**Memoirs of the American Mathematical Society**

2017;
215 pp;
Softcover

MSC: Primary 11; 12; 13; 14; 20;

**Print ISBN: 978-1-4704-2409-1
Product Code: MEMO/248/1176**

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

**Electronic ISBN: 978-1-4704-4054-1
Product Code: MEMO/248/1176.E**

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AMS Member Price: $45.00

MAA Member Price: $67.50

# Rationality Problem for Algebraic Tori

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*Akinari Hoshi; Aiichi Yamasaki*

The authors 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.

The authors 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 \(5\).

The authors 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, the suthors determine all the flabby and coflabby
\(G\)-lattices of rank up to \(6\) and verify that they
are stably permutation. The authors also show that the Krull-Schmidt
theorem for \(G\)-lattices holds when the rank \(\leq
4\), and fails when the rank is \(5\).

#### Table of Contents

# Table of Contents

## Rationality Problem for Algebraic Tori

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Chapter 2. Preliminaries: Tate cohomology and flabby resolutions 2330
- Chapter 3. CARAT ID of the โค-classes in dimensions 5 and 6 2936
- Chapter 4. Krull-Schmidt theorem fails for dimension 5 3340
- 4.0. Classification of indecomposable maximal finite groups ๐บโค\GL(๐,\bZ) of dimension ๐โค6 3744
- 4.1. Krull-Schmidt theorem (1) 4148
- 4.2. Krull-Schmidt theorem (2) 5057
- 4.3. Maximal finite groups ๐บโค\GL(๐,\bZ) of dimension ๐โค6 5966
- 4.4. Bravais groups and corresponding quadratic forms 6168

- Chapter 5. GAP algorithms: the flabby class [๐_{๐บ}]^{๐๐} 7986
- 5.0. Determination whether ๐_{๐บ} is flabby (coflabby) 7986
- 5.1. Construction of the flabby class [๐_{๐บ}]^{๐๐} of the ๐บ-lattice ๐_{๐บ} 8188
- 5.2. Determination whether [๐_{๐บ}]^{๐๐} is invertible 8491
- 5.3. Computation of ๐ธ with [[๐_{๐บ}]^{๐๐}]^{๐๐}=[๐ธ] 8693
- 5.4. Possibility for [๐_{๐บ}]^{๐๐}=0 8996
- 5.5. Verification of [๐_{๐บ}]^{๐๐}=0: Method I 93100
- 5.6. Verification of [๐_{๐บ}]^{๐๐}=0: Method II 97104
- 5.7. Verification of [๐_{๐บ}]^{๐๐}=0: Method III 105112

- Chapter 6. Flabby and coflabby ๐บ-lattices 113120
- Chapter 7. ๐ปยน(๐บ,[๐_{๐บ}]^{๐๐})=0 for any Bravais group ๐บ of dimension ๐โค6 125132
- Chapter 8. Norm one tori 129136
- Chapter 9. Tate cohomology: GAP computations 133140
- Chapter 10. Proof of Theorem 1.27 139146
- Chapter 11. Proof of Theorem 1.28 153160
- Chapter 12. Proof of Theorem 12.3 165172
- Chapter 13. Application of Theorem 12.3 173180
- Chapter 14. Tables for the stably rational classification of algebraic ๐-tori of dimension 5 179186
- Bibliography 211218
- Back Cover Back Cover1228