Induction theorems on the stable rationality of the center of the ring of generic matrices

Beneish, Esther

doi:10.1090/s0002-9947-98-02202-8

Cited by 16 publications

(11 citation statements)

References 12 publications

(4 reference statements)

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“…When n is 2, 3, or 4 then ZÔF, nÕ is rational, as proved (respectively) by Sylvester, Procesi and Formanek. When n is 5 or 7 Bessenrodt and Lebruyn showed in [BL91] that ZÔF, nÕ is stably rational, and a second, more elementary proof of this was given by Beneish [Ben98].…”

Section: Center Of Generic Matricesmentioning

confidence: 98%

Open problems on central simple algebras

Auel

Brussel

Garibaldi

et al. 2011

Transformation Groups

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Abstract. We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners. Motivated by these and other developments, we now present an updated list of open problems. Some of these problems are-or have become-special cases of much more general problems for algebraic groups. To keep our task manageable, we (mostly) restrict our attention to central simple algebras and PGL n . The following is an idiosyncratic list that originated during the Brauer group conference held at Kibbutz Ketura in January 2010. We thank the participants in that problem session for their contributions to this text. We especially thank Darrel Haile,

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Section: Center Of Generic Matricesmentioning

confidence: 98%

Open problems on central simple algebras

Auel

Brussel

Garibaldi

et al. 2011

Transformation Groups

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“…The following lemma was proved by Bessenrodt and Lebruyn but unpublished. It was proved in Beneish [Ben98]. Here is a simpler proof.…”

Section: Stable Rationality Of Exceptional Torimentioning

confidence: 75%

“…Note that, for a prime p, Beneish [Ben98] proved that the S p -lattice J Xp is flasque equivalent to…”

Section: Stable Rationality Of Exceptional Torimentioning

confidence: 99%

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Four-Dimensional Algebraic Tori

Lemire

2015

Preprint

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The study of the birational properties of algebraic k-tori began in the sixties and seventies with work of Voskresenkii, Endo, Miyata, Colliot-Thélène and Sansuc. There was particular interest in determining the rationality of a given algebraic k-tori. As rationality problems for algebraic varieties are in general difficult, it is natural to consider relaxed notions such as stable rationality, or even retract rationality. Work of the above authors and later Saltman in the eighties determined necessary and sufficient conditions to determine when an algebraic torus is stably rational, respectively retract rational in terms of the integral representations of its associated character lattice. An interesting question is to ask whether a stably rational algebraic k-torus is always rational. In the general case, there exist examples of non-rational stably rational k-varieties. Algebraic k-tori of dimension r are classified up to isomorphism by conjugacy classes of finite subgroups of GLr(Z). This makes it natural to examine the rationality problem for algebraic k tori of small dimensions. In 1967, Voskresenskii [Vos67] proved that all algebraic tori of dimension 2 are rational. In 1990, Kunyavskii [Kun87] determined which algebraic tori of dimension 3 were rational. In 2012, Hoshi and Yamasaki [HY12] determined which algebraic tori of dimensions 4 and 5 were stably (respectively retract) rational with the aid of GAP. They did not address the rationality question in dimensions 4 and 5. In this paper, we show that all stably rational algebraic k-tori of dimension 4 are rational, with the possible exception of 10 undetermined cases. Hoshi and Yamasaki found 7 retract rational but not stably rational dimension 4 algebraic k-tori. We give a non-computational proof of these results.(1) K/k (G m ) in the A 5 case. Hoshi and Yamasaki [HY12] used GAP to show that in fact R(1) K/k (G m ) is stably rational in this case. In this paper, we present a non-computational proof of that fact. The norm one torus R(1) K/k (G m ) corresponding to A 5 has character lattice J A 5 /A 4 . It is one of the ten algebraic k-tori of dimension 4 which is stably rational but whose rationality is unknown. Another of these exceptional stably rational algebraic k-torus of dimension 4 whose rationality is unknown is intimately related to this norm one torus. In fact its splitting group is A 5 × C 2 and its character lattice restricts to J A 5 /A 4 on A 5 . The character lattices of the remaining 8 stably rational algebraic k-tori whose rationality is unknown are also intimately related. See Section 5 for more details.

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“…Proof. Bessenrodt and Le Bruyn proved that C(M n × M n ) P GLn is stably rational for n = 5, 7 ([3, Theorem 1], see also [2]). They used this to deduce that if V is a finite dimensional complex representation of PGL n such that PGL n acts almost freely (i.e., there is a Zariski open subset of V such that every point of this open set has a trivial stabilizer), then for any h dividing 420 = 3.4.5.7, the field of invariants C(V ) PGL n is stably rational [3,Theorem 2].…”

Section: Thus We Have Dimmentioning

confidence: 99%