Electronic ISBN:  9781470440541 
Product Code:  MEMO/248/1176.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 248; 2017; 215 ppMSC: Primary 11; 12; 13; 14; 20;
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 KrullSchmidt theorem for \(G\)lattices holds when the rank \(\leq 4\), and fails when the rank is \(5\). 
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. KrullSchmidt 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 $5$


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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 KrullSchmidt theorem for \(G\)lattices holds when the rank \(\leq 4\), and fails when the rank is \(5\).

Chapters

1. Introduction

2. Preliminaries: Tate cohomology and flabby resolutions

3. CARAT ID of the $\mathbb {Z}$classes in dimensions $5$ and $6$

4. KrullSchmidt 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 $5$