DEPTH STABILITY OF COVER IDEALS MAI PHUOC BINH NGUYEN THU HANG TRUONG THI HIEN AND TRAN NAM TRUNG Abstract. LetRKx1 x r be a polynomial ring over a eld K. Let Gbe a_2

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DEPTH STABILITY OF COVER IDEALS
MAI PHUOC BINH, NGUYEN THU HANG, TRUONG THI HIEN, AND TRAN NAM TRUNG
Abstract. Let R=K[x1, . . . , xr] be a polynomial ring over a field K. Let Gbe a
graph with vertex set {1, . . . , r}and let Jbe the cover ideal of G. We give a sharp
bound for the stability index of symbolic depth function sdstab(J). In the case Gis
bipartite, it yields a sharp bound for the stability index of depth function dstab(J)
and this bound is exact if Gis a forest.
Introduction
Let R=K[x1, . . . , xr] be a polynomial ring over a field Kand let Ibe a homoge-
neous ideal of R. We call the functions depth R/Inand depth R/I(n), for n>1, the
depth function and the symbolic depth function of I, respectively.
It is a classical result of Brodmann [3] that the depth function of an ideal in a
Noetherian ring is asymptotically a constant function. The same result is not true
for the symbolic depth function (see [26, Theorem 4.4]), but it is true for square-free
monomial ideals (see [20,21]).
By virtue of these results we define the stability index of depth function of Ito be
dstab(I) = min{n0|depth R/In= lim
k→∞ depth R/Ik,for all n>n0}
and the stability index of symbolic depth function of Ito be
sdstab(I) = min{n0|depth R/I(n)= lim
k→∞ depth R/I(k),for all n>n0}
provided the second limit exists.
It is of great interest to bound effectively dstab(I) and sdstab(I) for square-free
monomial ideals. However these problems seem to be very difficult and are currently
solved for only few cases (see e.g. [11,13,14,16,17,18,20,25,30]).
In our paper, we investigate these indices on the class of square-free monomial ideals
of height 2, these ideals can be defined as cover ideals of graphs. Let G= (V, E) be
a graph with vertex set V={1, . . . , r}. For a subset τof V, denote the square-free
monomial xτto be the product of all variables xiwhere iτ. Then, the cover ideal
of Gis defined by:
J(G) = (xτ|τis a minimal vertex cover of G).
In order to find a good bounds for dstab(J(G)) and sdstab(J(G)) we use the theory
of matchings in graphs. We mainly deal with ordered matchings which are defined as
2020 Mathematics Subject Classification. 13A15, 13C15, 05C90, 13D45.
Key words and phrases. Ordered matching, Cover ideal, Depth function, Symbolic power.
1
arXiv:2210.09528v1 [math.AC] 18 Oct 2022
2 M.P. BINH, N.T. HANG, T.T. HIEN, AND T.N. TRUNG
follows. A matching M={{ui, vi} | i= 1, . . . , s}in a graph Gis called an ordered
matching if:
(1) {u1, . . . , us}is an independent set in G,
(2) {ui, vj} ∈ Eimplies i6j.
In this case, the set A={u1, . . . , us}is called the free parameter set of Gand
B={v1, . . . , vs}is called the partner set of A.
For an ordered matching Min a graph G, an M-alternating path is a path whose
edges are alternately in Mand E\M. This notion plays an important role in the
theory of matchings (see e.g. [2]). For an ordered matching Min G, let `(M) be the
length of a longest M-alternating path, and
`(G) = min{`(M)|Mis a maximum ordered matching in G}.
Although an M-alternating path may be not simple but these numbers are always
finite (see Lemmas 1.11 and 1.13).
Then the main result of our paper is the following theorem.
Theorem 1. (see Theorem 2.5) sdstab(J(G)) 6(`(G) + 1)/2 for any graph G.
Note that the bound in Theorem 1 improves the bound given in [13, Theorem 3.6]
for all graphs, and the one in [20, Theorem 3.4] for bipartite graphs.
Moreover, the equality holds for a broad class of graphs. Let ν(G) and ν0(G) denote
the matching and the ordered matching numbers of G, respectively. Then, we have
the following theorem (see Propositions 2.6,3.2 and 3.4).
