Let \(k\) be a field and \(A^n_k\) be the \(n\)-dimensional affine space over \(k\). The following two results are known to be fundamental in the study of torsors over \(k\):
(1) \(H^1_{\mathrm{Zar}}(A^n_k,\mathrm{GL}_n)=H^1_{\text{ét}}(A^n_k,\mathrm{GL}_n)=H^1_{\mathrm{fppf}}(A^n, \mathrm{GL}_n)\) and
(2) \(H^1_{\mathrm{Zar}}(A^n_k,\mathrm{GL}_n)=1\).
The first of these two results, a version of Hilbert 90, asserts that the Zariski topology - which is in general too coarse to deal with principal bundles - is fine enough to measure algebraic vector bundles. The second is equivalent to the theorem of Quillen and Suslin: All finitely generated projective modules over the polynomial ring \(k[t_1,t_2,\ldots,t_n]\) are free.
The main objective of the paper under review is to look at the above two questions in the case when \(k[t_1,t_2,\ldots,t_n]\) is replaced by the ring \(R_n=k[t_1^{\pm}{}^,\ldots,t_n{}^{\pm 1}]\) of Laurent polynomials in \(n\)-variables and \(\mathrm{GL}_n\) is replaced by some other affine smooth group scheme over \(R_n\). Assume \(k\) is algebraically closed of characteristic 0. Consider the class of Lie algebras \(L\) over \(R_n\) with the property that \(L\otimes_{R_n}S\simeq g\otimes_k S\) for some finite dimensional simple Lie algebra \(g\), and some cover \(S\) of \(R\) on the étale topology. Observe that in this way (for \(n=1\)) one obtains the affine Kac-Moody Lie algebras. All these algebras are parametrised by \(H^1_{\text{ét}}(R,\operatorname{Aut}(g))\), namely by torsors over \(R\) under \(\operatorname{Aut}(g)\). The main result of the paper is that any such torsor is isotrivial, i.e., trivialized by a finite étale extension of \(R_n\).