COHEN-MACAULAY AND GORENSTEIN COMPLEXES FROM A
TOPOLOGICAL POINT OF VIEW
DIETRICH NOTBOHM
Abstract.The main invariant to study the combinatorics of a simplicial c*
*omplex K is
the associated face ring or Stanley-Reisner algebra. Reisner respectivel*
*y Stanley explained
in which sense Cohen-Macaulay and Gorenstein properties of the face ring*
* are reflected
by geometric and/or combinatoric properties of the simplicial complex. W*
*e give a new
proof for these result by homotopy theoretic methods and constructions. *
*Our approach is
based on ideas used very successfully in the analysis of the homotopy th*
*eory of classifying
spaces.
1.Introduction
The main tool and invariant for understanding the combinatorics of a finite s*
*implicial
complex is the associated face ring or Stanley-Reisner algebra which is a quoti*
*ent of a
polynomial algebra generated by the vertices. It is of interest to which extend*
* algebraic
properties of the face ring are reflected by combinatorial or geometric propert*
*ies of the
simplicial complex. For example, Reisner characterized all simplicial complexes*
* whose face
ring is Cohen-Macaulay [18]. And Stanley proved a similar result for Gorenstein*
* face rings
[20]. In this paper we will look at these result with the eyes of a topologist *
*and reprove both
results with methods and ideas from homotopy theory, in particular from the hom*
*otopy
theory of classifying spaces. For the topological proof we introduce and discus*
*s some new
spaces associated with simplicial complexes, which, as we feel, deserve interes*
*t in their own
right.
Let K be an abstract simplicial complex with m vertices given by the set V = *
*{1, ..., m}
That is, K = {oe1, ..., oer} consists of a finite set of faces oei V , which i*
*s closed with respect
to formation of subsets. The dimension of K is denoted by dim K = n - 1. That i*
*s every
face oe of K has order ]oe n and there exists a face ~ of order ]~ = n. We co*
*nsider the
empty set ; as a face of K.
For a commutative ring R with unit we denote by R(K) the associated Stanley-R*
*eisner
algebra of K over the ring R. It is the quotient R[V ]=(voe: oe 62 K), where R*
*[V ] def=
Q
R[v1, ...vm ] is the polynomial algebra on m-generators and voedef=j2oevj. We c*
*an think of
R(K) as a graded object. Since we want to bring topology into the game we will *
*choose
the topological grading and give the generators of R(K) and R[V ] the degree 2.
___________
1991 Mathematics Subject Classification. 13F55, 55R35.
Key words and phrases. simplicial complex, Stanley-Reisner algebra, face ring*
*, Cohen-Macaulay, Goren-
stein, homology decomposition.
1
Each abstract simplicial complex K has a geometric realization, denoted by |K*
*|. Let
(V ) denote the full simplicial complex whose faces are given by all subsets o*
*f V . One
realization of (V ) is given by the standard m-1-dimensional simplex | (V )|de*
*f={ mi=1tiei:
0 ti 1, iti= 1} Rm , where e1, ..., em denotes the standard basis. And a *
*topological
realization of K is given by the subset |K| | (V )| defined by the subset rel*
*ation K
(V ). We define the homology H*(K) and cohomology H*(K) of K as the homology
H*(|K|) and cohomology H*(|K|) of the topological realization.
Before we can state the theorems of Reisner and Stanley, we have to recall so*
*me notions.
We call a simplicial complex K Cohen-Macaulay or Gorenstein over a field F if F*
*(K) is
a Cohen-Macaulay or Gorenstein algebra over F. We call K a Gorenstein* complex *
*if it
is Gorenstein and if V equals the union of all minimal missing simplices of K. *
*A subset
~ V is minimal missing if ~ 62 K and for each oe ( ~ we have oe 2 K. Moreover*
*, for any
face oe 2 K, the link of oe is defined as the simplicial complex
linkK(oe)def={o \ oe : oe o 2 K}.
