Theorem Subtori and Quotients via the Character Lattice
theorem admitted

Subtori and Quotients via the Character Lattice

Let \(T\) be a torus over \(k\) with character lattice \(X^*(T)\). Recall that a sublattice \(L \subseteq X^*(T)\) is saturated if \(X^*(T)/L\) is torsion-free.

For a \(\Theta\)-stable sublattice \(L \subseteq X^*(T)\) put \[ H_L \;=\; \bigcap_{\chi \in L} \ker \chi \;\subseteq\; T . \]

  1. \(H_L\) is a closed \(k\)-subgroup of multiplicative type with \(X^*(H_L) = X^*(T)/L\), and \(L \mapsto H_L\) is an inclusion-reversing bijection between \(\Theta\)-stable sublattices of \(X^*(T)\) and closed multiplicative-type \(k\)-subgroups of \(T\).
  2. \(H_L\) is a subtorus if and only if \(L\) is saturated.
  3. The quotient \(T/H_L\) is a torus with \(X^*(T/H_L) = L\), and the exact sequence \(1 \to H_L \to T \to T/H_L \to 1\) corresponds to the exact sequence \(0 \to L \to X^*(T) \to X^*(T)/L \to 0\) of \(\Theta\)-modules.
  4. Dually, quotient tori of \(T\) correspond to \(\Theta\)-stable saturated sublattices (their character lattices), and surjections of tori correspond to injections of character lattices.

In particular every subtorus and every quotient torus of \(T\) is again a torus, computed purely from the Galois lattice.

Proof

After base change to \(k_s\) the torus is diagonalizable and the anti-equivalence Diagonalizable Groups Are Dual to Character Groups is exact: a quotient of character groups \(X^*(T) \twoheadrightarrow X^*(T)/L\) corresponds to the closed subgroup \(D(X^*(T)/L) \subseteq D(X^*(T))\), which is exactly the common kernel \(H_L\) of the characters in \(L\); a subgroup \(L \subseteq X^*(T)\) corresponds to the quotient \(D(X^*(T)) \twoheadrightarrow D(L)\), whose kernel is \(H_L\). All constructions are functorial, so \(\Theta\)-stability of \(L\) is exactly what makes them descend to \(k\) by F-Tori Are Classified by Galois Modules.

For (2): \(H_L\) is a torus iff its character group \(X^*(T)/L\) is torsion-free, which is the definition of saturation. For (3): the characters of \(T\) vanishing on \(H_L\) are precisely those mapping to zero in \(X^*(H_L) = X^*(T)/L\), i.e. the elements of \(L\); the quotient group \(T/H_L\) is of multiplicative type with character group \(L\), and \(L\), being a subgroup of a lattice, is torsion-free, so \(T/H_L\) is a torus. (4) is (1)-(3) read through the anti-equivalence.

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