Rationality Problem for Algebraic Tori

DOI10.1090/memo/1176zbMath1411.11037arXiv1210.4525OpenAlexW2963695022MaRDI QIDQ5281642

Aiichi Yamasaki, Akinari Hoshi

Publication date: 26 July 2017

Published in: Memoirs of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1210.4525

zbMATH Keywords

Krull-Schmidt theoremTate cohomologyalgebraic torirationality problemstably rationalretract rationalflabby resolutionBravais group

Mathematics Subject Classification ID

Linear algebraic groups over arbitrary fields (20G15) Integral representations of finite groups (20C10) Actions of groups on commutative rings; invariant theory (13A50) Transcendental field extensions (12F20) Galois cohomology of linear algebraic groups (11E72) Rationality questions in algebraic geometry (14E08)

Related Items

Twisted forms of toric varieties, Explicit methods for the Hasse norm principle and applications toA_nandS_nextensions, Three-dimensional purely quasimonomial actions, Norm one Tori and Hasse norm principle, TORSORS AND STABLE EQUIVARIANT BIRATIONAL GEOMETRY, Explicit Tamagawa numbers for certain algebraic tori over number fields, Rationality problem for norm one tori for dihedral extensions, Equivariant Burnside groups and toric varieties, Algebraic construction of quasi-split algebraic tori, Chow's theorem for semi-abelian varieties and bounds for splitting fields of algebraic tori, Rationality problem for norm one tori, Rationality problem for norm one tori in small dimensions, The Hasse norm principle for \(A_n\)-extensions, On the Noether problem for torsion subgroups of tori, Norm one tori and Hasse norm principle. II: Degree 12 case, Multiplicative Invariant Fields of Dimension ≤6

Uses Software

• GAP
• CARAT
• HAP
• MersenneTwister
• Carat

Cites Work

• Retract rational fields
• Twisted symmetric group actions
• Rationality of three-dimensional quotients by monomial actions
• Rationality problem of three-dimensional monomial group actions
• Induced representations and projective modules
• Retract rational fields and cyclic Galois extensions
• Noether's problem over an algebraically closed field
• Multiplicative field invariants and the Brauer group
• Noether's problem for groups of order 32
• A note on monomial automorphisms
• On algebraic tori of norm type
• Finite unimodular groups of prime degree and circulants
• Principal homogeneous spaces under flasque tori; applications
• Multiplicative field invariants
• Rationality of finite groups of monomial automorphisms of k(x,y)
• Torsion free cancellation over orders
• Finite group actions on rational function fields
• Rational functions invariant under a finite Abelian group
• The permutation class group of a finite group
• Finite groups with trivial class groups
• Three-dimensional purely monomial group actions
• Multiplicative invariant theory
• Negative solutions to three-dimensional monomial Noether problem
• Rationality problem for generic tori in simple groups.
• Three-dimensional purely quasimonomial actions
• On representations of finite groups over valuation rings
• Noether's problem and unramified Brauer groups
• On the Krull-Schmidt theorem for integral group representations of rank 1
• The maximal finite groups of \(4\times 4\) integral matrices
• Unramified Brauer groups of finite and infinite groups
• The rationality problem for norm one tori
• The Bogomolov Multiplier of Finite Simple Groups
• Integral representations with trivial first cohomology groups
• Retract Rationality and Noether's Problem
• Noether's Problem and the Unramified Brauer Group for Groups of Order 64
• Rationality problem of three-dimensional purely monomial group actions: the last case
• Crystallographic Algorithms and Tables
• The transitive groups of degree up to eleven⁺
• Groups of Permutation Projective Dimension Two
• THE BRAUER GROUP OF QUOTIENT SPACES BY LINEAR GROUP ACTIONS
• On Maximal Finite Irreducible Subgroups of GL(n, Z ): III. The Nine Dimensional Case
• On a classification of the function fields of algebraic tori
• La $R$-équivalence sur les tores
• On Maximal Finite Irreducible Subgroups of GL(n, Z ): I. The Five and Seven Dimensional Cases
• On reducible and decomposable representations of orders.
• On the krull-schmidt-azumaya theorem for integral group rings
• Mersenne twister
• Irreducible Finite Integral Matrix Groups of Degree 8 and 10
• On certain lattices associated with generic division algebras
• Counting Crystallographic Groups in Low Dimensions
• BRAUER GROUPS OF FIELDS OF INVARIANTS OF ALGEBRAIC GROUPS
• Unramified Brauer groups of finite simple groups of Lie type A l
• The lattices of six-dimensional Euclidean space
• Cayley groups
• BIRATIONAL PROPERTIES OF LINEAR ALGEBRAIC GROUPS
• Corrections and Supplementaries to My Paper concerning Krull-Remak-Schmidt’s Theorem
• Quasi-permutation modules over finite groups
• Rationality of \(P\)-group actions
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