Table Of ContentEXPANSION OF A SIMPLICIAL COMPLEX
SOMAYEHMORADIANDFAHIMEHKHOSH-AHANG
6
1
0 Abstract. For a simplicial complex ∆, we introduce a simplicial complex
2 attached to ∆, called the expansion of ∆, which is a natural generalization
n of the notion of expansion in graph theory. We are interested in knowing
a how thepropertiesofasimplicialcomplexanditsStanley-Reisner ringrelate
J tothoseofitsexpansions. Itisshownthattakingexpansionpreservesvertex
decomposableandshellablepropertiesandinsomecasesCohen-Macaulayness.
4
AlsoitisprovedthatsomehomologicalinvariantsofStanley-Reisnerringofa
simplicialcomplexrelatetothoseinvariantsintheStanley-Reisnerringofits
]
C expansions.
A
.
h
t
a Introduction
m
Simplicialcomplexesarewidelyusedstructureswhichhavemanyapplicationsin
[
algebraictopologyandcommutativealgebra. Inparticular,inordertocharacterize
1 monomialquotientringswithadesiredproperty,simplicialcomplexisaverystrong
v tool considering the Stanley-Reisner correspondence between simplicial complexes
6
and monomial ideals. Characterizing simplicial complexes which have properties
5
like vertex decomposability, shellability and Cohen-Macaulayness are some main
4
0 problems in combinatorial commutative algebra. It is rather hopeless to give a
0 full classification of simplicial complexes with each of these properties. In this
1. regard, finding classes of simplicial complexes, especially independence complexes
0 of graphs with a desired property have been considered by many researchers (cf.
6 [9, 12, 15, 26, 29, 30]). Constructing new simplicial complexes from the existing
1
ones satisfying a desired property is another way to know more about the charac-
:
v terization. In the works [6, 7, 10, 19, 27], the idea of making modifications to a
i graph like adding whiskers and ears to the graph in order to obtain sequentially
X
Cohen-Macaulay,Cohen-Macaulayandvertexdecomposablegraphsisinvestigated.
r
a In [2], the authors developed a construction similar to whiskers to build a vertex
decomposable simplicial complex ∆ from a coloring χ of the vertices of a simpli-
χ
cial complex ∆, and in [1] for colorings of subsets of the vertices, necessary and
sufficientconditionsaregivenforthisconstructiontoproducevertexdecomposable
simplicial complexes.
Motivated by the aboveworksandthe conceptof expansionofa graphin graph
theory,inthispaper,weintroducetheconceptofexpansionofsimplicialcomplexes
which is a natural generalization of expansion of graphs. Also, we study some
propertiesofthisexpansiontoseehowtheyarerelatedtocorrespondingproperties
2010 Mathematics Subject Classification. Primary13D02,13P10;Secondary 16E05.
Key words and phrases. Cohen-Macaulay, edge ideal, expansion, projective dimension, regu-
larity,shellable,vertexdecomposable.
1
2 S.MORADIANDF.KHOSH-AHANG
of the initial simplicial complex. This tool allows us construct new vertex decom-
posableand shellable simplicialcomplexes fromvertexdecomposable andshellable
ones. Moreover, some families of Cohen-Macaulay simplicial complexes are intro-
duced. We are also interested in knowing how the homological invariants of the
Stanley-Reisner ring of a simplicial complex and its expansions are related.
The paper is organized as follows. In the first section, we review some pre-
liminaries from the literature. In Section 2, first in Theorem 2.7 we show that
for a simplicial complex ∆, vertex decomposability of ∆ is equivalent to vertex
decomposability of an expansion of ∆. Also it is proved that expansions of a
shellable simplicial complex are again shellable (see Theorem 2.12). Moreover, it
is shown that under some conditions, expansions of a simplicial complex inherit
Cohen-Macaulayness(see Corollaries2.10, 2.11, 2.13 and 2.15). Finally, in Section
3,for a shellable simplicialcomplex, the projective dimensionandthe regularityof
its Stanley-Reisnerring arecomparedwiththe correspondingonesinanexpansion
of ∆ (see Propositions 3.1 and 3.4).
1. Preliminaries
Throughout this paper, we assume that ∆ is a simplicial complex on the vertex
set V(∆) = {x ,...,x }. The set of facets (maximal faces) of ∆ is denoted by
1 n
F(∆).
