Let A be an Artinian local ring with algebraically closed residue field k, and let G be an affine smooth group scheme over A. The Greenberg functor F associates to G a linear algebraic group G := (F G)(k) over k, such that G ∼ = G(A). We prove that if G is a reductive group scheme over A, and T is a maximal torus of G, then T is a Cartan subgroup of G, and every Cartan subgroup of G is obtained uniquely in this way. Moreover, we prove that if G is reductive and P is a parabolic subgroup of G, then P is a selfnormalising subgroup of G, and if B and B ′ are two Borel subgroups of G, then the corresponding subgroups B and B ′ are conjugate in G.
Errata notesA previous version of the present paper has appeared in J. Pure Appl. Algebra., 216 (2012), 1092-1101. After the publication the author was notified by Cristian D. González-Avilés and Alessandra Bertapelle that the formula on p. 1094, l. 14 in the published version does not hold in general. More precisely, in [8], p. 636 Greenberg defined the local ring scheme A over k (using different notation). However, the formula A(R) = A ⊗ Wm(k) W m (R) does not hold for all k-algebras R. Nevertheless, it does hold when R is a perfect kalgebra. The formula occurs in [2], p. 276, l. -18, but was stated correctly by Loeser and Sebag [14], p. 318. Moreover, Nicaise and Sebag have given counter-examples for non-perfect algebras; see [16], 2.2.In the present paper we have corrected the statements involving the above formula by either removing the formula or adding the hypothesis that R is a perfect k-algebra. The corrections correspond to p. 1094, l. 14 and l. -17, as well as the second line of the proof of Lemma 2.3, in the published version. The formula is actually not necessary for the results of our paper, and can simply be ignored. All that is needed is that A is an affine local ring scheme over k, and that it is isomorphic to some affine space. As mentioned above, these facts were established by Greenberg.Apart from these modifications, the content of the present version remains identical to the published version.