Stability of extremal ideals

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Preliminaries: Associated primes of monomial ideals

Extremal ideals

Definition 261.1(Definition).

$S_q:=k[y_A:A\subseteq [q]]$.
${\mathcal{E}}_q:=({\epsilon }_1,\dots ,{\epsilon }_q)\subseteq S_q$, where

$${\epsilon }_i=\prod _{A\subseteq [q],i\in A}{y}_A.$$

Extremal= q-> (
    S=set {1};
    for i from 1 to q  do S=S+set{i};
    PS=subsets S;
    V={};
    for i from 1 to #PS-1 do V=V|{x_{PS#i}};
    R=QQ[V];
    Elist={};
    for i from 1 to q do (
	    e_i=1;
	    for j from 0 to #PS-1 do  if isSubset(set{i},PS#j) then
	        e_i=e_i*x_{PS#j};
        Elist=Elist|{e_i};
    );
    return monomialIdeal(Elist);
)

For example,

$${\mathcal{E}}_3=(y_1{y}_{12}{y}_{13}{y}_{123},y_2{y}_{12}{y}_{23}{y}_{123},y_3{y}_{13}{y}_{23}{y}_{123}).$$

Let $I$ be a squarefree monomial ideal with $q$ generators. Then, properties of $I$ can be gleaned from properties of ${\mathcal{E}}_q$. Here’s an example:

Theorem 261.2(Theorem).

$\text{astab}(I)\le \text{astab}({\mathcal{E}}_q)$, where $I$ is a squarefree monomial ideal with $q$ generators.

Our goal is to find $\text{astab}({\mathcal{E}}_q)$ for all $q$.

Tool to find non-associated primes of ${\mathcal{E}}_q^r$

Let $I$ be a monomial ideal. By Theorem 262.19, every $P\in \text{Ass}(I)$ is a monomial prime ideal, that is, is generated by single powers of variables. Suppose $P=(x_1,\dots ,x_n)\in \text{Ass}(I)$. Then, by Theorem 262.20, $P=(I:f)$ for some monomial $f$, and thus, by Theorem 262.15,

$$f\in (I:P)\setminus I=\left(\bigcap _{i=1}^n(I:x_i)\right)\setminus I.$$

(Note that $f\notin I$, since if it was, $I:f$ would be the entire ring). Hence we have the following lemma:

Lemma 261.3(Lemma).

Let $I$ be a monomial ideal and $x_1,\dots ,x_n$ be some variables. If $I:(x_1,\dots ,x_n)=I$, then $(x_1,\dots ,x_n)\notin \text{Ass}(I)$.

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Note that for ideals $A\subseteq B$, we have $(I:B)\subseteq (I:A)$, since $rB\subseteq I$ implies $rA\subseteq I$ for $r\in R$. Let $P:=(x_1,\dots ,x_n)$ be as in the above lemma. If $J$ is a prime ideal such that $P\subseteq J$, we have

$$(I:J)\subseteq (I:P)=I.$$

however, $I\subseteq (I:J)$ always holds, so $(I:J)=I$, and by Lemma 3, $J\notin \text{Ass}(I)$.

Theorem 261.4(Theorem).

Let $I$ be a monomial ideal and $x_1,\dots ,x_n$ be some variables. If $I:(x_1,\dots ,x_n)=I$, then no associated prime of $I$ contains $(x_1,\dots ,x_n)$.

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General observations

${\mathcal{E}}_q$ is a squarefree monomial ideal. There are finitely many monomial prime ideals in $S_q$. Precisely, $S_q$ has $2^{2^q-1}-1$ prime monomial ideals. By Theorem 262.19, all associated primes of ${\mathcal{E}}_q$ are monomial prime ideals.

Theorem 262.23 tells us that ${\mathcal{E}}_q$ is the intersection of the ideals in $\text{Min}({\mathcal{E}}_q)$. When this is regarded as a primary decomposition of ${\mathcal{E}}_q$, Definition 262.1 tells us that $\text{Ass}({\mathcal{E}}_q)=\text{Min}({\mathcal{E}}_q)$ for all $q$.

Proposition 261.5(Proposition).

$\text{Ass}({\mathcal{E}}_q)=\text{Min}({\mathcal{E}}_q)$ for all $q$.