In this section, we recall some preliminaries which are needed in the sequel. We
beginwith definitionofa vertexdecomposablesimplicialcomplex. To this aim, we
needtorecalldefinitionsofthelinkandthedeletionofafacein∆. Forasimplicial
complex ∆ and F ∈∆, the link of F in ∆ is defined as
lk (F)={G∈∆:G∩F =∅,G∪F ∈∆},
∆
and the deletion of F is the simplicial complex
del (F)={G∈∆:G∩F =∅}.
∆
Definition 1.1. A simplicial complex ∆ is called vertex decomposable if ∆ is
a simplex, or ∆ contains a vertex x such that
(i) both del (x) and lk (x) are vertex decomposable, and
∆ ∆
(ii) every facet of del (x) is a facet of ∆.
∆
A vertex x which satisfies condition (ii) is called a shedding vertex of ∆.
Remark 1.2. It is easily seen that x is a shedding vertex of ∆ if and only if no
facet of lk (x) is a facet of del (x).
∆ ∆
Definition 1.3. A simplicial complex ∆ is called shellable if there exists an
ordering F < ··· < F on the facets of ∆ such that for any i < j, there exists a
1 m
vertex v ∈F \F and ℓ<j with F \F ={v}. We call F ,...,F a shelling for
j i j ℓ 1 m
∆.
The above definition is referred to as non-pure shellable and is due to Bj¨orner
and Wachs [3]. In this paper we will drop the adjective “non-pure”.
Definition1.4. AgradedR-moduleM iscalledsequentiallyCohen–Macaulay
(over a field K) if there exists a finite filtration of graded R-modules
0=M ⊂M ⊂···⊂M =M
0 1 r
EXPANSION OF A SIMPLICIAL COMPLEX 3
such that each M /M is Cohen–Macaulay and
i i−1
dim(M /M )<dim(M /M )<···<dim(M /M ).
1 0 2 1 r r−1
For a Z-graded R-module M, the Castelnuovo-Mumford regularity (or
briefly regularity) of M is defined as
reg(M)=max{j−i: β (M)6=0},
i,j
and the projective dimension of M is defined as
pd(M)=max{i: β (M)6=0 for some j},
i,j
where β (M) is the (i,j)th graded Betti number of M.
i,j
Let V = {x ,...,x } be a finite set, and let E = {E ,...,E } be a family of
1 n 1 s
nonemptysubsets ofV. The pairH=(V,E) iscalleda simple hypergraph iffor
each i, |E | ≥ 2 and whenever E ,E ∈ E and E ⊆ E , then i = j. The elements
i i j i j
of V are called the vertices and the elements of E are called the edges of H. For a
hypergraph H, the independence complex of H is defined as
∆ ={F ⊆V(H): E *F, for each E ∈E(H)}.
H
A simple graph G = (V(G),E(G)) is a simple hypergraph with the vertices
V(G) and the edges E(G), where each of its edges has cardinality exactly two.
For a simple graph G, the edge ideal of G is defined as the ideal I(G) = (x x :
i j
{x ,x }∈E(G)). It is easy to see that I(G) can be viewed as the Stanley-Reisner
i j
idealofthe simplicialcomplex ∆ i.e., I(G)=I . Also, the big heightofI(G),
G ∆G
denoted by bight(I(G)), is defined as the maximum height among the minimal
prime divisors of I(G).
AgraphGiscalledvertexdecomposable,shellable,sequentiallyCohen-Macaulay
orCohen-Macaulayiftheindependencecomplex∆ isvertexdecomposable,shellable,
G
sequentially Cohen-Macaulay or Cohen-Macaulay.
AgraphGiscalledchordal,ifitcontainsnoinducedcycleoflength4orgreater.
Definition 1.5. A monomial ideal I in the ring R = K[x ,...,x ] has linear
1 n
quotientsifthereexistsanorderingf ,...,f ontheminimalgeneratorsofI such
1 m
that the colon ideal (f ,...,f ) : (f ) is generated by a subset of {x ,...,x }
1 i−1 R i 1 n
forall2≤i≤m. We show this orderingbyf <···<f andwecallit an order
1 m
of linear quotients on G(I). Also for any 1≤i≤m, set (f ) is defined as
I i
set (f )={x : x ∈(f ,...,f ): (f )}.
