Rationality Problem for Algebraic Tori

ISBN-13: 9781470424091 / Angielski

Akinari Hoshi;Aiichi Yamasaki

Opis

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 $5$.

Spis treści

• Introduction
• Preliminaries: Tate cohomology and flabby resolutions
• CARAT ID of the $\mathbb$-classes in dimensions $5$ and $6$
• Krull-Schmidt theorem fails for dimension $5$
• GAP algorithms: the flabby class $[M_G]^$
• Flabby and coflabby $G$-lattices
• $H^1(G,[M_G]^)=0$ for any Bravais group $G$ of dimension $n\leq 6$
• Norm one tori
• Tate cohomology: GAP computations
• Proof of Theorem 1.27
• Proof of Theorem 1.28
• Proof of Theorem 12.3
• Application of Theorem 12.3
• Tables for the stably rational classification of algebraic $k$-tori of dimension $5$
• Bibliography.