The prime spectrum of a distributive lattice

Christopher Tedd (University Of Manchester, School of Mathematics)

Frank Adams 2, Alan Turing Building,

The spectral spaces are the topological spaces that are sober, quasi-compact, that have a basis of quasi-compact open sets, and for which the quasi-compact open sets are closed under finite intersections. It is well known that given a ring R, then Spec R, the prime spectrum of R, is a spectral space; Hochster (1969) proved that every spectral space is homeomorphic to Spec R for some ring R.
Stone (1938) gave a topological proof that every bounded distributive lattice is isomorphic to a lattice of sets (ordered by inclusion): given a (bounded, distributive) lattice L we construct a topological space X from L such that L is isomorphic to the lattice of quasi-compact open sets of X. The space in question takes for its points the prime (lattice) ideals of L and is denoted the prime spectrum, Spec L. It turns out that the class of spaces so arising are exactly the spectral spaces. In fact, the construction on lattices gives a (dual) equivalence of categories between spectral spaces and lattices; something which is far from the case where rings are concerned.
In this talk I will introduce Stone's result, and indicate some of the possibilities and difficulties present in this connection between rings and lattices.

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