Monomial ideals whose depth function has any given number of strict local maxima

c 2013 by Institut Mittag-Leffler. All rights reserved

Monomial ideals whose depth function has any given number of strict local maxima

Somayeh Bandari, J¨urgen Herzog and Takayuki Hibi

In recent years there have been several publications concerning the stable set of prime ideals of a monomial ideal, see for example [4], [7], [10] and [9]. It is known by Brodmann [2] that for any graded idealI in the polynomial ringS (or any proper idealI in a local ring) there exists an integerk₀ such that Ass(I^k)=Ass(I^k+1) for k≥k₀. The smallest integerk₀with this property is called theindex of stability ofI and Ass(I^k0) is the set ofstable prime idealsofI. A prime idealP∈_∞

k=1Ass(I^k) is said to bepersistent with respect toI if whenever P∈Ass(I^k) thenP∈Ass(I^k+1), and the ideal I is said to satisfy the persistence property if all prime ideals P∈ _∞

k=1Ass(I^k) are persistent. It is an open question (see [6] and [13, Question 3.28]) whether any square-free monomial ideal satisfies the persistence property.

We call the numerical functionf(k)=depth(S/I^k) thedepth function ofI. It is easy to see that a monomial idealI satisfies the persistence property if all mono- mial localizations of I have non-increasing depth functions. In view of the above mentioned open question it is natural to ask whether all square-free monomial ideals have non-increasing depth functions. The situation for non-square-free monomial ideals is completely different. Indeed, in [8, Theorem 4.1] it is shown that for any non-decreasing numerical function f, which is eventually constant, there exists a monomial idealI such thatf(k)=depth(S/I^k) for allk. Note that a similar result for non-increasing depth functions is not known, even though it is expected that all square-free monomial ideals have non-increasing depth functions. In general the depth function of a monomial ideal does not need to be monotone. Examples of monomial ideals with non-monotone depth functions are given in [12, Example 4.18]

and [8]. The question arises which numerical functions are depth functions of mono- mial ideals. Since depth(S/I^k) is constant for allk0 (see [1]), any depth function is eventually constant. So the most wild conjecture one could make is that any numerical function which is eventually constant is indeed the depth function of a monomial ideal. In support of this conjecture we show in our theorem that for any given number nthere exists a monomial ideal whose depth function has precisely

nstrict local maxima. Sathaye in [11, Example, p. 2] gives an example of an ideal I in a graded ring R and a prime ideal P of R such that P∈Ass(I^k) for k even andP /∈Ass(I^k) fork odd, for all kup to any given bound. Our example has this property, too, but in contrast to Sathaye’s example it is defined in a regular ring, indeed in the polynomial ring. The price that we have to pay is that the number of variables needed to define our ideal withn strict local maxima is relatively large, namely 2n+4.

For the class of examples considered here the depth function is constant beyond the number of variables. In all other examples known to us, in particular those discussed in [8], this is also the case. Thus we are tempted to conjecture that for any monomial idealI in a polynomial ring innvariables depth(I^k) is constant for k≥n.

In the following theorem we present the monomial ideals admitting a depth function as announced in the title of the paper. For the calculation of some prelim- inary examples we used the computer algebra system CoCoA [5].

Theorem 1. Letn≥0be an integer andI⊂S=K[a, b, c, d, x₁, y₁, ..., x_n, y_n]be the monomial ideal in the polynomial ringS with generators

a⁶, a⁵b, ab⁵, b⁶, a⁴b⁴c, a⁴b⁴d, a⁴x1y₁², b⁴x²₁y1, ..., a⁴xny²_n, b⁴x²_nyn. Then

depth(S/I^k) =

⎧⎨

⎩

0, if kis odd and k≤2n+1, 1, if kis even and k≤2n, 2, if k>2n+1.

In particular, the depth function of this ideal has preciselyn strict local maxima.

