Wolfgang Rump

Pruefer domains and a fancy basis of the Euclidean square.

Abstract:

In his 1967 thesis, Melvin Hochster characterized the topological spaces underlying a scheme. In particular, his criterion yields a very simple characterization of the subspace of closed (i. e. visible) points, hence for the maximal ideal space of a commutative ring. Hochster tried to do the same for noetherian rings, without success. For special rings like Pruefer or Bezout domains (arising in number theory and functional analysis), only partial results could be obtained. Using an order-theoretic approach, we give a complete description for the latter two cases. For example, the unit square satisfies our criterion, that is, it arises as maximal spectrum of a Pruefer domain. Topologically, this means that there exists a special basis of simply connected open sets.