The behavior of depth functions of cover ideals of unimodular hypergraphs

Transcription

1 Ark. Mat., ), DOI: /ARKIV.2017.v55.n1.a4 c 2017 by Institut Mittag-Leffler. All rights reserved The behavior of depth functions of cover ideals of unimodular hypergraphs Nguyen Thu Hang and Tran Nam Trung Abstract. We prove that the cover ideals of all unimodular hypergraphs have the nonincreasing depth function property. Furthermore, we show that the index of depth stability of these ideals is bounded by the number of variables. Introduction Let R=k[x 1,..., x n ] be a polynomial ring over a given field k, andleti be a homogeneous ideal in R. It is known by Brodmann [3] that depthr/i s ) takes a constant value for large s. Moreover, lim depth s R/Is dim R li), where li) istheanalytic spread of I. Theindex of depth stability of I is defined by dstabi):=min { s 0 1 depth S/I s =depths/i s0 for all s s 0 }. Two natural questions arise from Brodmann s theorem: 1) What is the nature of the function s depth R/I s for s dstabi)? 2) What is a reasonable bound for dstabi)? On the nature of the function s depth R/I s for s 1, which is called the depth function of I, Herzog and Hibi [10] conjectured that the depth function of ideals can be any convergent nonnegative integer valued function. The answer is affirmative for bounded increasing functions see [10]) and non-increasing functions see [8]). The behavior of depth functions, even for monomial ideals, is complicated see e.g. [1]). Squarefree monomial ideals behave considerably better than monomial Key words and phrases: depth function, cover ideals, powers of ideals Mathematics Subject Classification: 13A15, 13C13.

2 90 Nguyen Thu Hang and Tran Nam Trung ideals in general, and Herzog and Hibi [10] asked whether depth functions of any squarefree monomial I is non-increasing, that is depth R/I s) depth R/I s+1) for all s 1. Kaiser, Stehlík and Škrekovski [18] gave a graph whose cover ideal is a counterexample to this question see also [9] for more counterexamples). It is an interesting problem to characterize squarefree monomial ideals I with non-increasing depth function. This problem is difficult, and the characterization now has focused on certain families of monomial ideals, such as: ideals with all powers having a linear resolution see [10]); cover ideals of bipartite graphs see [6]); and ideals with constant depth function see [8], [14] and[22]). In this paper we prove that the cover ideal of a unimodular hypergraph has non-increasing depth function. Before stating our result we recall some terminology from graph theory see [2] for more detail). Let V ={1,..., n}, andlete be a family of distinct nonempty subsets of V. The pair H=V, E) is called a hypergraph with vertex set V and edge set E. Notice that a hypergraph generalizes the classical notion of a graph; a graph is a hypergraph for which every E E has cardinality two. One may also define a hypergraph by its incidence matrix AH)=a ij ), with columns representing the edges E 1,E 2,..., E m and rows representing the vertices 1, 2,..., n where a ij =0 if i/ E j and a ij =1 if. A hypergraph H is said to be unimodular if its incident matrix is totally unimodular, i.e., every square submatrix of AH) has determinant equal to 0, 1or 1. A vertex cover of H is a subset of V which meets every edge of H; a vertex cover is minimal if none of its proper subsets is itself a cover. For a subset τ ={i 1,..., i t } of V, setx τ :=x i1...x it.thecover ideal of H is then defined by: JH):=x τ τ is a minimal vertex cover of H). It is well-known that there is one-to-one correspondence between squarefree monomial ideals of R and cover ideals of hypergraphs on the vertex set V. The first main result of this paper is the following theorem. Theorem 2.3. Let H be a unimodular hypergraph. Then, JH) has nonincreasing depth function. We next address the question of when depth R/I s becomes stationary. Effective bounds of dstabi) are known only for a few special classes of ideals I, suchas complete intersection ideals see [5]), squarefree Veronese ideals see [10]), polymatroidal ideals see [13]), edge ideals see [25]). For any cover ideal of a unimodular hypergraph H, we establish the universal and effective bound for dstabjh)). Namely,