I i k k 1 i−1 R i
We denote set (f ) by set(f ) if there is no ambiguity about the ideal I.
I i i
AmonomialidealI generatedbymonomialsofdegreedhasalinearresolution
ifβ (I)=0 for all j 6=i+d. Havinglinear quotients is a strongtoolto determine
i,j
some classes of ideals with linear resolution. The main tool in this way is the
following lemma.
Lemma 1.6. (See [9, Lemma 5.2].) Let I = (f ,...,f ) be a monomial ideal
1 m
with linear quotients such that all f s are of the same degree. Then I has a linear
i
resolution.
For a squarefree monomial ideal I =(x ···x ,...,x ···x ), the Alexan-
11 1n1 t1 tnt
der dual ideal of I, denoted by I∨, is defined as
I∨ :=(x ,...,x )∩···∩(x ,...,x ).
11 1n1 t1 tnt
4 S.MORADIANDF.KHOSH-AHANG
For a simplicial complex ∆ with the vertex set X ={x ,...,x }, the Alexander
1 n
dual simplicial complex associated to ∆ is defined as
∆∨ ={X\F : F ∈/ ∆}.
ForasubsetC ⊆X,byxC wemeanthemonomial xintheringK[x ,...,x ].
x∈C 1 n
One can see that (I )∨ = (xFc : F ∈ F(∆)), Qwhere I is the Stanley-Reisner
∆ ∆
ideal associated to ∆ and Fc =X \F. Moreover,one can see that (I∆)∨ =I∆∨.
The following theorem which was proved in [24], relates projective dimension
and regularity of a squarefree monomial ideal to its Alexander dual. It is one of
our tools in the study of the projective dimension and regularity of the ring R/I .
∆
Theorem 1.7. (See [24, Theorem 2.1].) Let I be a squarefree monomial ideal.
Then pd(I∨)=reg(R/I).
2. Expansions of a simplicial complex and their algebraic properties
In this section, expansions of a simplicial complex and their Stanley-Reisner
ringsarestudied. The main goalis to explorehow the combinatorialandalgebraic
properties of a simplicial complex ∆ and its Stanley-Reisner ring affects on the
expansions.
Definition 2.1. Let ∆ = hF ,...,F i be a simplicial complex with the vertex
1 m
set V(∆) = {x1,...,xn} and s1,...,sn ∈ N be arbitrary integers. For any Fi =
{x ,...,x }∈F(∆), where 1≤i <···<i ≤n and any 1≤r ≤s ,...,1≤
i1 iki 1 ki 1 i1
r ≤s , set
ki iki Fr1,...,rki ={x ,...,x }.
i i1r1 ikirki
Wedefinethe(s ,...,s )-expansionof∆tobeasimplicialcomplexwiththevertex
1 n
set {{x ,...,x ,x ,...,x ,...,x ,...,x } and the facets
11 1s1 21 2s2 n1 nsn
{x ,...,x } : {x ,...,x }∈F(∆), (r ,...,r )∈[s ]×···×[s ]}.
i1r1 ikirki i1 iki 1 ki i1 iki
We denote this simplicial complex by ∆(s1,...,sn).
Example2.2. Considerthesimplicialcomplex∆=h{x ,x ,x },{x ,x ,x },{x ,x }i
1 2 3 1 2 4 4 5
depicted in Figure 1. Then
∆(1,2,1,1,2) =h{x ,x ,x },{x ,x ,x },{x ,x ,x },{x ,x ,x },{x ,x },{x ,x }i.
11 21 31 11 22 31 11 21 41 11 22 41 41 51 41 52
x31 x21 x51
x3 x2 x5
31 21 51
13 12 15 x11 x41
11 41
11 14
x1 x4 52
22 x52
x22
∆ ∆(1,2,1,1,2)
Figure 1. The simplicial complex ∆ and the (1,2,1,1,2)-
expansion of ∆
EXPANSION OF A SIMPLICIAL COMPLEX 5
The following definition, gives an analogous concept for the expansion of a hy-
pergraph, which is also a generalizationof [11, Definition 4.2].
