Speaker: Ngo Viet Trung
Time: 9h00, Wednesday, January 24, 2018 Location: 611-612, Building A6, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi Abstract: We investigate the behavior of the function $depth R/I^{(t)}$, $t ge 1$, where $I^{(t)}$ is the $t$-th symbolic power of a homogeneous ideal $I$ in a polynomial ring $R$.
Our first result shows that if $I$ is an arbitrary squarefree monomial ideal, the function $depth R/I^{(t)}$ is almost non-increasing in the sense that $depth R/I^{(s)} ge depth R/I^{(t)}$ for all $s ge 1$ and $t in E(s)$, where $$E(s) = {t in NN| t = c(s-1)+r text{for} c ge 1, 1 le r le s-1, r le c}$$ (e.g. $depth R/I ge depth R/I^{(t)}$ for $t ge 2$, $depth R/I^{(2)} ge depth R/I^{(t)}$ for $t ge 3$, $depth R/I^{(3)} ge depth R/I^{(t)}$ for $t ge 5$, $depth R/I^{(4)} ge depth R/I^{(t)}$ for $t = 7,8$ or $t ge 10$, etc.) We also show that there exist squarefree monomial ideals $I$ such that $depth R/I^{(s)} < depth R/I^{(t)}$ if $t otin E(s)$.
If $I$ is an arbitrary monomial ideal, we know that the function $depth R/I^{(t)}$ is asymptotically periodic (i.e. periodic for $t gg 0$). However, it was not clear whether this function is always convergent. We show that if $I^{(t)}$ is integrally closed for $t gg 0$ (e.g. if $I$ is squarefree), then the function $depth R/I^{(t)}$ is convergent with $lim_{t to infty}depth R/I^{(t)} = liminf_{t to infty}depth R/I^{(t)}.$ On the other hand, we are able to construct examples of monomial ideals for which the function $depth R/I^{(t)}$ is not convergent.
Our last result shows that for any asymptotically periodic positive numerical function $phi(t)$, there exist a polynomial ring $R$ and a homogeneous ideal $I$ such that $depth R/I^{(t)} = phi(t)$ for all $t ge 1$. It is not known whether there exists any function $depth R/I^{(t)}$ which is not asymptotically periodic at all. |