3 The behavior of depth functions of cover ideals of unimodular hypergraphs 91 Theorem 3.2. Let H=V, E) be a unimodular hypergraph with the vertex set V ={1,..., n}. Then, depth R/JH) s = n l JH) ) for all s n. In particular, dstabjh)) n. When H is a bipartite graph G, this bound can be improved significantly by looking deeper into the structure of G. Letν 0 G) betheordered matching number of G see Definition 3.3). Then, we prove the following result. Theorem 3.6. Let G be a bipartite graph with n vertices. Then, depth R/JG) s = n ν 0 G) 1 for all s ν 0 G). In particular, dstabjg)) ν 0 G). Moreover, this bound is sharp see Proposition 3.7). It is worth mentioning that Herzog and Qureshi in [13] conjectured that dstabi)<li) for any squarefree monomial ideal I. Note that ν 0 G)=lJG)) 1 by [7, Corollary 2.9], so dstabjg))<ljg)) by Theorem 3.6. Thus, the conjecture holds for JG). Our approach is based on a generalized Hochster s formula for computing local cohomology modules of arbitrary monomial ideals formulated by Takayama [24]. Using this formula we are able to investigate the depth of powers of monomial ideals via the integer solutions of certain systems of linear inequalities. This allows us to use the theory of polytopes as the key role in this paper see e.g. [15] and[16] for this approach). The paper is organized as follows. In Section 1, we set up some basic notation and terminology for simplicial complex, the relationship between simplicial complexes and cover ideals of hypergraphs; and a generalization of Hochster s formula for computing local cohomology modules. In Section 2, we prove the non-increasing property for cover ideals of unimodular hypergraphs. In Section 3, we establish an upper bound for dstabjh)) of any unimodular hypergraph H. 1. Preliminary We first recall a relationship between cover ideals of hypergraphs and simplicial complexes. A simplicial complex on V ={1,..., n} is a collection of subsets of V such that if σ Δ andτ σ then τ Δ. Definition 1.1. The Stanley-Reisner ideal associated to a simplicial complex Δ is the squarefree monomial ideal I Δ := x τ τ/ Δ) R.

4 92 Nguyen Thu Hang and Tran Nam Trung Note that if I is a squarefree monomial ideal, then it is a Stanley-Reisner ideal of the simplicial complex ΔI):={τ V x τ / I}. If I is a monomial ideal maybe not squarefree) we also use ΔI) to denote the simplicial complex corresponding to the squarefree monomial ideal I. Let FΔ) be the set of facets of Δ. If FΔ)={F 1,..., F m }, we write Δ= F 1,..., F m. Then, I Δ has the primary-decomposition see [19, Theorem 1.7]): I Δ = x i i/ F ). F FΔ) For n 1, the n-th symbolic power of I Δ is I n) Δ = x i i/ F ) n. F FΔ) Let H=V, E) be a hypergraph. Then, the cover ideal of H canbewrittenas 1) JH)= E Ex i i E). By this formula, when consider JH) without loss of generality we may assume that H is simple, i.e., whenever E i,e j E and E i E j,thene i =E j.inthiscase,jh) is a Stanley-Reisner ideal with 2) Δ JH) ) = V \E E E. Let m:=x 1,..., x n ) be the maximal homogeneous ideal of R and I a monomial ideal in R. We can represent depth R/I via its local cohomology modules by depthr/i)=min { i H i mr/i) 0 }. Since R/I is an N n -graded algebra, HmR/I) i is a Z n -graded module over R/I for every i. For each degree α=α 1,..., α n ) Z n, in order to compute dim k HmR/I) i α we use a formula given by Takayama [24, Theorem 2.2] which is a generalization of Hochster s formula for the case I is squarefree [17, Theorem 4.1]. Set G α :={i α i <0}. For a subset F V, weletr F :=R[x 1 i i F G α ]. Define the simplicial complex Δ α I) by 3) Δ α I):= { F V\G α x α / IR F }. Lemma 1.2. [24, Theorem 2.2]) dim k H i mr/i) α =dim k Hi Gα 1Δ α I); k). If H is unimodular, the cover ideal JH) is normally torsion-free, i.e. JH) s) = JH) s for all s 1, by [12, Theorem 1.1]. Combining with [20, Lemma 1.3] we obtain:

5 The behavior of depth functions of cover ideals of unimodular hypergraphs 93 Lemma 1.3. Let H=V, E) be a unimodular hypergraph with V ={1,..., n}, and α=α 1,..., α n ) N n. Then, for every s 1 we have Δ α JH) s ) = V\E E E and α i s 1. i E For F V, sets:=k[x i i/ F ]andj :=JH)R F S. LetH be a hypergraph on the vertex set V :=V\F with the edge set E ={E E E F = }. By1) we obtain: 4) JH)R F S = J H ). In the sequel we also need the following two lemmas that will enable us to do induction on the number of variables. Lemma 1.4. [15, Lemma 1.3]) Let I be a monomial ideal of R and F {1,..., n} such that IR F R F. Let S:=k[x i i/ F ] and J :=IR F S. Then, depth R/I F +depths/j. Lemma 1.5. [15, Lemma 1.4]) Let I be a monomial ideal of R with depth R/I =d. Assume that H d mr/i) α 0 for some α=α 1,..., α n ) Z n. Let F := G α, S:=k[x i i/ F ] and J :=IR F S. Then, depth R/I =depths/j + F. 2. The non-increasing property of depth functions In this section we prove the non-increasing property of cover ideals of unimodular hypergraphs. Throughout this section, let e 1,..., e n be the canonical basis of R n. Let H=V, E) be a unimodular hypergraph on the vertex set V ={1,..., n}. Suppose that E ={E 1,..., E m }. Then, by 2) Δ JH) ) = V \E 1,..., V\E m. Now let p 0 andα=α 1,..., α n ) N n such that H p mr/jh) s ) α 0. By Lemma 1.2 we have dim k Hp 1 Δα JH) s ) ; k ) =dim k H p m R/JH) s ) α, so that H p 1 Δ α JH) s ); k) 0. In particular, Δ α JH) s ). Thus, by Lemma 1.3 we may assume that Δ α JH) s )= V \E 1,..., V\E r where 1 r m.

6 94 Nguyen Thu Hang and Tran Nam Trung For each t 1, let C t R n to be the solution of the following system of linear inequalities: 5) x i <t for j=1,..., r, x i t for j=r+1,..., m, x 1 0,..., x n 0. Then C t =tc 1.Sinceα C s,wehave1/s)α C 1. Hence, C 1, and hence C t. Moreover, by Lemma 1.3 one has Δ β JH) t ) = V \E 1,..., V\E r =Δ α JH) s ) for any β C t N r. Let C t be the closure of C t in R n with respect to the Euclidean topology. Then, C t =tc 1 and C 1. Moreover,C t is solutions in R n of the following system: 6) x i t for j=1,..., r, x i t for j=r+1,..., m, x 1 0,..., x n 0. In particular, C t is a convex polyhedron in R n. Lemma 2.1. C 1 is a polytope and dim C 1 =n. Proof. First we prove C 1 is bounded in R n. Indeed, let y=y 1,..., y n ) C 1.For every i=1,..., n, since Δ α JH) s )= V \E 1,..., V\E r is not acyclic, it is not a cone over i, hence for some j=1,..., r. Together with the system 5) weget 0 y i y q < 1, q E j so C 1 is bounded, as claimed. Therefore, C 1 is bounded too, and therefore C 1 is a polytope in R n. Now we prove dim C 1 =n. Since C 1, we can take a point β=β 1,..., β n ) C 1. From the system 5), there exists a real number ε>0 such that for all real numbers ε 1,..., ε n with 0 ε 1,..., ε n ε, wehaveβ+ε 1 e ε n e n C 1,so[β 1,β 1 +ε]... [β n,β n + ε] C 1 C 1. Thus, the polytope C 1 is full dimensional in R n, as required. We call a point of R n is an integer point if all its coordinates are integers. Lemma 2.2. Every vertex of C 1 is an integer point.