By Corollary 262.7, the associated primes of higher powers of ${\mathcal{E}}_q$ contain $\text{Min}({\mathcal{E}}_q)$, and maybe additional embedded primes from the pool of monomial prime ideals.

$|\text{Ass}({\mathcal{E}}_q^r)|$ sequences

1. ${\mathcal{E}}_2$: 2

2. ${\mathcal{E}}_3$: 8, 9

3. ${\mathcal{E}}_4$: 49, 80, 81

4. ${\mathcal{E}}_5$: 462, 2095, 2858, 2859

5. ${\mathcal{E}}_6$: 6424,

6. 2

7. 8, 9

8. 49, 80, 81

9. 462, 2095, 2858, 2859

10. 6424,

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Conjecture 261.6(Conjecture).

$\text{astab}({\mathcal{E}}_q)=q-1$.

Proposition 261.7(Proposition).

$|\text{Min}({\mathcal{E}}_q)|$ is the number of minimal covers of $n$ objects (https://oeis.org/A046165).

Finding $\text{astab}({\mathcal{E}}_2)$

${\mathcal{E}}_2=(y_1{y}_{12},y_2{y}_{12})$. It is easily seen that $\text{Min}({\mathcal{E}}_2)=\{(y_1,y_2),(y_{12})\}$ by Corollary 262.21 and Lemma 262.22. We can say more:

Claim 261.8(Claim).

$\text{Ass}({\mathcal{E}}_2^r)=\text{Min}({\mathcal{E}}_2)$ for all $r\ge 1$.

Proof.

It follows from Corollary 262.7 that$\text{Min}({\mathcal{E}}_2)\subseteq \text{Ass}({\mathcal{E}}_2^r)$. By Theorem 262.19, we only need to consider the monomial prime ideals of $k[y_1,y_2,y_{12}]$ for members of $\text{Ass}({\mathcal{E}}_2^r)$. The ones not already in $\text{Min}({\mathcal{E}}_2)$ are

$$(y_1),(y_2),(y_1,y_{12}),(y_2,y_{12}),(y_1,y_2,y_{12}).$$

Using Theorem 262.15 and Theorem 262.14, we have

$$\begin{array}{rl}{\mathcal{E}}_2^r:(y_2,y_{12})& =({\mathcal{E}}_2^r:(y_2))\cap ({\mathcal{E}}_2^r:(y_{12}))\\ & =(y_1^r{y}_{12}^r,y_2^{r-1}{y}_{12}^r)\cap (y_1^r{y}_{12}^{r-1},y_2^r{y}_{12}^{r-1})\\ & =(y_1^r{y}_{12}^r,y_2^r{y}_{12}^r)\\ & ={\mathcal{E}}_2^r.\end{array}$$

Thus, by Theorem 4, $(y_2,y_{12})$ and $(y_1,y_2,y_{12})$ are not in $\text{Ass}({\mathcal{E}}_2^r)$. By a symmetric argument, $(y_1,y_{12})$ is not in $\text{Ass}({\mathcal{E}}_2^r)$. As for $(y_1)$ and $(y_2)$, they are contained in the minimal prime $(y_1,y_2)$, and hence are not in $\text{Ass}({\mathcal{E}}_2^r)$¹. The claim follows.□

Finding $\text{astab}({\mathcal{E}}_3)$

Note that $\text{Ass}({\mathcal{E}}_3^2)=\text{Min}({\mathcal{E}}_3)\cup \{(y_{12},y_{13},y_{23})\}$.

[!Claim]
$(y_{12},y_{13},y_{23})\in \text{Ass}({\mathcal{E}}_3^r)$ for all $r\ge 2$.

[!Proof]-
Find $c_r$ such that $(x_{12},x_{23},x_{31})={\mathcal{E}}_3^r:(c_r)$.

Claim 261.9(Claim).

$\text{Ass}({\mathcal{E}}_3^r)=\text{Min}({\mathcal{E}}_3)\cup \{(y_{12},y_{13},y_{23})\}$ for all $r\ge 2$.

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Footnotes

1. It is clear from Definition 262.1 that an associated prime of $I$ must contain $I$; A prime ideal that is contained in a minimal prime cannot contain $I$. ↩