Comparison of stability indices of powers of graded ideals
Antonino Ficarra, Emanuele Sgroi
TL;DR
The paper analyzes the stability indices of powers of graded ideals, comparing the index of ass-stability $\text{astab}(I)$ with the index of $\mathrm{v}$-stability $\text{vstab}(I)$. It proves that in dimension $\dim(S)\le 2$ one has $\text{astab}(I)=1\le \text{vstab}(I)$ for all graded ideals and that $\text{vstab}(I)$ can be any positive integer, via an explicit planar example with $I=(x^{2b+1},x^2y^{2b-1},y^{2b+1})$. In contrast, for $\dim(S)\ge 3$ they construct, for any $a,b\ge1$, a graded monomial ideal $I$ with $(\text{astab}(I),\text{vstab}(I))=(a,b)$, using a layered sum of edge ideals and a final augmentation to control $v(I^k)$ piecewise, yielding $\text{vstab}(I)=b$. These results show that the two indices are not comparable in general, while in low dimension $\text{astab}$ is fixed and $\text{vstab}$ can be tuned arbitrarily.
Abstract
In this paper, we compare the index of ass-stability $\text{astab}(I)$ and the index of $\text{v}$-stability $\text{vstab}(I)$ of powers of a graded ideal $I$. We prove that $\text{astab}(I)=1\le\text{vstab}(I)$ for any graded ideal $I$ in a 2-dimensional polynomial ring, and that $\text{vstab}(I)$ can be any positive integer in this situation. Moreover, given any integers $a,b\ge1$, we construct a graded ideal $I$ in a $3(a+1)$-dimensional polynomial ring such that $(\text{astab}(I),\text{vstab}(I))=(a,b)$.