Definition 2.3. ForahypergraphH withthe vertexsetV(H)={x ,...,x }and
1 n
the edge set E(H), we define the (s ,...,s )-expansion of H to be a hypergraph
1 n
withthe vertexset{x ,...,x ,x ,...,x ,...,x ,...,x }andthe edgeset
11 1s1 21 2s2 n1 nsn
{{x ,...,x }: {x ,...,x }∈E(H), (r ,...,r )∈[s ]×···×[s ]}∪
i1r1 itrt i1 it 1 t i1 it
{{x ,x }: 1≤i≤n, j 6=k}.
ij ik
We denote this hypergraph by H(s1,...,sn).
Remark 2.4. From Definitions 2.1 and 2.3 one can see that for a hypergraph
H and integers s1,...,sn ∈ N, ∆H(s1,...,sn) = ∆(Hs1,...,sn). Thus the expansion of
a simplicial complex is the natural generalization of the concept of expansion in
graph theory.
Example 2.5. Let G be the following graph.
x1 x2
3 2
5
x5
1 4
x4 x3
The graph G(1,1,2,1,2) and the independence complexes ∆ and ∆ are
G G(1,1,2,1,2)
shown in Figure 2.
x31
x2 x3 x52 x21 3
2 3 4 x4 x11 x21 5 2
3 2 x11 1 5 x51
x1 1 5 x5 x31 x551 5 3
x411 4 3 x32 x52 x32 4 x41
(a)∆G (b)G(1,1,2,1,2) (c)∆G(1,1,2,1,2)
Figure 2. The graph G(1,1,2,1,2) and simplicial complexes ∆ and ∆
G G(1,1,2,1,2)
In the following proposition, it is shown that a graph is chordal if and only if
some of its expansions is chordal.
Proposition 2.6. For any s1,...,sn ∈ N, G is a chordal graph if and only if
G(s1,...,sn) is chordal.
Proof. If G(s1,...,sn) is chordal, then clearly G is also chordal, since it can be con-
sidered as an induced subgraph of G(s1,...,sn). Now, let G be chordal, V(G) =
6 S.MORADIANDF.KHOSH-AHANG
{x1,...,xn} and consider a cycle Cm :xi1j1,...,ximjm in G(s1,...,sn), where m≥4
and 1≤j ≤s for all 1≤k ≤m. We consider two cases.
k ik
Case 1. i = i for some distinct integers k and ℓ with 1 ≤ k < ℓ ≤ m. Then
k ℓ
by the definition of expansion, xikjkxiℓjℓ ∈ E(G(s1,...,sn)). Thus if xikjkxiℓjℓ is not
an edge of C , then it is a chordin C . Now, assume that x x is an edge of
m m ikjk iℓjℓ
C . Note that since x x ∈E(C ), either i =i or x x ∈E(G) (if
m iℓjℓ iℓ+1jℓ+1 m ℓ ℓ+1 iℓ iℓ+1
ℓ=m, then set ℓ+1:=1). Thus xikjkxiℓ+1jℓ+1 ∈E(G(s1,...,sn)) is a chord in Cm.
Case 2. i 6= i for any distinct integers 1 ≤ k,ℓ ≤ m. By the definition of
k ℓ
expansion,onecanseethatx ,...,x formsacycle oflengthminG. Soithasa
i1 im
chord. Let xikxiℓ ∈E(G) be a chordin this cycle. Then xikjkxiℓjℓ ∈E(G(s1,...,sn))
is a chord in Cm. Thus G(s1,...,sn) is also chordal. (cid:3)
The followingtheoremillustratesthatthe vertexdecomposabilityofasimplicial
complex is equivalent to the vertex decomposability of its expansions.
Theorem 2.7. Assume that s ,...,s are positive integers. Then ∆ is vertex
1 n
decomposable if and only if ∆(s1,...,sn) is vertex decomposable.
Proof. Assumethat∆isasimplicialcomplexwiththevertexsetV(∆)={x ,...,x }
1 n
and s ,...,s are positive integers. To prove the ‘only if’ part, we use gen-
1 n
eralized induction on |V(∆(s1,...,sn))| (note that |V(∆(s1,...,sn))| ≥ |V(∆)|). If
|V(∆(s1,...,sn))| = |V(∆)|, then ∆ = ∆(s1,...,sn) and so there is nothing to prove in
thiscase. Assumeinductivelythatforallvertexdecomposablesimplicialcomplexes
∆′ andallpositiveintegerss′1,...,s′n with|V(∆′(s′1,...,s′n))|<t,∆′(s′1,...,s′n) isvertex
decomposable. Now, we are going to prove the result when t = |V(∆(s1,...,sn))| >
|V(∆)|. Since |V(∆(s1,...,sn))|>|V(∆)|, there exists aninteger1≤i≤n suchthat
si >1. If∆=hFiisasimplex,weclaimthatxi1 isasheddingvertexof∆(s1,...,sn).
