Induced Torus

A torus \(T\) over \(k\) is induced (synonym in the literature: quasi-trivial) if the lattice \(X^*(T)\) — equivalently \(X_*(T)\) — has a \(\mathbb{Z}\)-basis permuted by \(\Theta\), i.e. is a permutation module.

Product description. \(T\) is induced if and only if \[ T \;\cong\; \prod_{i} R_{E_i/k}\,\mathbb{G}_m \] for finitely many finite separable extensions \(E_i/k\). Indeed, a permutation basis decomposes into \(\Theta\)-orbits; the stabilizer of a basis vector is an open subgroup \(\Theta_i \subseteq \Theta\) (openness by continuity of the action on the discrete lattice), with fixed field \(E_i\), and the orbit spans a copy of \(\mathbb{Z}[\Theta/\Theta_i] = X^*(R_{E_i/k}\mathbb{G}_m)\) by Weil Restriction of Tori. Conversely a product of such Weil restrictions has permutation character lattice.

In particular \(T(k) \cong \prod_i E_i^\times\), and induced tori are exactly the tori whose arithmetic reduces to that of \(\mathbb{G}_m\) over finite separable extensions — the reason they serve as the resolving class for arbitrary tori (see Resolution of a Torus by Induced Tori).

Terminology warning (following the source). Induced tori are sometimes called quasi-split tori in the literature. This knowledge base avoids that usage: every torus is a quasi-split reductive group, so "quasi-split" carries no information about a torus.