Algebraic tori revisited

Let $K/k$ be a finite Galois extension and \pi = \fn{Gal}(K/k). An algebraic torus $T$ defined over $k$ is called a $\pi$-torus if T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n for some integer $n$. The set of all algebraic $\pi$-tori defined over $k$ under the stably isomorphism form a semigroup, denoted by $T(\pi)$. We will give a complete proof of the following theorem due to Endo and Miyata \cite{EM5}. Theorem. Let $\pi$ be a finite group. Then $T(\pi)\simeq C(\Omega_{\bm{Z}\pi})$ where $\Omega_{\bm{Z}\pi}$ is a maximal $\bm{Z}$-order in $\bm{Q}\pi$ containing $\bm{Z}\pi$ and $C(\Omega_{\bm{Z}\pi})$ is the locally free class group of $\Omega_{\bm{Z}\pi}$, provided that $\pi$ is isomorphic to the following four types of groups : $C_n$ ($n$ is any positive integer), $D_m$ ($m$ is any odd integer $\ge 3$), $C_{q^f}\times D_m$ ($m$ is any odd integer $\ge 3$, $q$ is an odd prime number not dividing $m$, $f\ge 1$, and $(\bm{Z}/q^f\bm{Z})^{\times}=\langle \bar{p}\rangle$ for any prime divisor $p$ of $m$), $Q_{4m}$ ($m$ is any odd integer $\ge 3$, $p\equiv 3 \pmod{4}$ for any prime divisor $p$ of $m$).Comment: To appear in Asian J. Math. ; the title is change

Retract Rational Fields

Let $k$ be an infinite field. The notion of retract $k$-rationality was introduced by Saltman in the study of Noether's problem and other rationality problems. We will investigate the retract rationality of a field in this paper. Theorem 1. Let $k\subset K\subset L$ be fields. If $K$ is retract $k$-rational and $L$ is retract $K$-rational, then $L$ is retract $k$-rational. Theorem 2. For any finite group $G$ containing an abelian normal subgroup $H$ such that $G/H$ is a cyclic group, for any complex representation $G \to GL(V)$, the fixed field $\bm{C}(V)^G$ is retract $\bm{C}$-rational. Theorem 3. If $G$ is a finite group, then all the Sylow subgroups of $G$ are cyclic if and only if $\bm{C}_{\alpha}(M)^G$ is retract $\bm{C}$-rational for all $G$-lattices $M$, for all short exact sequences $\alpha : 0 \to \bm{C}^{\times} \to M_{\alpha} \to M \to 0$. Because the unramified Brauer group of a retract $\bm{C}$-rational field is trivial, Theorem 2 and Theorem 3 generalize previous results of Bogomolov and Barge respectively (see Theorem \ref{t5.9} and Theorem \ref{t6.1}).Comment: Several typos in the previous version were correcte

Insider patent holder licensing in an oligopoly market with different cost structures: Fixed-fee, royalty, and auction

The issue of the optimal licensing contract in firms having different cost structures is studied when the innovator is a producing patent holder who has three alternative licensing strategies, namely, the fixed-fee, royalty rate, and auction strategies. We conclude that the auction licensing strategy is not the best strategy when the innovator is a producing patent holder. This finding differs from that of Kabiraj (2004) where the auction licensing method is the optimal licensing strategy when the innovator is a non-producing patent holder. However, when we only compare two of the licensing methods, namely, the fixed-fee licensing method and the royalty licensing method, we conclude that if the inside innovator licenses to only some of the firms, then the royalty licensing method will be the best strategy. This result is different from that of Fosfuri and Roca (2004), who concluded that if only some of the licensees obtain a licensing contract, then the fixed-fee licensing method will be the best choice for a producing patent holder.Licensing strategy, Cost structure, Auction