Homology groups of simplicial complements: A new proof of Hochster theorem

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by  Jun Ma

HOMOLOGY GROUPS OF SIMPLICIAL COMPLEMENTS: A NEW
PROOF OF HOCHSTER THEOREM
5
1 JUN MA, FEIFEI FAN AND XIANGJUNWANG
0
2
Abstract. Inthispaper,weconsiderhomologygroupsinducedbytheexterioralgebra

n generated by a simplicial compliment of a simplicial complex K. These homology
a groupsareisomorphictotheTor-groupsTork[m](k(K),k)ofthefaceringk(K),which
J i,J
isveryusefulandmuchstudiedintorictopology. ByusingCˇechhomologytheoryand
8
Alexander duality theorem, we prove that these homology groups have dualities with

the simplicial cohomology groups of the full subcomplexes of K. Then we give a new
]
T proof of Hochster’s theorem.
A
.
h
t
a
m 1. Introduction
[ Throughout this paper, k is a field or the ring of integers Z. k[m] = k[v ,...,v ] is
1 m

1 the graded polynomial algebra on m variables, deg(vi) = 2. The face ring (also known
v as the Stanley-Reisner ring) of a simplicial complex K with m vertices is the quotient
7
ring
8
7 k(K) = k[m]/I
K
1
0 where I is the ideal generated by those square free monomials v ···v for which
K i1 is
1. {i1,...,is} is not a simplex in K.
0 For any simple polytope Pn, Davis and Januszkiewicz [5] introduced a Tm-manifold
5 Z with orbit space Pn. After that Buchstaber and Panov [3] generalized this definition
1 P
to any simplicial complex K with vertex set [m] = {1,2,...,m}, and named it the
:
v
moment-angle complex associated to K:
i
X
r ZK = [ D(σ),
a
σ∈K
where D(σ) = Y ×Y ×···×Y , Y = D2 if i ∈ σ and Y = S1 if i 6∈ σ. The follow-
1 2 m i i
ing theorem is proved by Buchstaber and Panov [3] for the case over a field by using
Eilenberg-Moore Spectral Sequence, [1] for the general case.
Theorem 1.1 (Buchstaber and Panov [7, Theorem 4.7]). Let K be a simplicial complex
with m vertices. Then the following isomorphism of algebras holds:
H∗(Z ;k) ∼= Tork[m](k(K),k).
K
2010 Mathematics Subject Classification. Primary 13F55, 18G15; Secondary 16E05, 55U10.
The authors are supported by NSFC grant No. 11261062, NSFC grant No. 11371093 and SRFDP
No.20120031110025.
1
2 J.MA,F.FAN&X.WANG
Since Tork[m](k(K),k) has a natural Z⊕Zm-bigrade. So the bigraded cohomology
ring can be decomposed as follows:
H∗(ZK;k) ∼= Tork[m](k(K),k) = M M Torki,[Jm](k(K),k).
i≥0 J⊆[m]
k[m]
Hochster [6] gave a combinatorial description of the Tor-groups Tor (k(K),k).
i,J
Theorem 1.2 (Hochster [6]).
Tork[m](k(K),k) ∼= H|J|−i−1(K ;k),
i,J J
e
where H−1(∅;k) = k, K is the full subcomplex of K corresponding to J.
J
e
Hochster proved this for the case k is a field by analyzing the betti number of both
sides. Buchstaber andPanov [4, Theorem 3.2.9] generalize this resultto k = Z by means
of the Koszul resolution Λ[m]⊗k[m] of k.
Recently, WangandZheng[8]gave anotherwaytocalculateTork[m](k(K),k)byusing
Taylor resolution on the Stanley-Reisner ring k(K). This method was presented firstly
by Yuzvinsky in [9]. In this paper we give a combinatorial description of this method,
and then we prove Hochster’s theorem in a different way.
Definition 1.3 (missing faces and simplicial complements). Let K be a simplicial com-
plex on the set [m]. A missing face of K is the subset τ ⊆ [m] where τ ∈/ K and every
