nLab Hochster duality

Idea

Given a coherent space, $X$, a new topology may be constructed by taking as basic open subsets the closed sets of $X$ with quasicompact complements. This space $X^{\vee}$ is called the Hochster dual of $X$. The space $X^{\vee}$ is also coherent and $X^{\vee \vee} = X$.

The Hochster dual of a distributive lattice is the opposite lattice. The Hochster dual of a coherent frame is its join completion.

References

The original source is

• M. Hochster, Prime ideal structure in commutative rings, Transactions of the American Mathematical Society, 142, (1969), 43–60

See also:

• Joachim Kock, Wolfgang Pitsch, Hochster duality in derived categories and point-free reconstruction of schemes, (arXiv: 1305.1503)