Proof. First of all, for each odd integer k=2t−1 with t≤n+1, we show that depth(S/I^k)=0. For this purpose we find a monomial belonging to (I^k:m)\I^k, wherem=(a, b, c, d, x1, y1, ..., xn, yn). We claim that the monomial

u=a⁴b⁴(a⁴x1y₁²)(b⁴x²₁y1)...(a⁴xt−1y²_t−1)(b⁴x²_t−1yt−1)xtyt...xnyn

satisfiesu∈(I^k:m)\I^k. Let

v₁=a⁵b·b⁶(a⁴x_t−1y_t²₋₁)

t−2

i=1

(a⁴x_iy_i²)(b⁴x²_iy_i),

v2=ab⁵·a⁶(b⁴x²_t−1yt−1)

t−2

i=1

(a⁴xiy_i²)(b⁴x²_iyi),

v3=a⁴b⁴c

t−1

i=1

(a⁴xiy²_i)(b⁴x²_iyi),

v4=a⁴b⁴d

t−1 i=1

(a⁴xiy_i²)(b⁴x²_iyi),

v2l+3=a⁶(b⁴x²_lyl)²

l−1

i=1

(a⁴xiy²_i)(b⁴x²_iyi)

t−1

i=l+1

(a⁴xiy_i²)(b⁴x²_iyi), 1≤l≤t−1,

v2l+4=b⁶(a⁴xly_l²)²

l−1

i=1

(a⁴xiy_i²)(b⁴x²_iyi)

t−1 i=l+1

(a⁴xiy²_i)(b⁴x²_iyi), 1≤l≤t−1,

v2l+3= (b⁴x²_lyl)

t−1

i=1

(a⁴xiy²_i)(b⁴x²_iyi), t≤l≤n,

v_2l+4= (a⁴x_ly_l²)

t−1

i=1

(a⁴x_iy²_i)(b⁴x²_iy_i), t≤l≤n.

Clearlyvi∈I^k for 1≤i≤2n+4. One easily sees that

v1|au, v2|bu, v3|cu and v4|du.

Since

a⁴b⁴(a⁴xly_l²)xl=a²(a⁶(b⁴x²_lyl))yl and a⁴b⁴(b⁴x²_lyl)yl=b²(b⁶(a⁴xly²_l))xl, it follows that

v2l+3|xluandv2l+4|ylu, 1≤l≤t−1.

Moreover,

v2l+3|xluandv2l+4|ylu, t≤l≤n.

Henceum⊆I^k. In other words,u∈I^k:m.

Now, we wish to prove thatu /∈I^k. Since neithercnorddividesu, it is enough to show thatu /∈I¯^k, where

I¯= (a⁶, a⁵b, ab⁵, b⁶, a⁴x1y₁², b⁴x²₁y1, ..., a⁴xny²_n, b⁴x²_nyn).

Suppose that there exists a monomialw=u1...uk∈I¯^k with each ui∈G( ¯I) such that wdividesu. Since deg_xi(u)=deg_yi(u)=1 fori=t, ..., n, eachui belongs to

M={a⁶, a⁵b, ab⁵, b⁶, a⁴x1y₁², b⁴x²₁y1, ..., a⁴xt−1y_t²₋₁, b⁴x²_t−1yt−1}.

Since deg_xi(u)=deg_yi(u)=3 fori=1, ..., t−1, it follows that, foruiandujbelonging to

N={a⁴x1y²₁, b⁴x²₁y1, ..., a⁴xt−1y²_t−1, b⁴x²_t−1yt−1}

with i=j, one has u_i=u_j. As|N |=2t−2=k−1, there exists 1≤j≤k with u_j∈N/ . Let ρdenote the number of integers 1≤j≤k with uj∈N/ . Since w divides u, one has deg_a(w)≤4t and deg_b(w)≤4t.

Case 1. ρ=1. Since |N |=k−1, each monomial belonging to N divides w.

Thus deg_a(w)=4(t−1)+c and deg_b(w)=4(t−1)+d, where (c, d) belongs to {(0,6),(1,5),(5,1),(6,0)}.

Hence one has either deg_a(w)>4t or deg_b(w)>4t, a contradiction.

Case 2. ρ=2. Then we may assume that deg_a(w)=4(t−1)+c₁+c₂ and deg_b(w)=4(t−2)+d1+d2, where each (ci, di) belongs to{(0,6),(1,5),(5,1),(6,0)}. Again, one has either deg_a(w)>4tor deg_b(w)>4t, a contradiction.

