Dimension of boolean valued lattices and rings

Abstract

A. Joyal initiated the dimension theory of rings in a topos. Joyal's notion of Krull dimension of lattices and rings was considered by the author, who has shown that dim K[ X] = 1 for any field K in a topos I . The basic aim of this paper is to prove that dim R[ X] = 1 for any regular ring R in I , that is by working in commutative algebra without choice and excluded middle. Given a regular ring R, let E be the boolean algebra of idempotents of R, and E = sh( E) the topos of sheaves over E with the finite cover topology. The Pierce representation R ̃ of R is a field in E , so that dim R ̃ [X] = 1 and this implies dim R[ X] = 1 by using preserving properties of the global sections functor Γ: E → I . Section 1 deals with lattices in the topos E = sh( E) of sheaves over a boolean algebra E with the finite cover topology. We characterize lattices in E as lattice homomorphisms E → D, and we consider the dimension of lattices in this form. In Section 2 we describe rings in E as boolean homomorphisms E → E( A). Here, we discuss the Pierce representation and polynomials. The spectrum of a ring is considered in Section 3, which ends with the aim theorem.