The main theme of the paper is to use group functors to prove the normality of elementary subgroups in a classical group or Chevalley group over a ring under certain assumptions on the base ring generalizing the theorem that the subgroup generated by the elementary matrices is normal in the group of all invertible matrices over a field.
Given a category \(\Lambda\) with objects being the pairs \((k_M,M)\) of commutative rings (with 1) \(k_M\) and \({\mathcal I}\)-graded \(k_M\)-modules \(M=\bigoplus_{i\in{\mathcal I}}M_i\) with a fixed index set \(\mathcal I\) (which corresponds to the set of roots in the case of Chevalley groups). The morphisms are pairs of mappings with the first being ring homomorphisms while the second being semilinear mappings.
The authors require the category \(\Lambda\) to be closed under the localization construction, i.e., if \((k_M,M)\) is in \(\Lambda\), so are \((S^{-1}k_M,S^{-1}M)\) and the natural morphism \((k_M,M)\to(S^{-1}k_M,S^{-1}M)\). A \(\Lambda\)-group functor \(G\) is a group functor from \(\Lambda\) to the category of groups such that for each \(i\in{\mathcal I}\) there is a fixed injective natural transformation \(\varepsilon_i:\lambda_i\to G\), where \(\lambda_i(M)=M_i\) defines a group functor from \(\Lambda\) to the category of Abelian groups. The group functor \(E\) is defined such that \(E(k_M,M)\) is the subgroup of \(G(k_M,M)\) generated by all \(\varepsilon_i(k_M,M)\) (the root subgroups in the case of Chevalley groups). The main theorem states that for each \((k_M,M)\), under certain conditions related to the localization at elements not in the Jacobson radical of \(k_M\), \(E(k_M,M)\) is normal in \(G(k_M,M)\). Then this result is applied to classical groups and Chevalley groups over rings, improving certain results obtained earlier by many people. It should be noted that the result of the paper does not prove the normality of the subgroup functor \(E\) in a group functor \(G\) unless the category \(\Lambda\) is carefully chosen (so that the conditions are satisfied for all objects and compatible with the morphisms).