7 The behavior of depth functions of cover ideals of unimodular hypergraphs 95 Proof. Let β=β 1,..., β n ) R n be a vertex of C 1. Note that C 1 is a polytope of full dimensional in R n. Moreover, C 1 is the solutions in R n of the system 6) with t=1. By [23, Formula 23 in p. 104], one has β must be the unique solution of a system of linear equations of the form { i E 7) j x i =1 for j S 1 {1,..., m}, x j =0 for j S 2 {1,..., n} where S 1 + S 2 =n. By Cramer s rule, we can represent coordinates of β as 8) β i = D i D for i =1,...n, where D, D 1,..., D n Z and D is the determinant of the matrix of this system. Since the matrix of the system: x i =1, j S 1 ; is a submatrix of AH) T, the transpose of AH). Thus, it is totally unimodular, and thus the matrix of the system 7) is totally unimodular. This yields either D= 1 or D=1. Consequently, β 1,..., β n N n, i.e. β is an integer point. We are now ready to prove the main result of this section. Theorem 2.3. Let H be a unimodular hypergraph. Then, JH) has nonincreasing depth function. Proof. Fix any s 1. We prove by induction on n= V that depth R/JH) s depth R/JH) s+1. If n 2, then depth R/JH) s =depthr/jh) s+1, so the theorem holds. Assume that n 3. Let d:=depth R/JH) s. Then, HmR/JH) d s ) α 0 for some α Z n.letf:=g α. We now consider two cases: Case 1. F ) Let V :=V\F and S:=k[x i i V ]. Let H be the hypergraph on the vertex set V with the edge set E ={E E E F = }. Then, by 4) wehave JH )=JH)R F S. SinceH is unimodular, H is unimodular. By Lemma 1.5 we have depth R/JH) s =depths/jh ) s + F. By Lemma 1.4 we have depth R/JH) s+1 depth S/JH ) s+1 + F. On the other hand, since V = V F < V, by the induction hypothesis we have depth S/JH ) s depth S/JH ) s+1. It follows that depth R/JH) s+1 depth S/J H ) s+1 + F depth S/J H ) s + F =depthr/jh) s.

8 96 Nguyen Thu Hang and Tran Nam Trung Case 2. F =, i.e. α=α 1,..., α n ) N n ) By Lemma 1.2 we have so H d 1 Δ α IH) s ); k) 0. dim k Hd 1 Δα IH) s ) ; k ) =dim k H d m R/JH) s ) α, Suppose that E ={E 1,..., E m }. By 2) and Lemma 1.2 we may assume that FΔ α JH) s )={V \E 1,..., V\E r } for some 1 r m. By Lemma 1.3 we have { i E 9) j α i <s for j=1,..., r, α i s for j=r+1,..., m. On the other hand, by Lemmas 2.1 and 2.2, the following system of linear inequalities x i 1 for j=1,..., r, 10) x i 1 for j=r+1,..., m, x 1 0,..., x n 0, has at least one integer solution, say β=β 1,..., β n ) N n. Let γ:=α+β so that γ N n.writeγ=γ 1,..., γ n ). From 9) and10) we yields { γ i <s+1 for j=1,..., r, γ i s+1 for j=r+1,..., m. Together with Lemma 1.3 we have Δ γ JH) s+1 )=Δ α JH) s ). Since H d 1 Δ α JH) s ); k) 0, we have H d 1 Δ γ JH) s+1 ); k) 0. Together with Lemma 1.2 we get HmR/JH) d s+1 ) γ 0, sohmr/jh) d s+1 ) 0. This implies depth R/JH) s+1 d=depthr/jh) s. The proof is complete. In the case H is a bipartite graph. Then, it is unimodular by [2, Theorem 5]. As a consequence of Theorem 2.3 we recover [6, Theorem 3.2]. Corollary 2.4. JG) has non-increasing depth function for any bipartite graph G. 3. The index of depth stability In this section we establish the upper bound of dstabjh)) for the cover ideal of any unimodular hypergraph H. Recall that the analytic spread of a homogeneous