Theorem 2. sdstab(J(G)) = (`(G) + 1)/2 if Gis one of the following graphs:
(1) Ghas a perfect ordered matching, i.e. |V(G)|= 2ν0(G), or
(2) ν(G) = ν0(G) and Gcontains no cycles of length 5, or
(3) Gis a forest.
It is worth mentioning that dstab(J(G)) and sdstab(J(G)) depend on the charac-
teristic of the base filed K(see Example 2.9). This means that we can not give a
combinatorial formula for these indices as in Theorem 2.
1. Preliminary
In this section, we recollect notation, terminology and basic results used in the
paper. We follow standard texts [2,4,23]. Throughout the paper, let Kbe a field,
let R=K[x1, . . . , xr], r >1 be a polynomial ring, and let m= (x1, . . . , xr) be the
maximal homogeneous ideal of R.
STABILITY INDEX OF DEPTH FUNCTION 3
1.1. Depth and regularity. The object of our work is the depth of graded modules
and ideals over R. This invariant can be defined via either the minimal free resolutions
or the local cohomology modules.
Let Mbe a nonzero finitely generated graded R-module and let
0M
jZ
R(j)βp,j (M)→ ··· → M
jZ
R(j)β0,j (M)0
be the minimal free resolution of M. The projective dimension of Mis the length of
this resolution
pd(M) = p,
and the depth of Mis given by Auslander-Buchsbaum formula
depth(M) = rp.
Another invariant measures the complexity of the resolution is the Castelnuovo–Mumford
regularity (or regularity for short) of Mwhich is defined by
reg(M) = max{ji|βi,j (M)6= 0}.
The depth and regularity of Mcan also be computed via the local cohomology
modules of M. Let Hi
m(M) be the i-th cohomology module of Mwith support in m.
Then,
depth(M) = min{i|Hi
m(M)6= 0},
and
reg(M) = max{j+i|Hi
m(M)j6=0,for i= 0,...,dim(M),and jZ}.
1.2. Graphs. Let Gbe a simple graph. We use the symbols V(G) and E(G) to
denote the vertex set and the edge set of G, respectively. In this paper we always
assume that E(G)6=unless otherwise indicated.
An edge eis incident to a vertex vif ve. The degree of a vertex is the number of
edges incident to u,uis a leaf if it has degree one. If e={u, v}, then uand vare ends
of eor endpoints of e,uand vare adjacent to each other. If there is no confusion, we
simply write e=uv.
A graph His called a subgraph of Gif V(H)V(G) and E(H)E(G). A graph
His called an induced subgraph of Gif the vertices of Hare vertices in G, and for
vertices uand vin V(H), {u, v}is an edge of Hif and only if {u, v}is an edge in G.
The induced subgraph of Gon a subset SV(G), denoted by G[S], is obtained by
deleting vertices not in Sfrom G(and their incident edges).
For a subset ME(G), denote V(M) to be all vertices of Gthat are endpoins of
edges in Mand denote G[M] to be G[V(M)].
Let p:v0, v1, . . . , vkbe a sequence of vertices of G. Then,
(1) pis called a path if {vi1, vi} ∈ E(G) for i= 1, . . . , k. In this case, we say that
pis a path from v0to vk.
(2) pis called a simple path if it is a path and every vertex appears exactly once.
4 M.P. BINH, N.T. HANG, T.T. HIEN, AND T.N. TRUNG
(3) pis called a cycle if k>3 and pis a path with distinct vertices except for
v0=vk.
In each case, kis called the length of p. A simple path is longest if it is among the
simple paths of largest length of G.
If p:v0, v1, . . . , vkis the path from v0to vkwe indicate it by v0v1→ ··· → vk,
and v0and vkare called the origin and the terminus of p, respectively.
A graph Gwith rvertices such that all edges lying on a simple path is called a path
with rvertices, denoted by Pr. Similarly, if all edges of Glying on a cycle, then Gis
called a cycle of rvertices, denoted by Cr. As usual, the cycle C3is call a triangle,
the cycle C4is called a quadrilateral, the cycle C5is called a pentagon, and so forth.
A graph is connected if there is a path from any point to any other point in the graph.
A graph that is not connected is said to be disconnected. A connected component of
a graph Gis a connected subgraph that is not part of any larger connected subgraph.
