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Register a new userThe saturation number of $cb$-bounded stable monomial ideals and their powers
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Let $S=K[x_1,ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$. In this paper, we compute the socle of $cb$-bounded strongly stable ideals and determine that the saturation number of strongly stable ideals and of equigenerated $cb$-bounded strongly stable ideals. We also provide explicit formulas for the saturation number $sat(I)$ of Veronese type ideals $I$. Using this formula, we show that $sat(I^k)$ is quasi-linear from the beginning and we determine the quasi-linear function explicitly.
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Let $A = K[X_1,ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n colon J^infty)$ be the $n^{th}$ symbolic power of $I$ wrt $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of quasi-polynomial type, say of period $g$ and degree $c$. For $n gg 0$ say [ f^I_J(n) = a_c(n)n^c + a_{c-1}(n)n^{c-1} + text{lower terms}, ] where for $i = 0, ldots, c$, $a_i colon mathbb{N} rt mathbb{Z}$ are periodic functions of period $g$ and $a_c eq 0$. In an earlier paper we (together with Herzog and Verma) proved that $dim I_n(J)/I^n$ is constant for $n gg 0$ and $a_c(-)$ is a constant. In this paper we prove that if $I$ is generated by some elements of the same degree and height $I geq 2$ then $a_{c-1}(-)$ is also a constant.
Given a number $q$, we construct a monomial ideal $I$ with the property that the function which describes the number of generators of $I^k$ has at least $q$ local maxima.
Let $S=K[x_1,ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $Isubset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u in S$ belonging to the socle $Soc(S/I^{k})$ of $S/I^{k}$, i.e., $u otin I^{k}$ and $ux_{i} in I^{k}$ for $1 leq i leq n$. We prove that if a monomial $x_1^{a_1}cdots x_n^{a_n}$ belongs to $Soc(S/I^{k})$, then $a_ileq k-1$ for all $1 leq i leq n$. We then discuss squarefree monomial ideals $I subset S$ for which $x_{[n]}^{k-1} in Soc(S/I^{k})$, where $x_{[n]} = x_{1}x_{2}cdots x_{n}$. Furthermore, we give a combinatorial characterization of finite graphs $G$ on $[n] = {1, ldots, n}$ for which $depth S/(I_{G})^{2}=0$, where $I_{G}$ is the edge ideal of $G$.
We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo-Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this we show the before-now unknown convergence of Stanley depths of integral closure powers. In addition, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.
This paper is concerned with the question of whether geometric structures such as cell complexes can be used to simultaneously describe the minimal free resolutions of all powers of a monomial ideal. We provide a full answer in the case of square-free monomial ideals of projective dimension one, by introducing a combinatorial construction of a family of (cubical) cell complexes whose 1-skeletons are powers of a graph that supports the resolution of the ideal.
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