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Rationality Problem for Algebraic Tori

Akinari Hoshi Niigata University, Niigata, Japan
Aiichi Yamasaki Kyoto University, Kyoto, Japan
Available Formats:
Electronic ISBN: 978-1-4704-4054-1
Product Code: MEMO/248/1176.E
List Price: $75.00 MAA Member Price:$67.50
AMS Member Price: $45.00 Click above image for expanded view Rationality Problem for Algebraic Tori Akinari Hoshi Niigata University, Niigata, Japan Aiichi Yamasaki Kyoto University, Kyoto, Japan Available Formats:  Electronic ISBN: 978-1-4704-4054-1 Product Code: MEMO/248/1176.E  List Price:$75.00 MAA Member Price: $67.50 AMS Member Price:$45.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 2482017; 215 pp
MSC: 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 Krull-Schmidt 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. Krull-Schmidt 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 $5$

• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2482017; 215 pp
MSC: 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 Krull-Schmidt 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. Krull-Schmidt 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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