Getting Stone duality from the duality between sober spaces and spatial frames

Solution 1:

The equivalence between spatial frames and sober spaces restricts to an equivalence between coherent spaces and coherent frames. The latter are in turn equivalent to distributive lattices, so you get an equivalence between distributive lattices and coherent spaces. This is the equivalence that restricts to Stone duality, for Boolean algebras are contained in distributive lattices and correspond precisely to Hausdorff coherent spaces.

Solution 2:

Coherent frames are precisely the ideal completions of distributive lattices. There is an equivalence between coherent frames and distributive lattices given by the functor which sends a distributive lattice to the lattice of its ideals and conversely the functor which sends a coherent frame to the distributive lattice of its compact elements.

Stone duality between Boolean algebras and Stone spaces can, in a way, be seen as a restriction of the duality between spatial locales and sober spaces, but your point about Boolean algebras not necessarily being frames is completely valid. To construct the dual of a spatial frame, you take completely prime filters, while to construct the dual of a Boolean algebra you take prime filters, so what's going on here is not a restriction in the obvious sense of the word.

You should think of Stone duality as the end point of this sequence of dualities:

spatial frames – sober spaces

coherent frames – coherent spaces (by restricting both sides)

distributive lattices – coherent spaces (by composing with the above equivalence, distributive lattices – coherent frames)

Boolean algebras – Stone spaces (by restricting both sides)