Case 3. ρ=h with h>2. Suppose that a⁴ divides each of the monomials u1, ..., us, wheres≤k−h. Let deg_a(w)=4s+c1+...+ch and deg_b(w)=4(k−h−s)+

d1+...+dh, where ci+di=6 for each 1≤i≤h. Since

deg_b(w) = 4(k−h−s)+(6−c₁)+...+(6−c_h)≤4t= 2(k+1), it follows that

deg_a(w) = 4s+c₁+...+c_h≥4(k−h)+6h−2(k+1) = 2k+2h−2.

However, sinceh>2, one has

2k+2h−2>2k+4−2 = 2k+2 = 2(k+1) = 4t.

Thus deg_a(w)>4t, a contradiction.

The above three cases complete the proof of u /∈I^k. Hence u belongs to (I^k:m)\I^k and depth(S/I^k)=0, as desired.

Now we are going to prove that depth(S/I^k)≥1 for any even number 0<k≤2n. For the proof we introduce the ideals J=(a⁶, a⁵b, ab⁵, b⁶, a⁴b⁴c, a⁴b⁴d) andL=(a⁴x1y²₁, b⁴x²₁y1, ..., a⁴xny_n², b⁴x²_nyn). ThenI=J+L, and hence

I^k=J^k+J^k−1L+...+J²L^k−2+J L^k−1+L^k.

We first show that fork≥2, the factor module (I^k:(c, d))/I^k is generated by the residue classes of the elements of the set

(1) Sk={a⁴b⁴v1...vk−1:vi∈G(L) andvi=vj fori=j}.

Observe that the minimal set of generators of J² only consists of monomials in aand b. Therefore, the only monomials inI^k which are divisible byc or d are the generators of J L^k−1. It follows that the generators of I^k:(c, d) which do not belong toI^k are the monomials of the forma⁴b⁴v₁...v_k−1 withv_i∈G(L).

Suppose that vi=vj for some i=j, sayvi=vj=a⁴xly²_l. We may assume that i=1 andj=2. Then

u=a⁴b⁴v₁...v_k−1=a¹²b⁴x²_ly_l⁴v₃...v_k−1= (a¹²)(b⁴x²_ly_l)v₃...v_k−1y³_l. Sincea¹²∈J² andb⁴x²_ly_l∈L, we see thatu∈J²L^k−2⊂I^k. This proves (1).

For a monomial u=a⁴b⁴v1...vk−1∈Sk, we set Zu={xl: deg_x

l(vi) = 1 for somei}∪{yl: deg_y

l(vi) = 1 for somei} and

Wu=

x_l∈/supp(u)

{x²_lyl, xly_l²}.

Note thatu+I^kis annihilated bya,b,candd, all variables inZuand all monomials inW_u. Indeed, it is obvious thata,b,c anddand all monomials inW_u annihilate u+I^k. Now letx_l∈Z_u; we shall show thatux_l∈I^k. We can assume thatv₁=a⁴x_ly_l². Hencea⁶(b⁴x²_lyl)v2...vk−1∈I^k anda⁶(b⁴x²_lyl)v2...vk−1|uxl, souxl∈I^k. Similarly for ys∈Zu, we show thatuys∈I^k.

It follows from this observation that (I^k:(c, d))/I^k is generated as aK-module by the residue classes of monomials uvw where u∈Sk, v is a monomial in the variables xi and yj belonging to Vu=supp(u)\Zu, and w is a monomial in the variablesx_iandy_jnot belonging to the support ofuand not divisible by a monomial inW_u.

Fixu=a⁴b⁴v1...vk−1∈Sk and let the residue class ofm=uvwbe a generator of (I^k:(c, d))/I^k as described in the preceding paragraph. Thenv is a monomial with deg_x

i(u)=deg_y

j(u)=2 for eachx_i, y_j∈supp(v). After relabeling of the variables we may assume that

supp(u) ={a, b, x1, y1, ..., xt, yt}. Then

(2) uv=a⁴b⁴ r i=1

(a⁴xiy²_i)(b⁴x²_iyi) s j=r+1

a⁴xjy_j^hj t l=s+1

b⁴x^g_l^lyl

withhj≥2 andgl≥2, andk−1=r+t.