9 The behavior of depth functions of cover ideals of unimodular hypergraphs 97 ideal I of R is defined by li):=dimri)/mri), where RI)= s=0 Is is the Rees ring of I. Lemma 3.1. Let H be a unimodular hypergraph. Then, lim depth s R/JH)s =dimr l JH) ). Proof. Since H is unimodular, JH) is totally torsion-free by [12, Theorem 1.1]. The lemma now follows from [11, Proposition and Theorem ]. We are in position to prove the main result of this section. Theorem 3.2. Let H=V, E) be a unimodular hypergraph with the vertex set V ={1,..., n}. Then, In particular, dstabjh)) n. depth R/JH) s = n l JH) ) for all s n. Proof. Since H is unimodular, by Theorem 2.3 and Lemma 3.1 we have 11) dstabjh)) = min { s 1 depth R/JH) s =dimr ljh) }. Thus, it remains to show that dstabjh)) n. We prove the assertion by induction on n. If n 2, then depth R/JH) s = depth R/JH) for all s 1, and then the assertion holds. Assume that n 3. Let s:=dstabjh)) and d:=n ljh)). Then, H d mr/ JH) s ) α 0 for some α Z n.letf :=G α. We now consider two cases: Case 1. F ) Let V :=V\F and S:=k[x i i V ]. Let H be the hypergraph on the vertex set V with the edge set E ={E E E F = }. Then, by 4) wehave JH )=JH)R F S. SinceH is unimodular, H is unimodular too. Note that depth R/JH) s =depths/jh ) s + F by Lemma 1.5. Now let p:= dstabjh )). Together with Lemma 1.4 and Theorem 2.3 we have depth S/J H ) p depth R/JH) p F depth R/JH) s F =depths/j H ) s. Together with Theorem 2.3, this fact follows that depth S/JH ) s =depths/jh ) p. Hence, the inequalities above yields depth R/JH) p =depthr/jh) s. By combining this with 11), we get s p. On the other hand, p V < V by the induction hypothesis. Thus, s<n, and the assertion holds for this case.

10 98 Nguyen Thu Hang and Tran Nam Trung Case 2. F =, i.e. α=α 1,..., α n ) N n ) By Lemma 1.2 we have so H d 1 Δ α IH) s ); k) 0. dim k Hd 1 Δα IH) s ) ; k ) =dim k H d m R/JH) s ) α, Suppose that E ={E 1,..., E m }. By 2) and Lemma 1.2 we may assume that FΔ α JH) s ))={V \E 1,..., V\E r } for some 1 r m. For every t 1, let C t the set of solutions in R n of the following system of linear inequalities x i <t for j=1,..., r, 12) x i t for j=r+1,..., m, x 1 0,..., x n 0. Let C t be closure of C t in R n. Then, by 12), C t is the set of solutions in R n of the following system of linear inequalities x i t for j=1,..., r, 13) x i t for j=r+1,..., m, x 1 0,..., x n 0. By Lemmas 2.1 and 2.2, we conclude that C 1 is a polytope of full dimensional in R n with all integer vertices. If C 1 has no supporting hyperplanes of the form x i =1 for all j=r+ 1,..., m. In this case, we imply that the zero vector β=0,..., 0) C 1,andsoβ C 1 N n. If C 1 has a supporting hyperplane of the form x i =1 for some j=r+ 1,..., m. We may assume such a supporting hyperplane is i E m x i =1; and let F be the facet of C 1 determined by this hyperplane. Now take n vertices of C 1 lying in F, say α 1,..., α n, such that they are affinely independent. Let β:=α α n )/n C 1. Then, β is a relative interior point of F, so that it does not belong to any another facet of C 1.Thus,By13), β satisfies the following system: x i <1 for j=1,..., r, 14) x i 1 for j=r+1,..., m, x 1 0,..., x n 0. This forces β C 1.Thus,nβ C n N n. Let γ=nβ C n N n. Then, by 14) and Lemma 1.3 we have Δ γ JH) n )= Δ α JH) s ), so H d 1 Δ γ JH) n ); k) 0. Together with Lemma 1.2 we deduce that

