Stability of depth and Stanley depth of symbolic powers of squarefree monomial ideals
Authors: Seyed Fakhari S. A. 
Published: October 2019

Let

K

\mathbb {K}

be a field and let

S
=

K

[

x
1

,
…
,

x
n

]

S=\mathbb {K}[x_1,\dots ,x_n]

be the polynomial ring in

n
n

variables over

K

\mathbb {K}

. Assume that

I
⊂
S

I\subset S

is a squarefree monomial ideal. For every integer

k
≥
1

k\geq 1

, we denote the

k
k

-th symbolic power of

I
I

by

I

(
k
)

I^{(k)}

. Recently, Montaño and Núñez-Betancourt (2018), and independently Nguyen and Trung (to appear), proved that for every pair of integers

k
,
i
≥
1

k, i\geq 1

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal d normal e normal p normal t normal h left-parenthesis upper S slash upper I Superscript left-parenthesis k right-parenthesis Baseline right-parenthesis less-than-or-equal-to normal d normal e normal p normal t normal h left-parenthesis upper S slash upper I Superscript left-parenthesis left ceiling StartFraction k Over i EndFraction right ceiling right-parenthesis Baseline right-parenthesis period">

d
e
p
t
h

(
S

/

I

(
k
)

)
≤

d
e
p
t
h

(
S

/

I

(
⌈

k
i

⌉
)

)
.

\begin{equation*} \mathrm {depth}(S/I^{(k)})\leq \mathrm {depth}(S/I^{(\lceil \frac {k}{i}\rceil )}). \end{equation*}

We provide an alternative proof for this inequality. Moreover, we re-prove the known results that the sequence

{

d
e
p
t
h

(
S

/

I

(
k
)

)

}

k
=
1

∞

\{\mathrm {depth}(S/I^{(k)})\}_{k=1}^{\infty }

is convergent and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="min Underscript k Endscripts normal d normal e normal p normal t normal h left-parenthesis upper S slash upper I Superscript left-parenthesis k right-parenthesis Baseline right-parenthesis equals limit Underscript k right-arrow normal infinity Endscripts normal d normal e normal p normal t normal h left-parenthesis upper S slash upper I Superscript left-parenthesis k right-parenthesis Baseline right-parenthesis equals n minus script l Subscript s Baseline left-parenthesis upper I right-parenthesis comma">

min
k

d
e
p
t
h

(
S

/

I

(
k
)

)
=

lim

k
→
∞

d
e
p
t
h

(
S

/

I

(
k
)

)
=
n
−

ℓ
s

(
I
)
,

\begin{equation*} \min _k\mathrm {depth}(S/I^{(k)})=\lim _{k\rightarrow \infty }\mathrm {depth}(S/I^{(k)})=n-\ell _s(I), \end{equation*}

where

ℓ
s

(
I
)

\ell _s(I)

denotes the symbolic analytic spread of

I
I

. We also determine an upper bound for the index of depth stability of symbolic powers of

I
I

. Next, we consider the Stanley depth of symbolic powers and prove that the sequences
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace normal s normal d normal e normal p normal t normal h left-parenthesis upper S slash upper I Superscript left-parenthesis k right-parenthesis Baseline right-parenthesis right-brace Subscript k equals 1 Superscript normal infinity">

{

s
d
e
p
t
h

(
S

/

I

(
k
)

)

}

k
=
1

∞

\{\mathrm {sdepth}(S/I^{(k)})\}_{k=1}^{\infty }

and

{

s
d
e
p
t
h

(

I

(
k
)

)

}

k
=
1

∞

\{\mathrm {sdepth}(I^{(k)})\}_{k=1}^{\infty }

are convergent and the limit of each sequence is equal to its minimum. Furthermore, we determine an upper bound for the indices of sdepth stability of symbolic powers.