Rationality problem for algebraic tori / Akinari Hoshi, Aiichi Yamasaki.

Format:

Book

Author/Creator:

Hoshi, Akinari, 1978- author.

Yamasaki, Aiichi, 1969- author.

Series:

Memoirs of the American Mathematical Society ; Volume 248, Number 1176.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 248, Number 1176

Language:

English

Subjects (All):

Representations of groups.

Affine algebraic groups.

Physical Description:

1 online resource (228 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2017.

Summary:

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.

Contents:

Cover

Title page

Chapter 1. Introduction

Chapter 2. Preliminaries: Tate cohomology and flabby resolutions

Chapter 3. CARAT ID of the ℤ-classes in dimensions 5 and 6

Chapter 4. Krull-Schmidt theorem fails for dimension 5

4.0. Classification of indecomposable maximal finite groups ≤\GL( ,\bZ) of dimension ≤6

4.1. Krull-Schmidt theorem (1)

4.2. Krull-Schmidt theorem (2)

4.3. Maximal finite groups ≤\GL( ,\bZ) of dimension ≤6

4.4. Bravais groups and corresponding quadratic forms

Chapter 5. GAP algorithms: the flabby class [ _{ }]^{ }

5.0. Determination whether _{ } is flabby (coflabby)

5.1. Construction of the flabby class [ _{ }]^{ } of the -lattice _{ }

5.2. Determination whether [ _{ }]^{ } is invertible

5.3. Computation of with [[ _{ }]^{ }]^{ }=[ ]

5.4. Possibility for [ _{ }]^{ }=0

5.5. Verification of [ _{ }]^{ }=0: Method I

5.6. Verification of [ _{ }]^{ }=0: Method II

5.7. Verification of [ _{ }]^{ }=0: Method III

Chapter 6. Flabby and coflabby -lattices

Chapter 7. ¹( ,[ _{ }]^{ })=0 for any Bravais group of dimension ≤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 -tori of dimension 5

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-4054-7