It can be easily checked that
lk∆(s1,...,sn)(xi1)=hF \{xi}i(s1,...,si−1,si+1,...,sn)
and
del∆(s1,...,sn)(xi1)=∆(s1,...,si−1,si−1,si+1,...,sn).
So, inductive hypothesis ensures that lk (x ) and del (x ) are
∆(s1,...,sn) i1 ∆(s1,...,sn) i1
vertex decomposable. Also, it can be seen that every facet of del (x ) is a
∆(s1,...,sn) i1
facet of ∆(s1,...,sn). This shows that ∆(s1,...,sn) is vertex decomposable in this case.
Now, if ∆ is not a simplex, it has a shedding vertex, say x . We claim that x is
1 11
a shedding vertex of ∆(s1,...,sn). To this end, it can be seen that
lk (x )=lk (x )(s2,...,sn)
∆(s1,...,sn) 11 ∆ 1
and
∆(s1−1,s2,...,sn) if s1 >1;
del (x )=
∆(s1,...,sn) 11 (cid:26) del (x )(s2,...,sn) if s =1.
∆ 1 1
Hence, inductive hypothesis deduces that lk (x ) and del (x )
∆(s1,...,sn) 11 ∆(s1,...,sn) 11
are vertex decomposable simplicial complexes. Now, suppose that Fj1,...,jk =
{x ,...,x } is a facet of lk (x ), where F = {x ,...,x } is a face
i1j1 ikjk ∆(s1,...,sn) 11 i1 ik
of ∆. Then since lk (x ) = lk (x )(s2,...,sn), F is a facet of lk (x ). So,
∆(s1,...,sn) 11 ∆ 1 ∆ 1
there is a vertex x ∈ V(∆) such that {x ,...,x ,x } is a face of del (x )
ik+1 i1 ik ik+1 ∆ 1
(see Remark 1.2). Hence {x ,...,x ,x } is a face of del (x ).
i1j1 ikjk ik+11 ∆(s1,...,sn) 11
This completes the proof of the first part.
EXPANSION OF A SIMPLICIAL COMPLEX 7
To prove the ‘if’ part, we also use generalized induction on |V(∆(s1,...,sn))|. If
|V(∆(s1,...,sn))| = |V(∆)|, then ∆ = ∆(s1,...,sn) and so there is nothing to prove
in this case. Assume inductively that for all simplicial complexes ∆′ and all pos-
itive integers s′1,...,s′n with |V(∆′(s′1,...,s′n))| < t such that ∆′(s′1,...,s′n) is vertex
decomposable, we have proved that ∆′ is also vertex decomposable. Now, we
are going to prove the result when t = |V(∆(s1,...,sn))| > |V(∆)|. Now, since
|V(∆(s1,...,sn))| > |V(∆)| and ∆(s1,...,sn) is vertex decomposable, it has a shedding
vertex, say x . If s >1, then
11 1
del (x )=∆(s1−1,s2,...,sn),
∆(s1,...,sn) 11
and the inductive hypothesis ensures that ∆ is vertex decomposable as desired.
Else, we should have s =1,
1
lk (x )=lk (x )(s2,...,sn)
∆(s1,...,sn) 11 ∆ 1
and
del (x )=del (x )(s2,...,sn).
∆(s1,...,sn) 11 ∆ 1
So,inductivehypothesisimpliesthatlk (x )anddel (x )arevertexdecomposable
∆ 1 ∆ 1
simplicial complexes. Now, assume that F = {x ,...,x } is a facet of del (x ).
i1 ik ∆ 1
Then{x ,...,x } is a facetof del (x ). Since x is a shedding vertex
i11 ik1 ∆(s1,...,sn) 11 11
of ∆(s1,...,sn), {xi11,...,xik1} is a facet of ∆(s1,...,sn). Hence, F is a facet of ∆ and
the proof is complete. (cid:3)
Remark 2.8. By the notations as in Definition 2.1, ∆ is pure if and only if
∆(s1,...,sn) is pure, since any facet Fr1,...,rki of ∆(s1,...,sn) has the same cardinal-
i
ity as F .
i
ThefollowingtheoremtogetherwithTheorem2.7helpustoseehowtheCohen-
Macaulaynesspropery in a vertex decomposable simplicial complex and its expan-
sions are related.
