NOETHER NORMALIZATIONS, REDUCTIONS OF IDEALS, AND MATROIDS
Abbreviated Journal Title
Proc. Amer. Math. Soc.
BRIANCON-SKODA THEOREM; JOINT REDUCTIONS; TIGHT CLOSURE; GRAPHS; MULTIPLICITIES; ALGEBRAS; RINGS; Mathematics, Applied; Mathematics
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.
Proceedings of the American Mathematical Society
"NOETHER NORMALIZATIONS, REDUCTIONS OF IDEALS, AND MATROIDS" (2011). Faculty Bibliography 2010s. 1121.