  We show that given a finitely generated standard-graded algebra of dimension
$d$ over an infinite field, its graded Noether normalizations obey a certain
kind of `generic exchange', allowing one to pass between any two of them in at
most $d$ steps. We prove analogous generic exchange theorems for minimal
reductions of an ideal, minimal complete reductions of a set of ideals, and
minimal complete reductions of multigraded $k$-algebras. Finally, we unify all
these results into a common axiomatic framework by introducing a new
topological-combinatorial structure we call a generic matroid, which is a
common generalization of a topological space and a matroid.

Commutative Algebra;Combinatorics