Theorem 2.9. A vertex decomposable simplicial complex ∆ is Cohen-Macaulay if
and only if ∆ is pure.
Proof. See [4, Theorem 11.3] and [28, Theorem 5.3.18]. (cid:3)
Corollary 2.10. Let ∆ be a vertex decomposable simplicial complex and s ,...,s
1 n
be positive integers. Then ∆ is Cohen-Macaulay if and only if ∆(s1,...,sn) is Cohen-
Macaulay.
Proof. By Theorem 2.7, ∆(s1,...,sn) is also vertex decomposable. Also, by Theorem
2.9, ∆, respectively ∆(s1,...,sn), is Cohen-Macaulay if and only if ∆, respectively
∆(s1,...,sn), is pure. Now, by Remark 2.8, the result is clear. (cid:3)
Corollary 2.11. Let G be a Cohen-Macaulay chordal graph or a Cohen-Macaulay
bipartite graph. Then G(s1,...,sn) is Cohen-Macaulay.
Proof. By [29, Corollary 7] and [25, Corollary 2.12] chordal graphs and Cohen-
Macaulay bipartite graphs are vertex decomposable. The result now follows from
Corollary 2.10. (cid:3)
In the following theorem, it is shown that shellability is preservedunder expan-
sion and from a shelling for ∆, a shelling for its expansion is constructed.
8 S.MORADIANDF.KHOSH-AHANG
Theorem 2.12. Let ∆ be a shellable simplicial complex with n vertices. Then
∆(s1,...,sn) is shellable for any s1,...,sn ∈N.
Proof. Use the notations as in Definition 2.1. Let ∆ be a shellable simplicial com-
plex with the shelling order F < ··· < F on the facets of ∆. Consider an order
1 m
on F(∆(s1,...,sn)) as follows. For two facets Fr1,...,rki and Fr1′,...,rk′j of ∆(s1,...,sn)
i j
(i) if i<j, set Fr1,...,rki <Fr1′,...,rk′j,
i j
(ii) if i=j, set Fr1,...,rki <Fr1′,...,rk′i, when (r ,...,r )< (r′,...,r′ ).
i i 1 ki lex 1 ki
Weshowthatthisorderingformsashellingorder. ConsidertwofacetsFr1,...,rki and
i
r′,...,r′
F 1 kj with i<j. Since F <F , there exists an integer ℓ<j and x ∈F \F
j i j jt j i
suchthatFj\Fℓ ={xjt}. Soxjtrt′ ∈Fjr1′,...,rk′j\Fir1,...,rki. LetFℓ ={xℓ1,...,xℓkℓ},
where ℓ <···<ℓ . Then there exist indices h ,...,h ,h ,...,h such that
1 kℓ 1 t−1 t+1 kj
j =ℓ ,...,j =ℓ ,j =ℓ ,...,j =ℓ . Thus
1 h1 t−1 ht−1 t+1 ht+1 kj hkj
Fjr1′,...,rk′j \Fℓr1′′,...,rk′′ℓ ={xjtrt′},
where r′′ = r′,...,r′′ = r′ ,r′′ =r′ ,...,r′′ = r′ and r′′ =1 for other
h1 1 ht−1 t−1 ht+1 t+1 hkj kj λ
indices λ. Since ℓ<j, we have Fr1′′,...,rk′′ℓ <Fr1′,...,rk′j.
ℓ j
Now assume that i=j and Fr1,...,rki <Fr1′,...,rk′i. Thus
i i
(r ,...,r )< (r′,...,r′ ).
1 ki lex 1 ki
Let 1≤t≤ki be an integer with rt <rt′. Then xitrt′ ∈Fir1′,...,rk′i \Fir1,...,rki,
Fir1′,...,rk′i \Fir1′,...,rt′−1,rt,rt′+1,...,rk′i ={xitrt′}
and
(r′,...,r′ ,r ,r′ ,...,r′ )< (r′,...,r′ ).
