TITLE: Continuous closure, axes closure, and natural closure
ABSTRACT: Let R be a finitely generated algebra over the complex number field. We consider three operations on ideals of R: continuous closure I cont (based on continuous maps on algebraic varieties), axes closure I ax (based on maps to so-called axes rings; coordinate rings of the union of the coordinate axes in Cn), and natural closure I natural (which can be seen in a number of different ways, e.g. using valuation theory). These apparently disparate operations are surprisingly closely related. One always has I natural is contained in I cont is contained in I ax. Brenner asked whether continuous and axes closures (both of which he originally defined) were always equal, showing some cases where they are. We show many cases where two or all of these closures coincide, but we ultimately give counterexamples showing that Brenner’s question has a negative answer and indeed, that all three closures are in general distinct. Along the way, we give connections with the Jacobian ideal (of first-order partial derivatives), monomial ideals, unique factorization, and seminormality. This work is joint with Mel Hochster.