Max Zeuner: Locally ringed lattices - a constructive approach to quasi-compact, quasi-separated schemes

Abstract

In this talk we introduce the category of locally ringed lattices. We will show that this is equivalent to a subcategory of locally ringed spaces, which contains quasi-compact, quasi-separated schemes as a full subcategory. We will then show how this category of locally ringed lattices can be used as the basis of a point-free and constructive development of algebraic geometry. In particular, we will give a constructive proof that the functor "Spec" taking a commutative ring R to the locally ringed lattice corresponding to the affine scheme Spec(R) is left adjoint to the global sections functor. We will also give a definition of quasi-compact, quasi-separated schemes in this setting and discuss examples of non-affine schemes if time permits.