1 t−1 t t+1 ki lex 1 ki
Thus Fr1′,...,rt′−1,rt,rt′+1,...,rk′i <Fr1′,...,rk′i. The proof is complete. (cid:3)
i i
The following corollary is an immediate consequence of Theorem 2.12, Remark
2.8 and [28, Theorem 5.3.18].
Corollary 2.13. Let ∆ be a pure shellable simplicial complex. Then ∆(s1,...,sn) is
Cohen-Macaulay for any s1,...,sn ∈N.
Theorem 2.14. Let ∆ be a pure one dimensional simplicial complex. Then the
following statements are equivalent.
(i) ∆ is connected.
(ii) ∆ is vertex decomposable.
(iii) ∆ is shellable.
(iv) ∆ is sequantially Cohen-Macaulay.
(v) ∆ is Cohen-Macaulay.
Proof.(i⇒ii) Suppose that ∆=hF ,...,F i. We use induction on m. If m=1,
1 m
∆ is clearly vertex decomposable. Suppose inductively that the result has
been proved for smaller values of m. We consider two cases. If ∆ has a
free vertex(a vertexwhichbelongstoonlyone facet),thenthere isafacet,
EXPANSION OF A SIMPLICIAL COMPLEX 9
say F ={x,y}, of∆ suchthat x6∈ m−1F . In this case lk (x)=h{y}i,
m i=1 i ∆
which is clearly vertex decomposableS. Also, since ∆ is connected,
del (x)=hF ,...,F i
∆ 1 m−1
is a pure one dimensional connected simplicial complex. So, by inductive
hypothesis del (x) is also vertex decomposable. Moreover each facet of
∆
del (x) is a facet of ∆. This shows that ∆ is vertex decomposable. Now,
∆
suppose that ∆ doesn’t have any free vertex. So, each vertex belongs to
at least two facets. Hence, there is a vertex x such that del (x) is also
∆
connected and one dimensional. (Note that since ∆ is connected and one
dimensional, it may be illustrated as a connected graph. Also, from graph
theory, we know that every connected graph has at least two vertices such
thatbydeletingthem,westillhaveaconnectedgraph). Now,byinduction
hypothesis we have that del (x) is vertex decomposable. Also, lk (x) is
∆ ∆
a discrete set and so vertex decomposable. Furthermore, in view of the
choice of x, it is clear that every facet of del (x) is a facet of ∆. Hence, ∆
∆
is vertex decomposable as desired.
(ii⇒iii) follows from [4, Theorem 11.3].
(iii⇒iv) is firstly shown by Stanley in [23].
(iv ⇒v) TheresultfollowsfromthefactthateverypuresequantiallyCohen-Macaulay
simplicial complex is Cohen-Macaulay.
(v ⇒i) follows from [28, Corollary 5.3.7].
(cid:3)
Corollary 2.15. Let∆beaCohen-Macaulay simplicial complex ofdimension one.
Then ∆(s1,...,sn) is Cohen-Macaulay for any s1,...,sn ∈N.
Proof. Since ∆ is Cohen-Macaulayofdimension one, Theorem2.14 implies that ∆
is pure shellable. Hence, Corollary 2.13 yields the result. (cid:3)
Theevidencesuggestswhen∆isCohen-Macaulay,itsexpansionsarealsoCohen-
Macaulay. Corollaries2.10, 2.11, 2.13and2.15aresomeresultsinthis regard. But
in general, we did not get to a proof or a counter example for this statement. So,
we just state it as a conjecture as follows.
Conjecture. If ∆ is a Cohen-Macaulay simplicial complex, then ∆(s1,...,sn) is
Cohen-Macaulay for any s1,...,sn ∈N.
3. Homological invariants of expansions of a simplicial complex
We begin this section with the next theorem which presents formulas for the
projective dimension and depth of the Stanley-Reisner ring of an expansion of a
shellablesimplicialcomplexintermsofthecorrespondinginvariantsoftheStanley-
Reisner ring of the simplicial complex.
Theorem3.1. Let∆beashellablesimplicialcomplexwiththevertexset{x ,...,x },
1 n
s1,...,sn ∈N and R=K[x1,...,xn] and R′ =K[x11,...,x1s1,...,xn1,...,xnsn]
be polynomial rings over a field K. Then
pd(R′/I )=pd(R/I )+s +···+s −n
∆(s1,...,sn) ∆ 1 n
and
depth(R′/I )=depth(R/I ).
