Counter-example to a Strong Matroid Minor Conjecture

Abstract The main result of this note asserts that a strong form of the Matroid Minor Conjecture due to Draisma is not true, that is, there exist properly ascending chains of $S_\infty $-stable ideals in the affine coordinate ring of the affine infinite Grassmannian, where $S_\infty $ is the infinite symmetric group. In fact, we explicitly construct such an ascending chain. His conjectures on topological Noetherian property for the affine infinite Grassmannian remain open though.