Now the theorems of Reisner and Stanley read as follows:
Theorem 1.1. (Reisner [18]) Let F be a field and K a simplicial complex. Then,*
* K is
Cohen-Macaulay over F if and only if for each face oe of K, including the empty*
* face,
Hei(linkK(oe); F) = 0 fori < dim linkK(oe)
Theorem 1.2. (Stanley [20]) Let F be a field and K a simplicial complex. Then,*
* K is
Gorenstein* over F if and only if
ae
Hei(linkK(oe); F) ~= F fori = dim linkK(oe)
0 fori 6= dim linkK(oe)
For a definition of Cohen-Macaulay and Gorenstein properties see either [2], *
*[20] or
Section 4. There also exists a version of the second statement which deals wit*
*h general
Gorenstein complexes. But for simplification we formulated the result for Gore*
*nstein*
complexes.
Reisner used methods of commutative algebra, in particular the machinery of l*
*ocal co-
homology of modules, to prove his theorem. Another proof by Hochster (unpublish*
*ed, see
[20]) is based on similar methods and a detailed analysis of the Poincar'e seri*
*es of R(K).
Similar ideas were used by Stanley in his approach towards Gorenstein complexes*
*. In this
paper we want to give a different proof for both results. Our proof is based on*
* topological
constructions related to and based on topological interpretations of the combin*
*atorial data
of K. For example, there exists a topological space c(K) such that H*(X; Z) ~=*
* Z(K)
as a graded algebra. These spaces can be constructed as the Borel construction*
* of toric
manifolds [4], as a (pointed) colimit of a particular diagram [3] or as the hom*
*otopy col-
imit of the same diagram [15]. This last construction is the most appropriate *
*for doing
homotopy theory and is the one which we will use in this paper. Using the homo*
*topy
colimit construction, if dim K = n - 1, one can construct a very interesting no*
*n trivial
map f : c(K) -! BU(n) [17]. The Chern classes of the associated vector bundle a*
*re given
by the elementary symmetric polynomials in the generators of Z(K) (see Section *
*3). The
2
homotopy fibre XK is a finite CW -complex, which contains a large amount of inf*
*ormation
about the associated Stanley-Reisner algebra.
Theorem 1.3.
(i) A simplicial complex K is Cohen-Macaulay over F, if and only if H*(XK ; F) *
*is concen-
trated in even degrees.
(ii) K is Gorenstein over F if and only if H*(XK ; F) is a Poincar'e duality al*
*gebra and
concentrated in even degrees.
This theorem translates Cohen-Macaulay and Gorenstein properties of F(K) into*
* con-
ditions on XK and is the key result necessary for our proof of the results of R*
*eisner and
Stanley.
The paper is organised as follows. In the next two sections we provide the ba*
*sic topolog-
ical ingredients necessary for the proof of Theorem 1.1 and Theorem 1.2. In par*
*ticular, we
recall the above mentioned homotopy colimit construction for the space c(K) in *
*Section
2 and discuss the map fK : c(K) -! BU(n) in Section 3. In Section 4 we provide *
*defi-
nitions for Cohen-Macaulay and Gorenstein algebras appropriate for our purpose,*
* express
these properties in terms of the homotopy fibre XK and prove Theorem 1.3. In Se*
*ction 5
we discuss homotopy fixed point sets and study them for particular actions of e*
*lementary
abelian groups on XK . The final three sections are devoted to a proof of Theor*
*em 1.1 and
Theorem 1.2.
We will fix the following notation throughout. K always denotes a (n - 1)-dim*
*ensional
abstract finite simplicial complex with m-vertices. The set of vertices will be*
* denoted by
V def=VK def={1, ..., m}. We denote by ab the category of abelian groups, by *
*Top the
category of topological spaces and by Top + the category of pointed topological*
* spaces.
Mostly, we are working over commutative rings with unit or fields. In particula*
*r, R will
always denote such a commutative ring and F a field. When we deal with torsion *
*groups,
we will use the topological convention for the grading. Projective resolutions *
*are considered
as non positively graded cochain complexes and torsion groups are non positivel*
*y graded
objects denoted by TorjA(M, N), where j 0.
