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

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

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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 Thm 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 Thm 262.20, $P=(I:f)$ for some monomial $f$, and thus, by Thm 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 Lem 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 Thm 262.19, all associated primes of ${\mathcal{E}}_q$ are monomial prime ideals.

Thm 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 ${\mathcal{E}}_q$, Def 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 Cor 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.

Proposition 261.6(Proposition).

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

$$|\text{Min}({\mathcal{E}}_q)|=\sum _{k=0}^q\frac{1}{k!}\sum _{i=0}^k(-1{)}^i(\genfrac{}{}{0ex}{}{k}{i})(2^k-1-i{)}^n.$$

Data

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

1. ${\mathcal{E}}_2$: 2; 2
2. ${\mathcal{E}}_3$: 8, 9; 8, 1
3. ${\mathcal{E}}_4$: 49, 80, 81; 49, 31, 1
4. ${\mathcal{E}}_5$: 462, 2095, 2858, 2859; 462, 1633, 763, 1
5. ${\mathcal{E}}_6$: 6424

Data

(# IPI, # Ass)

⬇️ q ➡️ r	1	2	3	4	5
2	2, 2
3	8, 8	17, 9
4	49, 49	152, 80	333, 81
5	462, 462	3024, 2095	10784, 2858	28257, 2859

Irredundant primary intersection is given by irreducibleDecomposition.

IPI sequences for $q=2,3,4$

2, 3, 4, 5, 6, 7, 8, 9
8, 17, 28, 41, 56, 73, 92, 113, 136, 161, 188, 217, 248, 281
49, 152, 333, 615, 1021, 1574

Conjecture 261.7(Conjecture).

$|\text{IPI}({\mathcal{E}}_q^r)|$ is a polynomial sequence in $r$ for fixed $q$.

$$\begin{array}{rl}|\text{IPI}({\mathcal{E}}_2^r)|& =r+1\\ |\text{IPI}({\mathcal{E}}_3^r)|& =(r+3{)}^2-8\\ |\text{IPI}({\mathcal{E}}_2^r)|& =1+\left(\frac{169}{6}\right)r+16r^2+\left(\frac{23}{6}\right)r^2.\end{array}$$

[!Data]- Irredundant primary intersection (${\mathcal{E}}_4^2$)

1, 2, 3, 4s
1, 2, 3s, 4
1, 2s, 3, 4
1s, 2, 3, 4
$P_1{P}_4,P_2{P}_4,P_3{P}_4,P_4^2$

12, 3, 4s
14, 24 | 34 | 4s
2, 13, 4s
1, 23, 4s
12, 4, 3s
13, 4, 2s
4, 23, 1s
2, 41, 3s
3, 41, 2s
1, 42, 3s
1, 42, 1s
1, 43, 2s
2, 43, 1s

12, 13, 4s
24 | 34 | 4s
12, 23, 4s
13, 23, 4s
12, 41, 3s
13, 41, 2s
12, 42, 3s
42, 23, 1s
41, 42, 3s
13, 43, 2s
23, 43, 1s
41, 42, 2s
42, 41, 1s

123, 4s
14, 24, 34 | 4s
412, 3s
413, 2s
423, 1s

3, 4, 12s
13, 23 | 14, 24 | 1s, 2s, 12
2, 4, 13s
1, 4, 23s
2, 3, 41s
1, 3, 42s
1, 2, 43s

4, 12, 13s
34 | 23 | 1s, 3s
4, 13, 12s
4, 12, 23s
4, 13, 23s
4, 23, 12s
4, 23, 13s
3, 12, 41s
2, 13, 41s
3, 41, 12s
2, 41, 13s
3, 12, 42s
1, 23, 42s
3, 41, 42s
3, 42, 12s
1, 42, 23s
3, 42, 41s
2, 13, 43s
1, 23, 43s
2, 41, 43s
1, 42, 43s
2, 43, 13s
1, 43, 23s
2, 43, 41s
1, 43, 42s

4, 12s, 13s, 23s
14, 24, 34 | 13 | 12 | 23
3, 12s, 41s, 42s
2, 13s, 41s, 43s
1, 23s, 42s, 43s

4, 123s
14, 24, 34 | 1s, 2s, 3s, 12, 13, 23
3, 412s
2, 413s
1, 423s

12, 13, 41s

12, 41, 13s
13, 41, 12s
12, 23, 42s
12, 42, 23s
23, 42, 12s
13, 23, 43s
41, 42, 43s
13, 43, 23s
23, 43, 13s
43, 41, 42s
42, 43, 41s

23, 41s
41, 23s
13, 42s
42, 13s
12, 43s
43, 12s

123, 41s
123, 42s
412, 13s
412, 23s
123, 43s
412, 43s
413, 12s
413, 23s
413, 42s
423, 12s
423, 13s
423, 41s

41, 123s
42, 123s
13, 412s
23, 412s
43, 123s
43, 412s
12, 413s
32, 413s
42, 413s
12, 423s
13, 423s
41, 423s

3, 123s, 41s, 42s
2, 123s, 41s, 43s
1, 123s, 42s, 43s
2, 412s, 13s, 43s
2, 412s, 23s, 43s

123, 412s
412, 123s
123, 413s
412, 413s
413, 123s
413, 412s
123, 423s
412, 423s
413, 423s
423, 123s
423, 412s
423, 413s

4, 13s, 23s, 412s
4, 12s, 23s, 413s
3, 12s, 42s, 413s
1, 23s, 42s, 413s
4, 12s, 13s, 423s
3, 12s, 41s, 423s
2, 13s, 41s, 423s

12s, 13s, 23s, 41s, 42s, 43s

123s, 41s, 42s, 43s
412s, 13s, 23s, 43s
413s, 12s, 23s, 42s
423s, 12s, 13s, 41s

123s, 412s, 43s
123s, 42s, 413s
23s, 412s, 413s
123s, 41s, 423s
13s, 412s, 423s
12s, 413s, 423s

123s, 412s, 413s
123s, 412s, 423s
132s, 413s, 423s
412s, 413s, 423s

1234

Conjecture 261.8(Conjecture).

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

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 Cor 262.21 and Lem 262.22. We can say more:

Claim 261.9(Claim).

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

Proof.

It follows from Cor 262.7 that$\text{Min}({\mathcal{E}}_2)\subseteq \text{Ass}({\mathcal{E}}_2^r)$. By Thm 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 Thm 262.15 and Thm 262.14, we have

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

Thus, by Thm 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 261.10(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 Def 262.1 that an associated prime of $I$ must contain $I$; A prime ideal that is contained in a minimal prime cannot contain $I$. ↩