Claim 2. None of the monomialsm=uvw belong toI^k.

For the proof of Claim2 we first observe the following claim.

Claim 3. If w₁...w_s divides m with w₁, ..., w_s∈G(L), then s≤k−1 and after renumbering of thevi we havewi=vi fori=1, ..., s.

Proof. Indeed we may assume that w1=a⁴xjy_j². It follows from (2) that xjy²_j appears in one of the vi. Hence after renumbering we may assume that w1=v1. Thenw2...wsdivides m/v1. Induction on kcompletes the proof.

Proof of Claim 2. We assume on the contrary that m∈I^k. Then there exist w_i∈G(I) such thatw₁...w_k dividesm. We may assume thatw₁, ..., w_s∈G(L) and w_s+1, ..., w_k∈G(J). By Claim3we may assume that w_i=v_i fori=1, ..., s. We next need the following claim.

Claim 4. s=k−1.

Proof. For the proof we consider the following two cases.

Case1. s=k−2. Then,wk−1wkdividesa⁴b⁴vk−1vw. However, sincewk−1, wk∈ G(J), we have

deg_a(wk−1wk)>deg_a(a⁴b⁴vk−1vw) or deg_b(wk−1wk)>deg_b(a⁴b⁴vk−1vw), a contradiction.

Case 2. s=k−h with h>2. Thenwk−h+1...wk divides a⁴b⁴vk−h+1...vk−1vw.

Sincew_k−h+1, ..., w_k∈G(J), it follows that

deg_a(wk−h+1...wk)+deg_b(wk−h+1...wk)≥6h.

On the other hand,

deg_a(a⁴b⁴vk−h+1...vk−1vw)+deg_b(a⁴b⁴vk−h+1...vk−1vw) = 4h+4.

Now sinceh>2, it follows that 4h+4<6h. This means that deg_a(a⁴b⁴v_k−h+1...v_k−1v)+deg_b(a⁴b⁴v_k−h+1...v_k−1v)

<deg_a(wk−h+1...wk)+deg_b(wk−h+1...wk), a contradiction. This concludes the proof.

We now continue with the proof of Claim 2. As we know that s=k−1, it follows thatw_k divides a⁴b⁴vw. This is a contradiction, sincew_k∈G(J). Thus the proof of Claim2is complete.

From Claim2it follows that depth(S/I^k)>0 for even 0<k≤2n. Indeed suppose that depth(S/I^k)=0. ThenI^k:m=I^k. Since I^k:m⊂I^k:(c, d), it follows that there exists a monomialm=uvw∈I^k:m of the form as described above. Now since k is even and 0<k≤2nand v_i=v_j for i=j, the setV_u=∅. It follows thatmv∈/I^k for anyv∈Vu, a contradiction.

In the next step we show that depth(S/I^k)≤1 (and hence depth(S/I^k)=1) for evenkwith 0<k≤2n. Indeed, we claim that

P= (a, b, c, d, x₁, y₁, ..., x_n−1, y_n−1, x_n) belongs to Ass(I^k) for evenkwith 0<k≤2n. Then, since

depth(S/I^k)≤min{dim(S/Q) :Q∈Ass(I^k)} (see [3, Proposition 1.2.13]), the required inequality follows.

To show this we note that P∈Ass(I^k) if and only if depth(S(P)/I(P)^k)=0, see for example [9, Lemma 2.3]. HereS(P) is the polynomial ring in the variables which generateP, and I(P) is obtained fromI by the substitutiony_n→1.

In our caseI(P) is generated by

a⁶, a⁵b, ab⁵, b⁶, a⁴b⁴c, a⁴b⁴d, a⁴x1y²₁, b⁴x²₁y1, ..., a⁴xn−1y²_n−1, b⁴x²_n−1yn−1, a⁴xn, b⁴x²_n.