11 The behavior of depth functions of cover ideals of unimodular hypergraphs 99 HmR/JH) d n ) 0. Consequently, depth R/JH) n d. Thus, depth R/JH) n =d, and thus s n by 11), as required. In the case H is a bipartite graph we will establish a better bound for dstabjh)). We first recall some terminology from the graph theory. Let G= V G),EG)) be a simple graph. Let υg) denote the number of vertices of G. A set of vertices S V G) is called independent if {v, w} / EG) for any v, w S. AsetM EG) is a matching of G if any two distinct edges of M have no vertex in common. Let M ={{a i,b i } i=1,..., m} be a nonempty matching of G. According to [7], we say that M is an ordered matching if: {a 1,..., a m } is a set of independent vertices, {a i,b j } EG) implies i j. Definition 3.3. The ordered matching number of G is: ν 0 G):=max { M M EG) is an ordered matching of G }. Assume that G is bipartite with V G)={1,..., n}. To describe Δ α JG) s )for α=α 1,..., α n ) N n and s 1, in more explicit way, for every edge {i, j} of G we let F ij :={1,..., n}\{i, j}. Then, F Δ JG) )) = { F ij {i, j} EG) } and by Lemma 1.3 we obtain 15) Δ α JG) s ) = F ij α i +α j s 1 and{i, j} EG). Lemma 3.4. Let G be a bipartite graph with bipartition X, Y ). If υg)= 2ν 0 G), then G has an ordered matching M ={{a i,b i } i=1,..., m} where m=ν 0 G), such that X = {a 1,..., a m } and Y = {b 1,..., b m }. Proof. We prove by induction on ν 0 G). If ν 0 G)=1, then G is an edge, and then the lemma is obvious. Assume that ν 0 G)=m 2. Let {{x i,y i } i=1,..., m} be an ordered matching of G such that {x 1,..., x m } is an independent set of G; {x i,y j } EG) implies i j. This implies deg G x m )=1. Let H :=G\{x m,y m }. Then, H is a bipartite graph with so that υh)=υg) 2=2m 1) and ν 0 H)=m 1. We now distinguish two cases:

