Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and it’s particularly evident when you realize that a spectral space is the same thing as a Priestly space — a Stone space equipped with a Stone partial ordering of its points (recording the constructively topology and the specialization order).

Under Hochster duality, many concepts in topology and algebraic geometry have dual concepts. This comes up for example in tensor triangulated geometry, where the Balmer spectrum seems to live on the other side of Hochster duality from the Zariski spectrum of the unit. For example, I think that if $A$ is an integral domain, then $Spec(A)$ is Hochster dual to some $Spec(B)$ where $B$ is a local ring.

Question: What are some more examples of Hochster dual concepts?

I’d like to create a big list.

Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and it’s particularly evident when you realize that a spectral space is the same thing as a Priestley space — a Stone space equipped with a Stone partial ordering of its points (recording the constructively topology and the specialization order).

Under Hochster duality, many concepts in topology and algebraic geometry have dual concepts. This comes up for example in tensor triangulated geometry, where the Balmer spectrum seems to live on the other side of Hochster duality from the Zariski spectrum of the unit. For example, I think that if $A$ is an integral domain, then $Spec(A)$ is Hochster dual to some $Spec(B)$ where $B$ is a local ring.

Question: What are some more examples of Hochster dual concepts?

I’d like to create a big list.