Klyachko, A. A. \(K\)-theory of Demazure models. (English. Russian original) Zbl 0796.14031 Selected papers in \(K\)-theory. Transl., Ser. 2, Am. Math. Soc. 154, 37-46 (1992); translation from Arithmetic of Algebraic Varieties, Issled. Teor. Chisel, Saratov 8, 61-72 (1982). The Grothendieck ring \(K(x)\) for complete, nonsingular toric varieties (Demazure models) \(X\) is studied. At the beginning of the work, an explicit description of \(K(x)\) is given in combinatorial terms of the corresponding fan \(\Sigma(x)\) over an algebraically closed ground field. After that, the author studies the action of the Galois group on the ring \(K(x)\), if \(X\) is a smooth complete variety over a nonclosed field, which is a form of some Demazure model.For the entire collection see [Zbl 0782.00016]. Reviewer: V. Iskovskikh (R. Zh. Mat. 1983, 3A417) MSC: 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14L30 Group actions on varieties or schemes (quotients) 19E08 \(K\)-theory of schemes 14G27 Other nonalgebraically closed ground fields in algebraic geometry Keywords:Demazure models; fan; action of the Galois group; complete variety over a nonclosed field PDFBibTeX XMLCite \textit{A. A. Klyachko}, Transl., Ser. 2, Am. Math. Soc. 154, 1 (1982; Zbl 0796.14031); translation from Arithmetic of Algebraic Varieties, Issled. Teor. Chisel, Saratov 8, 61--72 (1982)