On linear systems and the divisor class group of a real variety (Q1061189)

Let k be a field of characteristic zero and non necessarily algebraically closed. If I is an ideal of the polynomial ring \(k[X_ 1,...X_ n]\), \(Z_ k(I)\) the set of zeroes of I in \(k^ n\), \(I_ k(V)\) the ideal of the polynomials vanishing on \(V\subseteq k^ n\), we call I ''ideal of definition'' (or ''defining ideal'') if \(I=I_ k(Z_ k(I))\). If A is the coordinate domain of an affine or projective algebraic variety, we denote by \(Cl_ k(A)\) the subgroup of Cl(A) \((=divisor\) class group of A) generated by the classes of prime ideals of definition. The main results of the paper are: (i) A is factorial if and only if every prime ideal of definition in A of height 1 is principal. (ii) If \(k={\mathbb{R}} = field\) of real numbers, dim V\(>1\), A normal, then \(Cl(A)=Cl_{{\mathbb{R}}}(A)\). - In other words, the author shows that the rôle played by all prime ideals of height one, in some arguments and in the case of an algebraically closed field k, when k is not algebraically closed, it is played by all prime ideals of definition of height one.