Rationality problem for algebraic tori
著者
書誌事項
Rationality problem for algebraic tori
（Memoirs of the American Mathematical Society, no. 1176）
American Mathematical Society, 2017
大学図書館所蔵 全10件
注記
"Volume 248, number 1176 (second of 5 numbers), July 2017"
Bibliography: p. 211-215
内容説明・目次
内容説明
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 $5$.
目次
Introduction
Preliminaries: Tate cohomology and flabby resolutions
CARAT ID of the $\mathbb{Z}$-classes in dimensions $5$ and $6$
Krull-Schmidt theorem fails for dimension $5$
GAP algorithms: the flabby class $[M_G]^{fl}$
Flabby and coflabby $G$-lattices
$H^1(G,[M_G]^{fl})=0$ 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.
「Nielsen BookData」 より