bundle
with fibre
and the compatibility of the glueing data amounts to the condition
.
For a treatment of forms of Lie algebras (over fields) cf. [a7], for Lie algebras over characteristic zero fields and the modular case (i.e. over fields of characteristic ) cf. [a5]. For a quite comprehensive treatment of descent and forms cf. [a1].
A form of an object is also occasionally called a twisted form.
In the case of descent with respect to a Galois field extension (or
) one speaks of Galois descent.
References
[a1] | M.-A. Knus, M. Ojanguren, "Théorie de la descent et algèbres d'Azumaya" , Springer (1974) MR417149 |
[a2] | A. Grothendieck, "Revêtements étales et groupe fondamental" , SGA 1960–1961 , Exp. VI: Categories fibrées et descente , IHES (1961) MR2017446 MR0354651 MR0217088 MR0217087 Zbl 1039.14001 |
[a3] | J.P. Murre, "Lectures on an introduction to Grothendieck's theory of the fundamental group." , Tata Inst. Fund. Res. (1967) pp. Chapt. VII MR302650 |
[a4] | J.-P. Serre, "Cohomologie Galoisienne" , Springer (1973) MR0404227 Zbl 0259.12011 |
[a5] | G.B. Seligman, "Modular Lie algebras" , Springer (1967) pp. Chapt. IV MR0245627 Zbl 0189.03201 |
[a6] | J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) pp. Chapt. V, Sect. 20 MR0103191 |
[a7] | N. Jacobson, "Lie algebras" , Dover, reprint (1979) pp. Chapt. X ((also: Dover, reprint, 1979)) MR0559927 Zbl 0333.17009 Zbl 0215.38701 Zbl 0144.27103 Zbl 0121.27601 Zbl 0121.27504 Zbl 0109.26201 Zbl 0198.05404 Zbl 0064.27002 Zbl 0064.03503 Zbl 0046.03402 Zbl 0043.26803 Zbl 0039.02803 Zbl 0063.03015 Zbl 0025.30302 Zbl 0025.30301 Zbl 0022.19801 Zbl 0019.19402 Zbl 0018.10302 Zbl 0017.29203 Zbl 0016.20001 |
Form of an (algebraic) structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Form_of_an_(algebraic)_structure&oldid=15826