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OverviewThe 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 $5$. Full Product DetailsAuthor: Akinari Hoshi , Aiichi YamasakiPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.340kg ISBN: 9781470424091ISBN 10: 1470424096 Pages: 215 Publication Date: 30 June 2017 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsIntroduction 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.ReviewsAuthor InformationAkinari Hoshi, Niigata University, Japan. Aiichi Yamasaki, Kyoto University, Japan. Tab Content 6Author Website:Countries AvailableAll regions |
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