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Rationality Problem for Algebraic Tori
 
Akinari Hoshi :  Niigata University, Niigata, Japan
Aiichi Yamasaki :  Kyoto University, Kyoto, Japan
Rationality Problem for Algebraic Tori
eBook 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
Add to cart
Rationality Problem for Algebraic Tori
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Rationality Problem for Algebraic Tori
Akinari Hoshi :  Niigata University, Niigata, Japan
Aiichi Yamasaki :  Kyoto University, Kyoto, Japan
eBook 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
Add to cart
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 248; 2017; 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 \(5\).
  • 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. 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 $5$
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Memoirs of the American Mathematical Society
Volume: 248; 2017; 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 $5$
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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