Rousseau  has shown that classical or standard function theory of n variables is no other than intuitionistic function theory of one variable over _C_n–1. Similar works have been done by Nishimura  in the realm of Sato hyperfunctions and by Takeuti and Titani  in the realm of complex manifolds. The main purpose of this paper is to pursue similar results in the arena of algebraic geometry. Since we would like to do so in an intuitionistically valid manner, we reconstruct some rudiments of algebraic geometry, using the complete Heyting algebra of radical ideals in place of the space of prime ideals with Zariski topology as the starting point of our scheme theory.
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Hirokazu Nishimura, Some Connections between Heyting Valued Set Theory and Algebraic Geometry —Prolegomena to Intuitionistic Algebraic Geometry—. Publ. Res. Inst. Math. Sci. 23 (1987), no. 3, pp. 501–515DOI 10.2977/PRIMS/1195176443