12 100 Nguyen Thu Hang and Tran Nam Trung Case 1. x m X) Then, y m Y and H has a bipartition X \{x m },Y\{y m }). Since ν 0 H)=m 1<m, by the induction hypothesis we imply that H has an ordered matching {a i b i i=1,..., m 1} such that X \{x m } = {a 1,..., a m 1 } and Y \{y m } = {b 1,..., b m 1 }. Note that deg G x m )=1, x m X and y m Y. Therefore, {a 1 b 1,..., a m 1 b m 1,x m y m } is a desired ordered matching. Case 2. x m Y )Theny m X and H has a bipartition Y \{x m },X\{y m }). Since ν 0 H)=m 1<m, by the induction hypothesis we imply that H has an ordered matching {a i b i i=1,..., m 1} such that Y \{x m } = {a 1,..., a m 1 } and X \{y m } = {b 1,..., b m 1 }. Note that deg G x m )=1, x m X and y m Y. Thus, {a 1 b 1,..., a m 1 b m 1,x m y m } is an ordered matching of G. Thisimpliesthat {y m x m,b m 1 a m 1,..., b 1 a 1 } is a desired ordered matching, and the proof of the lemma is complete. Lemma 3.5. [15, Lemma 3.5]) Let G be a graph with 2m vertices. Then, depth R/JG)=m 1 if and only if G consists of m disjoint edges. If G be a bipartite graph, then by Theorem 3.2 we have dstabjg)) n. This bound is refined as dstabjg)) 2ν 0 G) 1 by[15, Theorem 3.4]. By the same method we improve this bound as follows. Theorem 3.6. Let G be a bipartite graph with n vertices. Then, depth R/JG) s = n ν 0 G) 1 for all s ν 0 G). In particular, dstabjg)) ν 0 G). Proof. Let m:=ν 0 G) and let {{a i,b i } i=1,..., m} be an ordered matching of G. LetH be the induced subgraph of G on the vertex set {a i,b i i=1,..., m}. Then H is a bipartite graph with ν 0 H)=m and υh)=2ν 0 H). By Lemma 3.4, H has an ordered matching {{x i,y i } i=1,..., m} such that {x 1,..., x m }, {y 1,..., y m })isa bipartition of H. Without loss of generality we may assume that x i =i and y i =m+i for all i=1,..., m. LetF :={2m+1,..., n} and S:=k[x 1,..., x 2m ]. We now define α=α 1,..., α 2m ) N 2m as follows { j 1 for j=1,..., m, α j := 2m j for j =m+1,..., 2m.

13 The behavior of depth functions of cover ideals of unimodular hypergraphs 101 Since H has an ordered matching {{i, m+i} i=1,..., m} and a bipartition ) {1,..., m}, {m+1,..., 2m}, we imply that Δ α JH) m ) = F i,m+i i =1,..., m. Let Λ:=Δ α JH) m ). By Alexander duality see [4, Lemma 5.5.3]) we have Since Λ= F i,m+i i=1,..., m, one has dim k Hm 2 Λ; k)=dim k Hm 1 Λ ; k ). Λ = {1}, {m+1} {2}, {m+2}... {m}, {2m}. Thus dim k Hm 1 Λ ; k) 0, and thus dim k Hm 2 Λ; k) 0. On the hand, by Lemma 1.2 we have dim k Hn m 1 S/JH) m ) α =dim H k m 2 Λ; k) where n=x 1,..., x 2m ) S, hence Hn m 1 S/JH) m ) 0, and hence 16) depth S/JH) m m 1. Let F :={2m+1,..., n}. Then, JH)=JG)R F S and JH) m =JG) m R F S. Together Inequality 16) and Lemma 1.4 we have depth R/JG) m F +depths/jh) m n 2m)+m 1) = n ν 0 G) 1. On the other hand, by [7, Corollary 2.9] we have ljg))=ν 0 G)+1. Together with Theorems 2.3 and 3.2, this fact gives depth R/JG) m n ν 0 G) 1. Thus, depth R/JG) m =n ν 0 G) 1. Finally, by Theorems 2.3 and 3.2 we have the sequence {depth R/JG) s } s 1 is non-increasing and bounded below by n ν 0 G) 1, we obtain depth R/JG) s = n ν 0 G) 1, for all s m = ν 0 G), as required. Finally, we construct an example to show that the bound in Theorem 3.6 is sharp.

