Finite numbers of initial ideals in non-Noetherian polynomial rings

TL;DR

This paper extends the finiteness of initial ideals from Noetherian polynomial rings to certain non-Noetherian rings with invariant ideal chains, revealing finiteness for $c=1$ and infinitude for larger $c$, addressing a question by Hillar, Kroner, Leykin.

Contribution

It generalizes the finiteness of initial ideals to non-Noetherian polynomial rings with invariant ideal chains under $Inc(N)$ action, providing a complete classification for $c=1$ and showing infinitude for $c>1$.

Findings

Finiteness of initial ideal chains for $c=1$.

Infinitude of compatible term orders for $c>1$.

Addresses a question by Hillar, Kroner, Leykin.

Abstract

In this article, we generalize the well-known result that ideals of Noetherian polynomial rings have only finitely many initial ideals to the situation of ascending ideal chains in non-Noetherian polynomial rings. More precisely, we study ideal chains in the polynomial ring $R=K[x_{i,j}\,|\,1\leq i\leq c,j\in N]$ that are invariant under the action of the monoid $Inc(N)$ of strictly increasing functions on $N$, which acts on $R$ by shifting the second variable index. We show that for every such ideal chain, the number of initial ideal chains with respect to term orders on $R$ that are compatible with the action of $Inc(N)$ is finite. As a consequence of this, we will see that $Inc(N)$-invariant ideals of $R$ have only finitely many initial ideals with respect to $Inc(N)$-compatible term orders. The article also addresses the question of how many such term orders exist. We give a complete list of the $Inc(N)$-compatible term orders for the case $c=1$ and show that there are infinitely many for $c >1$. This answers a question by Hillar, Kroner, Leykin.