# memo_has_moved_text();Rationality Problem for Algebraic Tori

Akinari Hoshi and Aiichi Yamasaki

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.

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Chapters

• Chapter 1. Introduction
• Chapter 2. Preliminaries: Tate cohomology and flabby resolutions
• Chapter 3. CARAT ID of the $\mathbb Z$-classes in dimensions $5$ and $6$
• Chapter 4. Krull-Schmidt theorem fails for dimension $5$
• Chapter 5. GAP algorithms: the flabby class $[M_G]^fl$
• Chapter 6. Flabby and coflabby $G$-lattices
• Chapter 7. $H^1(G,[M_G]^fl)=0$ for any Bravais group $G$ of dimension $n\leq 6$
• Chapter 8. Norm one tori
• Chapter 9. Tate cohomology: GAP computations
• Chapter 10. Proof of Theorem 1.27
• Chapter 11. Proof of Theorem 1.28
• Chapter 12. Proof of Theorem 12.3
• Chapter 13. Application of Theorem 12.3
• Chapter 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 and over a field . In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank and is given. We show that there exist exactly (resp. , resp. ) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension , and there exist exactly (resp. , resp. ) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension . We make a procedure to compute a flabby resolution of a -lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a -lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby -lattices of rank up to and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for -lattices holds when the rank , and fails when the rank is . Indeed, there exist exactly (resp. ) -lattices of rank (resp. ) which are decomposable into two different ranks. Moreover, when the rank is , there exist exactly -lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that for any Bravais group of dimension where is the flabby class of the corresponding -lattice of rank . In particular, for any maximal finite subgroup where . As an application of the methods developed, some examples of not retract (stably) rational fields over are given.