proper subset of τ is a simplex of K. Denote by MF(K) the missing face set of K.
Clearly MF(K) is uniquely determined by K.
A simplicial complement P of K is a subsetof {τ ⊆ [m]| τ 6∈ K}so that MF(K) ⊆ P.
Usually P is not unique. Denote by P(K) the set of simplicial complements of K.
Given a simplicial complement P of K, one can define an exterior algebra Λ∗[P] gen-
erated by all elements of P. For a monomial u = τ τ ···τ ∈ Λ∗[P], let
i1 i2 in
Su = τi1 ∪τi2 ∪···∪τin
be the total set of u. So Λ∗[P] has a natural Z⊕Zm-bigrade, which means
Λ∗[P]= M M Λi,J[P]
i≥0 J⊆[m]
where Λi,J[P] is generated by monomials u satisfying Su = J and deg(u) = i.
One can make Λ∗[P] into a chain complex by defining a differential d on it. The
differential d: Λn,∗[P]→ Λn−1,∗[P] is generated by
n
d(u) = X(−1)j+1∂ju·δj,
j=1
where u = τ τ ···τ , ∂ u = τ ···τ ···τ (particularly, if n= 1, ∂ (τ )= 1), δ = 1
i1 i2 in j i1 ij in 1 i1 j
if Su = S∂i(u) and zero otherwise. b
Theorem 1.4 (see [8, Theorem 2.6 and Theorem 3.2]). Let K be a simplicial complex
on the set [m]. Given a simplicial complement P ∈ P(K). Then
Tork[m](k(K),k) ∼= H Λ∗,J[P],d .
i,J i(cid:0) (cid:1)
SIMPLICIAL COMPLEMENTS 3
Remark 1.5. From Theorem 1.4, we know that the homology groups H Λ∗,J[P],d
i(cid:0) (cid:1)
of a simplicial complement P is not depend on the choice of P. It just depend on the
simplicial complex K.
If we can prove that
H Λ∗,J[P],d ∼= H|J|−i−1(K ;k),
i(cid:0) (cid:1) J
e
then the Hochster theorem is proved. The following theorem ensure these isomorphisms
hold.
Theorem 1.6. Let K be a simplicial complex on the set [m], and let P be one of the
simplicial compliments of K. Then we have the following group isomorphisms:
H Λ∗,[m][P],d ∼= Hm−i−1(K;k).
i(cid:0) (cid:1)
e
Corollary 1.7. Let everything be as before. Then
H Λ∗,J[P],d ∼= H|J|−i−1(K ;k).
i(cid:0) (cid:1) J
e
2. Combinatorial description of homology groups of simplicial
complements
If K is a simplex, the theorem is trivial. So in this paper, we assume that K is a
simplicial complex on the set [m] but not a simplex. Given a simplicial complement P
of K, suppose
P = {τ ,τ ,...,τ }.
1 2 r
For any τ ∈ P, we have a simplicial complex
i
star∂∆m−1τi := {τ ∈ ∂∆m−1 | τ ∪τi ∈ ∂∆m−1}.
Clearly star∂∆m−1τi is a triangulation of Dm−2. We denote by Ui = int(cid:12)star∂∆m−1τi(cid:12), the
(cid:12) (cid:12)
interior of the geometric realization of star∂∆m−1τi.
Proposition 2.1. Let U(K) := |∂∆m−1|\|K|. P and U are defined as above. Then
i
U= U is a open cover of U(K).
(cid:8) i(cid:9)i=1,2,...,r
Proof. If x ∈ |∂∆m−1|\|K|, there is a simplex τ ∈ 2[m] \K satisfying x in the relative
interior of τ. From the definition of a simplicial complement, there exists a simplex
τ ∈ P, such that τ ∈ τ. Thus, x ∈ U . (cid:3)
i i i
Definition 2.2. For any topological space X, Let U= U be an open cover of the
(cid:8) i(cid:9)i∈I
space X indexed by a set I. We define the nerve N(U) to be the abstract simplicial
complex on the set I
N(U) = (i ,i ...,i ) ⊆ I | U ∩U ∩...∩U 6= ∅ .
(cid:8) 1 2 n i1 i2 in (cid:9)
For a simplex σ = (i ,i ,...,i ) ∈N(U), denote by U = U ∩U ∩...∩U .
1 2 n σ i1 i2 in
The simplicial homology groups of N(U) are called the homology groups of the open
cover U, and denoted by
Hˇ (X,U;k) := H (N(U);k).
∗ ∗
The following theorem is a canonical result of Cˇech homology theory which will be