The components of any graph partition its vertices into disjoint sets, and are the
induced subgraphs of those sets.
The graph Gis bipartite if V(G) can be partitioned into two subsets Xand Ysuch
that every edge has one end in Xand another end in Y; such a partition (X, Y ) is
called a bipartition of the graph. Note that Gis bipartite if and only if it has no
cycle of odd length (see [2, Theorem 4.7]). A connected graph without cycles is a tree.
Obviously, a tree is bipartite. A graph is a forest if every its connected component is
a tree.
Amatching in the graph Gis a set of pairwise non adjacent edges. If Mis a
matching, the two ends of each edge of Mare said to be matched under M, and each
vertex incident with an edge of Mis said to be covered by M. A perfect matching is
one which covers every vertex of the graph, a maximum matching one which covers as
many vertices as possible. The number of edges in a maximum matching in a graph
Gis called the matching number of Gand denoted ν(G).
A matching Mof Gis called an induced matching if the graph G[M] is just disjoint
edges. The induced matching number of G, denoted by ν0(G), is the maximum size of
an induced matching in G.
An independent set in Gis a set of vertices no two of which are adjacent to each
other. According to Constantinescu and Varbaro [6], we define an ordered matching
as follows.
Definition 1.1. A matching M={{ui, vi} | i= 1, . . . , s}in a graph Gis called an
ordered matching if:
(1) {u1, . . . , us}is an independent set in G,
(2) {ui, vj} ∈ E(G) implies i6j.
In this case, the set A={u1, . . . , us}is called the free parameter set of Gand
B={v1, . . . , vs}is called the partner set of A. For simplicity, we also say Aand B
are free parameter set and partner set for M.
The ordered matching number of G, denoted by ν0(G) is the maximum size of an
ordered matching in G.
STABILITY INDEX OF DEPTH FUNCTION 5
An ordered matching in Gis a perfect matching is called a perfect ordered matching.
Thus, Ghas a perfect order matching if and only if |V(G)|= 2ν0(G).
For example, the graph Gdepicted in Figure 1 has a perfect ordered matching
M={{1,5},{2,6},{3,7},{4,8}}
which are bold edges in the figure. In this case, A={1,2,3,4}and B={5,6,7,8}.
11
55
22
66
33
77
44
88
Figure 1. A graph with a perfect ordered matching
Lemma 1.2. If a graph Ghas a perfect ordered matching, then it has a unique perfect
matching.
Proof. Let M={aibi|i= 1, . . . , s}be a perfect ordered matching in G. Let M0be
a perfect matching in G. We will prove by induction on sthat M0=Mas subsets of
E(G). Indeed, if s= 1, then Gis just the edge a1b1, and the lemma is obvious true.
Assume that s>2. Since Mis a perfect ordered matching, we have asis a leaf.
Recall that M0is a perfect matching, so asbsM0.
Since M\ {asbs}and M0\ {asbs}are a perfect ordered matching and a perfect
matching in G\{as, bs}, respectively, by the induction hypothesis we have M\{asbs}=
M0\ {asbs}. Thus, M=M0, as required.
Let Mbe a matching of G. An M-alternating path is a path whose edges are
alternately in Mand E\M. This notion plays an important role in the theory of
matchings (see [2]).
Definition 1.3. Let Mbe an ordered matching in G. Let
`(M) = max{length(p)|pis an M-alternating path},
and
`(G) = min{`(M)|Mis a maximum ordered matching of G}.
Example 1.4. Let Gbe a graph depicted in Figure 1. Then, Ghas a perfect ordered
matching M={{1,5},{2,6},{3,7},{4,8}}. We can see that
48372615627384,
is an M-alternating path. Note that it is not simple.
摘要:

DEPTHSTABILITYOFCOVERIDEALSMAIPHUOCBINH,NGUYENTHUHANG,TRUONGTHIHIEN,ANDTRANNAMTRUNGAbstract.LetR=K[x1;:::;xr]beapolynomialringovera eldK.LetGbeagraphwithvertexsetf1;:::;rgandletJbethecoveridealofG.Wegiveasharpboundforthestabilityindexofsymbolicdepthfunctionsdstab(J).InthecaseGisbipartite,ityieldsash...

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