In 1988 it was proved by the first author that the closure of a partially semialgebraic set is partially semialgebraic. The essential tool used in that proof was the regular separation property. Here we give another proof without using this tool, based on the semianalytic L-cone theorem (Theorem 2), a semianalytic analog of the Cartan-Remmert-Stein lemma with parameters.
Let be a coherent subsheaf of a locally free sheaf and suppose that has pure codimension. Starting with a residue current obtained from a locally free resolution of we construct a vector-valued Coleff-Herrera current with support on the variety associated to such that is in if and only if . Such a current can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed. By a construction...