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2 Theorems and proofs

You can state and prove results, and refer to them, for instance Lemma 1 below.

Lemma 1

Splittings of \(0\to \mathbf{G}_m\to \tilde{G}\to G\to 0\) canonically biject with twisting elements for \(G\).


To give a splitting is to give a map \(\tilde{G}\to \mathbf{G}_m\) such that the composite \(\mathbf{G}_m\to {\tilde{G}}\to \mathbf{G}_m\) is the identity; then the induced map \({\tilde{G}}\to G\times \mathbf{G}_m\) is an injection with trivial kernel so is an isomorphism for dimension reasons. If \(\chi :{\tilde{G}}\to \mathbf{G}_m\) is such a character then \(\chi \) gives rise to an element of \(X^*({\tilde{T}})\) which is Galois-stable, whose image in \(\mathbf{Z}\) is 1, and which pairs to zero with each coroot (because \(\chi \) factors through the maximal torus quotient of \({\tilde{G}}\)). Conversely to give such a character is to give a splitting. Now one checks that \(\theta -\chi \) has image in \(\mathbf{Z}\) equal to zero so gives rise to an element of \(X^*(T)\) which is Galois-stable, and pairs with each simple coroot to 1—but this is precisely a twisting element for \(G\). Conversely if \(t\) is a twisting element for \(G\) then \(\theta -t\) gives a splitting of the exact sequence.