It is my pleasure to thank N. Ray and T. Panov for introducing me to this sub*
*ject as
well as for plenty of helpful discussions.
2. Pointed and unpointed homotopy colimits
Given a category c and functor F : c -! Top , the colimit colimcF behaves par*
*ticu-
larly poorly in the context of homotopy theory, since object-wise equivalent di*
*agrams may
well have homotopy inequivalent colimits. The standard procedure for dealing w*
*ith this
situation is to introduce the left derived functor, known as the homotopy colim*
*it. Following
[11], for example, hocolimc F may be described by the two-sided bar constructio*
*n. In a
similar fashion, we can construct a pointed homotopy colimit hocolim+cG for a f*
*unctor
G : c -! Top +. Composing G with the forgetful functor OE : Top + -! Top induc*
*es an
identity colimcOEG = colim+cG and a cofibre sequence
Bc -! hocolimcOEG -! hocolim+cG
3
[1]. Here, Bc is the classifying space of the category c, that is the topologic*
*al realization
of the nerve N(c). For details of the homotopy colimit construction see [1] [6]*
* or [7].
The cohomology of (pointed) homotopy colimits can be calculated with the help*
* of the
Bousfield-Kan spectral sequence [1]. And this tool is more important for our pu*
*rpose than
the actual construction of the homotopy colimit. In both cases, this is a firs*
*t quadrant
spectral sequence and has in the case of the unpointed homotopy colimit the form
Ei,j2def=limicopHj(F ) =) Hi+j(hocolimc F )
and in the case of pointed homotopy colimit the form
Ei,j2def=limicopeHj(F ) =) eHi+j(hocolimc F )
In both cases, differentials dr : E*,*r-! E*,*rhave the degree (r, 1 - r).
Let OE : cop- ! ab be a (covariant) functor, e.g. OE def=H*(F ). Higher deriv*
*ed limits of
OE can be thought of as the cohomology groups of a certain cochain complex (C*(*
*c, F ), ffi).
The groups are defined as
Y
Cr(c; F )def= F (c0) forr 0.
co!c1!...!cr
Here, the morphism are morphisms in c and notPin cop. The differential ffi : Cn*
*(c; F ) -!
Cn+1(c; F ) is given by the alternating sum rk=0(-1)kffik where ffik is defin*
*ed on u 2 Cn(OE)
by (
u(c0 ! . .!.bck! . .!.cn+1) for k 6= 0
ffik(u)(c0 ! . .!.cn+1)def=
OE(co ! c1)u(c1 ! . .!.cn) for k = 0.
We may and will replace C*(c; F ) by it's quotient N*(c, F ) of normalised coch*
*ains, given
by the product over chains c0 ! c1... ! cr with distinct objects.
In most cases we consider, the Bousfield-Kan spectral sequence has a particul*
*ar simple
form. All higher limits will vanish. Following Dwyer [7] we turn this propert*
*y into the
following definition.
Definition 2.1. Let F : c -! Top be a functor and R a commutative ring with un*
*it. We
call a map f : hocolimcF -! Y a sharp R-homology decomposition, if limicopH*(F *
*; R) =
0 for i 1 and if f induces an isomorphism limcopH*(F ; R) ~=H*(Y ; R). If F t*
*akes values
in Top +, then we replace cohomology by reduced cohomology.
Using the subset relation on the faces, a simplicial complex K can be interpr*
*eted as a
poset and therefore as a category which we denote by cat(K). This category cont*
*ains the
empty set ; as an initial object. If we want to exclude ; we denote this by Kx *
*respectively
by cat(Kx ). In these cases the classifying space Bcat (K) is equivalent to the*
* cone of the
topological realization |K| and Bcat (Kx ) ' |K|.
For a pointed topological space Y we can define functors
Y K : cat(K) -! Top +, Y K : cat(K) -! Top ,
which assigns the Cartesian product Y oeto each face oe of K. The value of Y K *
*on oe o
is the inclusion Y oe Y owhere the superfluous coordinates are set to *. We n*
*ote that
4
Y K(;)def=*. Moreover, this functor comes with a natural transformation Y K -! *
*1Y Vwhere
1Y Vdenotes the constant functor mapping each face to Y V = Y m and each morphi*
*sm to
the identity map.
