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Rationality Problem for Algebraic Tori

About this Title

Akinari Hoshi, Department of Mathematics, Niigata University, Niigata 950-2181, Japan and Aiichi Yamasaki, Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 248, Number 1176
ISBNs: 978-1-4704-2409-1 (print); 978-1-4704-4054-1 (online)
DOI: https://doi.org/10.1090/memo/1176
Published electronically: March 9, 2017
Keywords: Rationality problem, algebraic tori, stably rational, retract rational, flabby resolution, Krull-Schmidt theorem, Bravais group, Tate cohomology.
MSC: Primary 11E72, 12F20, 13A50, 14E08, 20C10, 20G15.

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Preliminaries: Tate cohomology and flabby resolutions
  • 3. CARAT ID of the $\mathbb {Z}$-classes in dimensions $5$ and $6$
  • 4. Krull-Schmidt theorem fails for dimension $5$
  • 5. GAP algorithms: the flabby class $[M_G]^{fl}$
  • 6. Flabby and coflabby $G$-lattices
  • 7. $H^1(G,[M_G]^{fl})=0$ for any Bravais group $G$ of dimension $n\leq 6$
  • 8. Norm one tori
  • 9. Tate cohomology: GAP computations
  • 10. Proof of Theorem
  • 11. Proof of Theorem
  • 12. Proof of Theorem
  • 13. Application of Theorem
  • 14. Tables for the stably rational classification of algebraic $k$-tori of dimension $5$

Abstract

We give the complete stably rational classification of algebraic tori of dimensions $4$ and $5$ over a field $k$. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank $4$ and $5$ is given. We show that there exist exactly $487$ (resp. $7$, resp. $216$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $4$, and there exist exactly $3051$ (resp. $25$, resp. $3003$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $5$. We make a procedure to compute a flabby resolution of a $G$-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a $G$-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby $G$-lattices of rank up to $6$ and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for $G$-lattices holds when the rank $\leq 4$, and fails when the rank is $5$. Indeed, there exist exactly $11$ (resp. $131$) $G$-lattices of rank $5$ (resp. $6$) which are decomposable into two different ranks. Moreover, when the rank is $6$, there exist exactly $18$ $G$-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that $H^1(G,F)=0$ for any Bravais group $G$ of dimension $n\leq 6$ where $F$ is the flabby class of the corresponding $G$-lattice of rank $n$. In particular, $H^1(G,F)=0$ for any maximal finite subgroup $G\leq \textrm {GL}(n,\mathbb {Z})$ where $n\leq 6$. As an application of the methods developed, some examples of not retract (stably) rational fields over $k$ are given.

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