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Register a new userOn the fiber cone of monomial ideals
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We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $Isubset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for interesting classes of height $2$ monomial ideals $Isubset K[x,y]$, which are generated by $4$ elements. For these classes of ideals we also show that $F(I)$ is Cohen--Macaulay if and only if the defining ideal $J$ of $F(I)$ is generated by at most 3 elements. In all the cases a minimal set of generators of $J$ is determined.
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We prove a characterization of the j-multiplicity of a monomial ideal as the normalized volume of a polytopal complex. Our result is an extension of Teissiers volume-theoretic interpretation of the Hilbert-Samuel multiplicity for m-primary monomial ideals. We also give a description of the epsilon-multiplicity of a monomial ideal in terms of the volume of a region.
Let $K$ be a field and $S = K[x_1,dots,x_n]$ be a polynomial ring over $K$. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals. More precisely, we give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of such a class of squarefree monomial ideals.
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.
An explicit combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic $0$ is defined canonically, without any choices, using higher-dimensional generalizations of combined spanning trees for cycles and cocycles (hedges) in the upper Koszul simplicial complexes of $I$ at lattice points in $mathbb{Z}^n$. The differentials in these sylvan resolutions are expressed as matrices whose entries are sums over lattice paths of weights determined combinatorially by sequences of hedges (hedgerows) along each lattice path. This combinatorics enters via an explicit matroidal expression for the Moore-Penrose pseudoinverses of the differentials in any CW complex as weighted averages of splittings defined by hedges. This Hedge Formula also yields a projection formula from CW chains to boundaries. The translation from Moore-Penrose combinatorics to free resolutions relies on Wall complexes, which construct minimal free resolutions of graded ideals from vertical splittings of Koszul bicomplexes. The algebra of Wall complexes applied to individual hedgerows yields explicit but noncanonical combinatorial minimal free resolutions of arbitrary monomial ideals in any characteristic.
Let I=I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial characterization for the unmixed property of I, when D is bipartite, D is a whisker or D is a cycle. Finally, we study the Cohen-Macaulay property of I.
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