Since cat (K) has an initial object, the classifying space Bcat (K) is contra*
*ctible [1]
and the above cofibre sequence degenerates to a homotopy equivalence
hocolimcat(K)Y K -'! hocolim+cat(K)Y K.
We want to specialise further. Let T def=S1 denote the 1-dimensional torus an*
*d BT =
CP 1 the classifying space of T respectively the infinite dimensional complex p*
*rojective
space. For the functor BT K : cat (K) - ! Top , in fact for all functors of th*
*e form
Y K, the higher derived limits of the Bousfield-Kan spectral sequence vanish an*
*d the spec-
tral sequence collapses at the E2-term. Only the actual inverse limit contribu*
*tes some-
thing non trivial, namely the associated Stanley-Reisner algebra [15]. Definin*
*g c(K) def=
hocolimcat (K)BT K we can formulate this as follows.
Theorem 2.2. [15]
(i) H*(hocolimcat (K)BT K; R) ~=R(K).
(ii) hocolimcat (K)BT K -! c(K) is a sharp R-homology decomposition as well as
hocolim+cat(K)BT K -! c(K).
(iii) The natural map c(K) -! BT m realizes the algebra map Z[V ] -! Z(K).
Proof.Part (i) and (iii) and the first half of Part (ii)are already proven in [*
*15]. The second
claim of Part (ii) follows from the equivalence between the pointed and unpoint*
*ed homotopy
colimit, from the fact that reduced cohomology is a natural retract from cohomo*
*logy and
from part (i).
For our purpose we will also need another homology decomposition for our spac*
*e c(K).
For two simplicial complexes K and L we define the join product by K * L def={(*
*oe, o) :
oe 2 K, o 2 L}. Then, by construction, we have c(K * L) = c(K) x c(L). We also *
*notice
that, for the full simplex (V ) def={oe V } on a vertex set V , we have c( (*
*V )) ' BT V.
This follows from the fact that cat ( (V )) has V as a terminal object and that*
* therefore
hocolimcat ( (V ))BT (V )' BT K(V ) = BT V. For every face oe 2 K we denote by*
* st(oe)def=
stK(oe)def={o 2 K : oe [ o 2 K} the star of oe. This is again a simplicial comp*
*lex and, since
st(oe) = (oe) * link(oe) we have c(st(oe)) ' BT oex c(link(oe)).Moreover, for *
*oe o 2 K,
we have stK(o) stK(oe) which induces a pointed map c(stK(o)) -! c(stK(oe)). *
*This
establishes a functor
cstK : cat(K)op- ! Top +
defined by cstK(oe) def=c(stK(oe)). Since the category cat (K)op has an termin*
*al object,
namely ;, we have an obvious homotopy equivalence hocolim+cat(K)opcstK' cstK(;)*
* =
c(K). But restricting cstK to the full subcategory cat (Kx )opproduces a map
hocolim+cat(Kx)opcstK-! c(K) which turns out to be an equivalence as well.
5
Theorem 2.3.
(i) hocolim+cat(Kx)opcstK -! c(K) is a homotopy equivalence and a sharp R-homol*
*ogy
decomposition.
(ii) There exists a cofibration
|K| -! hocolimcat(Kx)op cstK-! c(K).
(iii) (
i R(K) eH0(K; R) for i = 0
lim H*(cstK; R) ~=
cat(Kx)op Hi(K; R) for i > 0.
The rest of this section is devoted to a proof of this theorem. We will compa*
*re the two
homotopy colimits, hocolim+cat(Kx)opcstKand hocolim+cat(K)BT K and do this in s*
*everal
steps. First we show that is sufficient to take the pointed homotopy colimit of*
* BT K over
the category cat (Kx ).
Proposition 2.4. We have an equivalence
hocolim+cat(Kx)BT K -'! hocolim+cat(K)BT K.
