Questions tagged [monomial-ideals]
A monomial ideal in a polynomial ring is an ideal generated by monomials.
30 questions
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Deformations of subalgebras of polynomial ring modulo monomial ideal
Let $V$ be a vector space of dimension $n$ over $\mathbb C$ with fixed basis $v_1,\dots,v_n$, and let $A$ and $B$ be associative algebras corresponding to a multiplicative structure on $V$. In the ...
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Canonical module of a squarefree monomial ideal
I am trying to understand Theorem 4 from the paper
Gräbe, Hans-Gert, The canonical module of a Stanley-Reisner ring, J. Algebra 86, 272-281 (1984). ZBL0533.13003.
This theorem describes the canonical ...
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Projective dimension and certain subideals
My question is related to this one. I thought mine could be very elementary but I'm not sure how to look into it.
Let $J$ be an ideal of $R=k[\mathbf{x}]$ where $\mathbf{x}=\{x_1,\dots,x_n\}$. Let $\{...
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Symmetric algebra of monomial ideal modulo a binomial ideal isomorphic to a quotient of a polynomial ring
Note: I posted this question on MSE but haven't received any response.
I’m trying to understand the proof of Corollary 1.9 in “Binomial ideals” by David Eisenbud and Bernd Sturmfels.
Notation: Let $S= ...
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Is Koszul homology of a monomial ideal always generated by the "obvious" things?
Let $R = k[x_1 , \dots , x_n]$ be a polynomial ring over a field and $I$ a monomial ideal in $R$. Then, is it true that the Koszul homology of $R/I$ is always generated by elements of the form
$$r e_{...
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Are there characteristic-dependent Betti numbers in characteristic not equal to two?
Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\...
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Reference request: Matroid cryptomorphisms for arbitrary monomial ideals
For a matroid $M$ let $C$ be the circuit ideal of $M$, that is, the Stanley-Reisner ideal of independence complex of $M$. Then there are simple ideal-theoretic operations that take $C$ to the facet ...
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Infinitely many initial ideals for non-Artinian monomial orders?
Consider the polynomial ring $R=\mathbb Z[x_1,\ldots,x_n]$ and an ideal $I\subset R$. Let $<$ be a monomial order, i.e. a total order on the set of monomials in $R$ such that for any monomials $a$, ...
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Reflection-invariant monomial ideals and Alexander duality
First we give some definitions from Section 3 of the paper Monomials, Binomials, and Riemann-Roch by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally ...
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Betti numbers of a Cohen-Macaulay Module in small projective dimension
I am trying to compute the Betti numbers of some Stanley-Reisner ring $R_\Delta$, where the underlying complex $\Delta$ is shellable and the projective dimension of the $R_\Delta$ is $3\text{ or }4$. ...
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Gröbner bases of resultants and their monomial ideals
$\newcommand{QQ}{\mathbb{Q}}$
Consider the ring $R = \QQ[x, a_1,\ldots,a_m]$ for a certain integer $m$ and the homogeneous polynomial
$$
f = x^{m+1} + \sum_{i=1}^m a_i^i x^{m+1 - i}
$$
Now let
$$
...
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Which ideals have standard Hilbert series?
Let $m$ and $d$ be two positive integers.
Consider the polynomial ring $R = \mathbb{C}[x_1 , \dots , x_m]$. Let $I$ be an ideal of $R$ generated by a finite family of polynomials of degree $d$, and ...
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The $k$ th symbolic power of a square free monomial ideal $\rm I$ is $\rm I^{(k)}= \cap_{p\in Min(I)}p^{k}.$
Let $\rm I$ is a square free monomial ideal in $K[X_1,\ldots,X_n].$ The $k$ th symbolic power of $\rm I$, denoted by $\rm I^{(k)}$ defined to be the intersection of all primary components of $\rm I^{k}...
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Monomial algebras and depth
Let $R:=k[x_1,\ldots, x_n]$ be the standard polynomial ring. Let $I\subseteq R$ be a monomial ideal of height $\ge 2,$ and $\{\ell_1, \ldots, \ell_{t}\}\subseteq R_1$ an $R$-regular sequence.
Assume $...
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Cut ideal of two graphs?
Consider a connected graph $G$ and a connected graph $H$. Their graph ideals are their path ideals. The Alexander duality of the graph ideals give the cut ideals. $G$ and $H$ are not connected to each ...
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