CiNii Books - Rationality problem for algebraic tori

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 $5$.

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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.

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Rationality problem for algebraic tori

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