∆(s1,...,sn) ∆
10 S.MORADIANDF.KHOSH-AHANG
Proof. Let ∆ be a shellable simplicial complex. Then it is sequentially Cohen-
Macaulay. By Theorem 2.12, ∆(s1,...,sn) is also shellable and then sequentially
Cohen-Macaulay. Thus by [20, Corollary 3.33],
pd(R′/I )=bight(I )
∆(s1,...,sn) ∆(s1,...,sn)
and
pd(R/I )=bight(I ).
∆ ∆
Letk =min{|F|: F ∈F(∆)}. Itiseasytoseethatmin{|F|: F ∈F(∆(s1,...,sn))}=
k. Then bight(I )=n−k and
∆
bight(I∆(s1,...,sn))=|V(∆(s1,...,sn))|−k=s1+···+sn−k =s1+···+sn+pd(R/I∆)−n.
The second equality holds by Auslander-Buchsbaum formula, since depth(R′) =
s +···+s . (cid:3)
1 n
Inthefollowingexample,wecomputetheinvariantsinTheorem3.1andillustrate
the equalities.
Example 3.2. Let ∆ = h{x ,x ,x },{x ,x ,x },{x ,x }i. Then ∆ is shellable
1 2 3 1 2 4 4 5
with the order as listed in ∆. Then
∆(1,1,2,1,2) =h{x ,x ,x },{x ,x ,x },{x ,x ,x },{x ,x },{x ,x }i.
11 21 31 11 21 32 11 21 41 41 51 41 52
computationsbyMacaulay2[14],showthatpd(R/I )=3andpd(R′/I )=
∆ ∆(s1,...,sn)
5=pd(R/I )+s +···+s −n=3+1+1+2+1+2−5. Alsodepth(R′/I )=
∆ 1 n ∆(s1,...,sn)
depth(R/I )=2.
∆
The following result, which is a special case of [22, Corollary 2.7], is our main
tool to prove Proposition 3.4.
Theorem 3.3. (See [22, Corollary 2.7].) Let I be a monomial ideal with linear
quotients with the ordering f <···<f on the minimal generators of I. Then
1 m
|set (f )|
I t
β (I)= .
i,j
(cid:18) i (cid:19)
deg(Xft)=j−i
Proposition 3.4. Let ∆ = hF ,...,F i be a shellable simplicial complex with
1 m
the vertex set {x1,...,xn}, s1,...,sn ∈ N and R = K[x1,...,xn] and R′ =
K[x ,...,x ,
11 1s1
...,x ,...,x ] be polynomial rings over a field K. Then
n1 nsn
(i) if s ,...,s >1, then reg(R′/I )=dim(∆)+1=dim(R/I );
1 n ∆(s1,...,sn) ∆
(ii) if for each 1≤i≤m, λ =|{x ∈F : s >1}|, then
i ℓ i ℓ
reg(R′/I )≤reg(R/I )+max{λ : 1≤i≤m}.
∆(s1,...,sn) ∆ i
Proof. Without loss of generality assume that F < ··· < F is a shelling for
1 m
∆. We know that I∆∨ has linear quotients with the ordering xF1c < ··· < xFmc
on its minimal generators (see [17, Theorem 1.4]). Moreover by Theorem 1.7,
reg(R′/I∆(s1,...,sn))=pd(I∆(s1,...,sn)∨)andby[5,Theorem5.1.4]wehavedim(R/I∆)=
dim(∆) + 1. Thus, to prove (i), it is enough to show that pd(I∆(s1,...,sn)∨) =
dim(∆)+1. By Theorem 3.3, pd(I∆∨) = max{|set(xFic)| : 1 ≤ i ≤ m}. For
any 1 ≤ i ≤ m, set(xFic) ⊆ Fi, since any element xℓ ∈ set(xFic) belongs to
(xFjc) :R (xFic) for some 1 ≤ j < i. Thus xℓ = xFjc/gcd(xFjc,xFic) = xFi\Fj.
Let Fi = {xi1,...,xiki} and set(xFic) = {xiℓ : ℓ ∈ Li}, where Li ⊆ {1,...,ki}.