We claim that fork=2twitht≤nthe monomial

u=a⁸b⁴(a⁴x1y₁²)(b⁴x²₁y1)...(a⁴xt−1y²_t−1)(b⁴x²_t−1yt−1)xtyt...xn−1yn−1xn

satisfiesu∈(I(P)^k:m(P))\I(P)^k. This shows that depth(S(P)/I(P)^k)=0. Let

v_i= (a⁴xn)vi fori= 1, ...,2n+2 and v_2n+3=a⁶(b⁴x²_n)

t−1

i=1

(a⁴xiy_i²)(b⁴x²_iyi),

whereviis defined as in the first part of the proof. Clearlyv_i∈I(P)^kfor 1≤i≤2n+3.

One easily sees that

v₁|au, v₂|bu, v₃|cu and v₄|du. Moreover

v_2l+3 |xlu, v_2l+4|ylu, 1≤l≤n−1, and v_2n+3|xnu. Henceu∈(I(P)^k:m(P)).

With the same argument as in the first part of the proof one can easily see that u∈/I(P)^k. Thereforeu∈(I(P)^k:m(P))\I(P)^k, so depth(S(P)/I(P)^k)=0, as desired.

Finally we show that depth(S/I^k)=2 fork>2n+1. Since the only generators of I^k which are divisible by c are among the generators of J L^k−1, we see that I^k:(c)/I^k is generated by the residue classes of the set of monomials

u∈Sk{u, ud}. Since k>2n+1, it follows that Sk=∅. Hence I^k:(c)=I^k for k>2n+1. Similarly, I^k:(d)=I^k for k>2n+1. It follows that c, d is a regular sequence on S/I^k for k>2n+1. This implies that depth(S/I^k)≥2 for allk>2n+1.

LetS¯=K[a, b, x₁, y₁, ..., x_n, y_n] and

I¯= (a⁶, a⁵b, ab⁵, b⁶, a⁴x1y₁², b⁴x²₁y1, ..., a⁴xny²_n, b⁴x²_nyn)⊂S.¯ Then (S/I^k)/(c, d)(S/I^k)=S/¯ I¯^k.

We claim thatw=a⁵b^6k−6x₁y₁x₂y₂...x_ny_n∈( ¯I^k:n)\I¯^k fork≥2, wherenis the graded maximal ideal of S. The claim implies that depth((S/I¯ ^k)/(c, d)(S/I^k))=0 for allk≥2. In particular it follows that depth(S/I^k)=2 for allk>2n+1, as desired.

To prove the claim we notice thataw is divisible by (a⁶)(b⁶)^k−1∈I¯^k, andbw is divisible by (a⁵b)(b⁶)^k−1∈I¯^k. Hence aw, bw∈I¯^k.

Next observe thatxiw is divisible by (a⁵b)(b⁶)^k−2(b⁴x²_iyi)∈I¯^k and yiw is di- visible by (b⁶)^k−1(a⁴xiy²_i)∈I¯^k. This implies that xiw, yiw∈I¯^k for all i. Thus we have shown thatw∈I¯^k:n.

It remains to be shown that w /∈I¯^k. Indeed, none of the generators of L di- videsw, because each of these generators hasxi-degree oryi-degree 2. Therefore, if w∈I¯^k, it follows thatwis divisible by a monomial inaandbof degree 6k. However, a⁵b^6k−6 has only degree 6k−1, a contradiction.

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2083, pp. 63–94, Springer, Berlin–Heidelberg, 2013 Somayeh Bandari

Department of Mathematics Az-Zahra University Vanak

1983 Tehran Iran

somayeh.bandari@yahoo.com J¨urgen Herzog

Fachbereich Mathematik Universit¨at Duisburg-Essen Campus Essen

DE-45117 Essen Germany

juergen.herzog@uni-essen.de

Takayuki Hibi

Department of Pure and Applied Mathematics

Graduate School of Information Science and Technology

Osaka University

Toyonaka, Osaka 560-0043 Japan

hibi@math.sci.osaka-u.ac.jp

Received May 12, 2012

in revised form September 10, 2012 published online September 28, 2013