View source for Rationality theorems

''for algebraic groups''

Statements about the rationality (unirationality) or non-rationality of various algebraic group varieties (cf. [[Rational variety|Rational variety]], [[Unirational variety|Unirational variety]]). Since Abelian varieties can never be rational, the main interest is in rationality theorems for linear algebraic groups. Here, the rationality problem has two essentially different features: geometrical and arithmetical, according as whether the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r0776701.png" /> is algebraically closed or not. The first rationality theorems over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r0776702.png" /> of complex numbers were in fact proved by E. Picard and, in contemporary terminology, establish the unirationality of varieties of connected complex groups. The rationality problem for group varieties was clearly stated by C. Chevalley [[#References|[1]]] as late as 1954. Progress in this direction is closely connected with achievements in the structure theory of algebraic groups. Thus, the Levi decomposition enables one to reduce the rationality problem to the case of reductive groups, and the [[Bruhat decomposition|Bruhat decomposition]] is the key to proving the rationality of varieties of reductive groups over any algebraically

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r0776703.png" /></td> </tr></table>

closed field (cf. [[Reductive group|Reductive group]]). Thus, in the geometrical case definitive results have been obtained.

The situation for algebraically non-closed fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r0776704.png" /> turns out to be much more complicated. Examples of non-rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r0776705.png" />-varieties are supplied by algebraic tori; for example, a three-dimensional torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r0776706.png" />, corresponding to the biquadratic extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r0776707.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r0776708.png" /> (see [[#References|[1]]]). This example is minimal, for tori of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r0776709.png" /> are rational. Algebraic tori are always unirational. Arbitrary connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767010.png" />-groups are not necessarily unirational [[#References|[3]]], but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767011.png" /> is perfect or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767012.png" /> is reductive, unirationality can be proved (see [[#References|[1]]]–[[#References|[4]]]). Thus, the rationality problem for group varieties has the character of the [[Lüroth problem|Lüroth problem]] over an algebraically non-closed field.

Since an arbitrary reductive group is the almost-direct product of a [[Torus|torus]] and a [[Semi-simple group|semi-simple group]], one can naturally distinguish two main cases: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767013.png" /> is a torus; or 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767014.png" /> is a semi-simple group. The first case is investigated using various cohomological invariants (for semi-simple groups, these invariants turn out to be ineffective). Fairly complete results are obtained for tori which split over an Abelian extension of the ground field (see [[#References|[5]]]). The first example of a non-rational variety in the class of semi-simple groups was a non-simply-connected group, whose construction is actually contained in [[#References|[10]]]. The resulting conjecture, that varieties of simply-connected groups are always rational, was solved negatively by V.P. Platonov, using reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767016.png" />-theory (see [[#References|[6]]], [[#References|[7]]]). It was found that the reduced [[Whitehead group|Whitehead group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767017.png" /> of a finite-dimensional central simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767018.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767019.png" /> is trivial if the variety determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767020.png" /> is rational over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767021.png" />. These results carry over to unitary groups [[#References|[12]]]. There are results on the rationality of the spinor varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767023.png" /> is a non-degenerate quadratic form in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767024.png" /> variables over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767026.png" />). Spinor varieties are rational if either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767027.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767028.png" /> is locally compact and non-discrete, or is the field of rational numbers (see [[#References|[8]]], [[#References|[9]]], [[#References|[11]]]); for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767029.png" /> there are spinor varieties that are not rational [[#References|[8]]]. The last result is astonishing, in that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767030.png" /> is a two-sheeted covering of the rational variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767031.png" />.

The term "rationality theorem" is sometimes used in the theory of algebraic groups in a somewhat different sense, in connection with assertions about the properties of groups over a, not necessarily algebraically closed, field, such as the Rosenlicht–Grothendieck theorem, which states that any connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767032.png" />-group possesses a maximal torus defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767033.png" /> (see [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Chevalley, "On algebraic group varieties" ''J. Math. Soc. Japan'' , '''6''' : 3/4 (1954) pp. 303–324 {{MR|0067122}} {{ZBL|0057.26301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Demazure, A. Grothendieck, "Schemas en groupes II" , ''Lect. notes in math.'' , '''152''' , Springer (1970) {{MR|0274458}} {{MR|0274459}} {{MR|0274460}} {{ZBL|0209.24201}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M. Rosenlicht, "Some rationality questions on algebraic groups" ''Ann. Mat. Pura Appl.'' , '''43''' (1957) pp. 25–50 {{MR|0090101}} {{ZBL|0079.25703}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.E. Voskresenskii, "Algebraic tori" , Moscow (1977) (In Russian) {{MR|0506279}} {{MR|0472845}} {{ZBL|0379.14001}} {{ZBL|0367.14007}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> V.P. Platonov, "Reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767034.png" />-theory and approximation in algebraic groups" ''Proc. Steklov Inst. Math.'' , '''142''' (1976) pp. 213–224 ''Trudy Mat. Inst. Steklov.'' , '''142''' (1976) pp. 198–207 {{MR|568310}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.P. Platonov, "Birational properties of the reduced Whitehead group" ''Dokl. Akad. Nauk BSSR'' , '''21''' : 3 (1977) pp. 197–198 (In Russian) {{MR|0432603}} {{ZBL|0791.20045}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V.P. Platonov, "On the problem of rationality of spinor varieties" ''Soviet Math. Dokl.'' , '''20''' : 5 (1979) pp. 1027–1031 ''Dokl. Akad. Nauk SSSR'' , '''248''' : 3 (1979) pp. 524–527 {{MR|553481}} {{ZBL|0435.20028}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> V.P. Platonov, "Birational properties of spinor varieties" ''Proc. Steklov Inst. Math.'' , '''157''' (1981) pp. 173–182 ''Trudy Mat. Inst. Steklov.'' , '''157''' (1981) pp. 161–169 {{MR|0651765}} {{ZBL|0496.14008}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0181643}} {{ZBL|0143.05901}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> V.I. Chernousov, "On the rationality of spinor varieties over the rational number field" ''Dokl. Akad. Nauk BSSR'' , '''25''' : 4 (1981) pp. 293–296 (In Russian) (English abstract) {{MR|0615460}} {{ZBL|}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> V.I. Yanchevskii, "Reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767035.png" />-theory. Applications to algebraic groups" ''Math. USSR Sb.'' , '''38''' : 4 (1981) pp. 533–548 ''Mat. Sb.'' , '''110''' : 4 (1979) pp. 579–596 {{MR|1331389}} {{MR|0919253}} {{MR|0772116}} {{MR|0684770}} {{MR|0549289}} {{MR|0562210}} {{MR|0509375}} {{MR|0508832}} {{ZBL|}} </TD></TR></table>



====Comments====
Unirationality results imply density results for rational points. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767036.png" /> is a connected reductive group over an infinite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767037.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767039.png" />-rational points is Zariski-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077670/r07767040.png" />.

The Rosenlicht–Grothendieck theorem is often simply called the Grothendieck theorem.