Rationality problem for Algebraic Tori

by Hoshi, Akinari

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Title Statements

Full title :Rationality problem for Algebraic Tori / Akinari Hoshi, Aiichi Yamasaki.

Summary

"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 ≤ 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 H1(G, F) = 0 for any Bravais group G of dimension n ≤ 6 where F is the flabby class of the corresponding G-lattice of rank n. In particular, H1(G, F) = 0 for any maximal finite subgroup G ≤ GL(n, Z) where n ≤ 6. As an application of the methods developed, some examples of not retract (stably) rational fields over k are given."--Page v.

Edition Statements

Publication Date:2017

Publication Statements

Publisher(s):American mathematical Society
Place of Publication:Providence, Rhode Island
Language:eng
Publication Date:2017

Notes

Titre de l'écran-titre (visionné le 5 décembre 2017).

In AMS.

"Volume 248, number 1176 (second of 5 numbers), July 2017."

The Physical Object

Description:1 ressource en ligne (v, 215 pages) : : fichier PDF

Edition Identifiers

ISBN:9781470440541, 1470440547
oclc:on1028650261
LC Classification:QA176
Dewey Decimal:512/.2

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 toriChapter 9. Tate cohomology: GAP computations; Chapter 10. Proof of Theorem 1.27; Chapter 11. Proof of Theorem 1…