14 102 Nguyen Thu Hang and Tran Nam Trung Proposition 3.7. Let m 1. {1,..., 2m} and the edge set Let G be a graph with the vertex set V G)= EG)= { {i, m+j} 1 i j m }. Then ν 0 G)=m and depth R/JG) s =2m ν 0 G) 1 == s ν 0 G). Proof. The equality ν 0 G)=m is clear. By Theorem 3.6 it suffices to show that if depth R/JG) s =m 1, then s m. Since depth R/JG) s =m 1, there is α=α 1,..., α 2m ) Z 2m such that 17) Hm m 1 R/JG) s ) α 0. We first claim that α N 2m. Indeed, let F :=G α and S:=k[x i i/ F ]. Let H be an induced subgraph of G on the vertex set [2m]\F. Then, JH)=JG)R F S. By Lemma 1.5 we have depth S/JH) s =depthr/jg) s F = m 1 F =dims m 1. Together with Theorems 2.3 and 3.2 we deduce that dim S m 1 dim S ν 0 H) 1, and so ν 0 H) m. This inequality forces ν 0 H)=m. In particular, V H)=V G) andg α =. This yields α N 2m, as claimed. Let G be a subgraph of G consists of all edges {i, j} of G such that α i +α j s 1. By Lemma 15 we have Δ α JG) s )= F ij {i, j} EG ). Since H m 2 Δ α JG) s ); k) 0 by 17) and Lemma 1.2, wehaveδ α JG) s )is not a cone, and so V G )=V G). Let β=0, 0,..., 0) N 2m. Then, Δ β J G )) = F ij {i, j} E G ) =Δ α JG) s ), so H m 2 Δ β JG )); k) 0. Together with Lemma 1.2, this fact gives Hm m 1 R/J G )) 0, β whence depth R/JG ) m 1. On the other hand, as ν 0 G ) m, by Theorems 2.3 and 3.2 we imply that depth R/JG ) m 1. Hence, depth R/JG )=m 1, and hence G consists of m disjoint edges by Lemma 3.5. From the structure of G we imply that G is just m disjoint edges: {1,m+1}, {2,m+2},..., {m, 2m}.

15 The behavior of depth functions of cover ideals of unimodular hypergraphs 103 Therefore, 18) Δ α JG) s ) = F 1,m+1,F 2,m+2,..., F m,2m. For each i=1,..., m 1, since F i,m+i Δ α JG) m )andf i,m+i+1 / Δ α JG) m ), by 15) wehaveα i +α m+i s 1 andα i +α m+i+1 s, and hence α m+i <α m+i+1.it follows that α m+1 <α m+2 <... < α 2m. In particular, α 2m m 1. As F m+1,2m Δ α JG) s ) we obtain s 1 α m+1 +α 2m m 1. Therefore s m, as required. Acknowledgement. We would like to thank the referee for many helpful comments. This work is partially supported by NAFOSTED Vietnam) under the grant number The first author is also partially supported by Thai Nguyen university of Science under the grant number ÐH2016-TN References 1. Bandari, S., Herzog, J. and Hibi, T., Monomial ideals whose depth function has any given number of strict local maxima, Ark. Mat ), Berge, C., Hypergraphs: Combinatorics of Finite Sets, North-Holland, New York, Brodmann, M., The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc ), Brun, W. and Herzog, J., Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, Cowsik, R. C. and Nori, M. V., Fibers of blowing up, J. Indian Math. Soc. N.S.) ), Constantinescu, A., Pournaki, M. R., Seyed Fakhari, S. A., Terai, N. and Yassemi, S., Cohen-Macaulayness and limit behavior of depth for powers of cover ideals, Comm. Algebra ), Constantinescu, A. and Varbaro, M., Koszulness, Krull dimension, and other properties of graph-related algebras, J. Algebraic Combin ), Hà, H. T., Trung, N. V. and Trung, T. N., Depth and regularity of powers of sums of ideals, Math. Z ), Hà, H. T. and Sun, M., Squarefree monomial ideals that fail the persistence property and non-increasing depth, Acta Math. Vietnam ), Herzog, J. and Hibi, T., Thedepthofpowersofanideal,J. Algebra ), Herzog, J. and Hibi, T., Monomial Ideals, GTM 260, Springer, 2010.

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