2017/08/25 by Lindner, Felicitas
#05E40 #13P10 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1708.07680
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[xi,j | 1≤ i≤ c,j∈ 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.