References

A partial realisation of Hilbert’s programme has proved successful in commutative algebra [1]. One of the key tools is Joyal’s point-free version of the Zariski spectrum as a distributive lattice. Extending this to algebraic geometry requires to reformulate Grothendieck’s language of schemes with, in Hilbert’s sense, finite methods. It turns out that distributive lattices even suffice for all the schemes whose underlying topological spaces are spectral. This includes pivotal cases such as the projective spectrum of a graded ring [2], and simplifies the preceding approach undertaken in formal topology [3, 4, 5]