used in the sequel.
4 J.MA,F.FAN&X.WANG
Theorem 2.3 (see [2, Corollary 13.3]). Let U = U be an open cover of the space
(cid:8) i(cid:9)i∈I
X having the property that H (U ) = 0 for all σ ∈ N(U). Then there is a canonical
∗ σ
e
isomorphism
H (X;k) ∼= Hˇ (X,U;k).
∗ ∗
Theorem 2.4. Let K be a simplicial complex on [m], P = {τ ,τ ,...,τ } be a simplicial
1 2 r
complement of K. By Proposition 2.1, U = U forms an open cover of the
(cid:8) i(cid:9)i=1,2,...,r
topological space U(K). Then we have the following isomorphisms:
H Λ∗,[m][P],d ∼= Hˇ (U(K),U;k) := H (N(U);k).
n(cid:0) (cid:1) n−2 n−2
e e
Before proving Theorem 2.4, we work on the following lemma first.
Lemma 2.5. We use notations as above. Then
(1) If τi∪τj 6= [m], then Ui∩Uj = int(cid:12)star∂∆m−1τi∪τj(cid:12).
(2) If τ ∪τ = [m], then U ∩U = ∅.(cid:12) (cid:12)
i j i j
Proof. Note that for any simplex τ of ∂∆m−1,
◦
int(cid:12)star∂∆m−1τ(cid:12)= [ |σ|,
(cid:12) (cid:12)
τ⊆σ
◦
where|σ|istherelativeinteriorofthegeometricrealization ofσ. Thenthelemmafollows
by a direct checking. (cid:3)
Proof of Theorem 2.4. From Definition 2.2, we know that
N(U) = (i ,i ...,i ) ∈ 2[r] |U ∩U ∩...∩U 6= ∅ .
(cid:8) 1 2 n i1 i2 in (cid:9)
Lemma 2.5 shows that U ∩U ∩...∩U 6= ∅ if and onely if τ ∪τ ...∪τ 6= [m].
i1 i2 in i1 i2 in
So the nerve can be written as
N(U) = (i ,i ...,i ) ∈2[r] | τ ∪τ ...∪τ 6= [m] .
(cid:8) 1 2 n i1 i2 in (cid:9)
Let Λ∗[P] be the exterior algebra generated by P= {τ ,τ ,...,τ }. We define another
1 2 r
differential ∂ : Λn[P]→ Λn−1[P ] generated by
0
n
∂(u) = X(−1)j+1∂ju,
j=1
where∂ u = τ ···τ ···τ foranymonomialu = τ τ ···τ . Defineahomomorphism
j i1 ij in i1 i2 in
b
Φ : C N(U);k −→ Λ∗+1[P],
∗(cid:0) (cid:1)
e
generated by Φ (i ,i ,...,i ) := τ τ ···τ ∈ Λn[P], where C N(U);k is the re-
(cid:0) 1 2 n (cid:1) i1 i2 in ∗(cid:0) (cid:1)
duced simplicial chain complex of N(U). Obviously, Φ is aemonomorphism. Let
Γ = ImΦ. Then we get a short exact sequence of chain complexes,
0 → C (N(U);k , ∂) → Λ∗+1[P], ∂ → Λ∗+1[P]/Γ, ∂) → 0.
(cid:0) ∗ (cid:1) (cid:0) (cid:1) (cid:0)
e
Apparently, Λ∗+1[P]/Γ is generated by all monomials u ∈ Λ∗+1,[m][P] (i.e., Su = [m]).
It is easy to see that there is a chain isomorphism
Λ∗+1[P]/Γ, ∂ ∼= Λ∗+1,[m][P], d ,
(cid:0) (cid:1) (cid:0) (cid:1)
SIMPLICIAL COMPLEMENTS 5
where Λ∗+1,[m][P], d is as in Theorem 1.4.
(cid:0) (cid:1)
It is easy to see that H Λ∗[P], ∂ = 0. Thus from the long exact sequence induced
∗(cid:0) (cid:1)
by the short exact sequence above, we get that
H Λ∗,[m][P], d ∼= H (N(U);k).
n(cid:0) (cid:1) n−2
e
(cid:3)
Proof of Theorem 1.6. Note first that ∂∆m−1 ∼= Sm−2, and (cid:12)star∂∆m−1τ(cid:12) ∼= Dm−2 for
each τ ∈ ∂∆m−1, τ 6= ∅. We combine the results of Theor(cid:12)em 2.3, The(cid:12)orem 2.4 and
Lemma 2.5 to get that
Hn(cid:0)Λ∗,[m][P], d(cid:1) ∼= Hn−2(U(K);k).
e
From Alexander duality theorem, we have
H (U(K);k) ∼= Hm−n−1(K;k).
n−2
e e
(cid:3)
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(J. Ma) School of Mathematics and Statistics, Fudan Univ., P. R. China
E-mail address: [email protected]
(F. Fan) School of Mathematics and Statistics, Nankai Univ., P. R. China
E-mail address: [email protected]
(X. Wang) School of Mathematics and Statistics, Nankai Univ., P. R. China
E-mail address: [email protected]