Moreover, hocolim+cat(Kx)BT K -! c(K) is also a sharp R-homology decomposition.
Proof.Since eH*(*) = 0, we have an isomorphism
N(cat (Kx )op, eH*(BT K)) ~=N(cat (K)op, eH*(BT K))
of normalised cochain complexes. This shows that the map between both pointed h*
*omotopy
colimits produces an isomorphism between higher limits, an isomorphism between *
*the
Bousfield-Kan spectral sequences and therefore an isomorphism in cohomology. Mo*
*reover,
both homotopy colimits are simply connected. Hence, by the Whitehead theorem, t*
*he map
also induces an isomorphism between the homotopy groups and is therefore a homo*
*topy
equivalence, which proves the second part.
Now, we construct a category c which contains both, cat (Kx )op and cat (Kx )*
*. This
will allow to compare the pointed homotopy colimits hocolim+cat(Kx)BT K and
hocolim+cat(Kx)opcstK. To do this we will distinguish between the objects of ca*
*t (Kx )op,
denoted by oop, and the objects of cat (Kx ), denoted by o. The objects of c ar*
*e given by
the union of the objects of cat (Kx )op and cat (Kx ). That is each face o of K*
* generates
two objects in c, namely o and oop. There exists at most one morphism between *
*two
objects. And there are morphism ae ! o, oop ! aeopand o ! aeopif and only if ae*
* o. We
have obvious inclusions OE : cat(Kx )op- ! c and _ : cat(Kx ) -! c.
Since for ae o 2 K, the set o is a face of stK(ae), there exists a well def*
*ined pointed map
BT o-! c(stK(ae)). These maps are compatible with the pointed inclusions BT ff *
* BT fias
well as with the pointed maps c(stK(fi)) ! c(stK(ff)) for ff fi. Therefore, w*
*e can define
a functor F : c -! Top + such that F(o) def=BT oand F(oop) def=cstK(o). In par*
*ticular,
F _ = BT K and F OE = cstK.
6
Given a functor : d0 -! d, for each object d 2 D, we can define the over ca*
*tegory
# d. The objects are given by morphisms i0: (d0) ! d in D, where d0is an obj*
*ect of d0.
And a morphism (i0: (d0) ! d) ! (i00: (d00) ! d) is given by a morphism j : d*
*0! d00
of d0 such that i00 (j) = i0. The under category d # ae is defined similarly. A*
*s usual, we
say that is left cofinal if all over categories # d and right cofinal if al*
*l under categories
d # ae are contractible; i.e. the classifying spaces are contractible.
The following series of statement shows how to compare the two homotopy colim*
*its in
question.
Lemma 2.5.
(i) For each object o 2 cat(Kx ) c, the under category o # OE is contractible.
(ii) For each object oop 2 cat (Kx )op c , there exists an isomorphism of cat*
*egories
cat (linkK(o)) ~=_ # oop induced by ae 7! (ae [ o ! oop).
Proof.The first claim follows from the fact that o ! oop is a terminal object of
o # OE. The second claim follows from an easy straight forward calculation.
Proposition 2.6.
(i) hocolim+cat(Kx)opcstK' hocolim+cF.
(ii) limicat(Kx)eH*(cstK) ~=limicopeH*(F )
Proof.Since for every object oop 2 c the under category oop # OE is obviously c*
*ontractible,
Lemma 2.5 implies that the inclusion functor cat (Kx )op ! c is right cofinal. *
*Since the
restriction of F |cat (Kx)op= cstK, the equivalence between the pointed homotop*
*y colimits
in part (i) follows from [1].
For the isomorphism in (ii) we need that (cat (Kx )op)op! copis left cofinal *
*[1], i.e. all
under categories c # OEop~= (OE # c)opare contractible. But this follows as abo*
*ve.
Proposition 2.7.
(i) hocolim+cat(Kx)BT K ' hocolim+cF.
(ii) limicat(Kx)opeH*(BT K) ~=limicopeH*(F ). In particular, for i 1,
limicopeH*(F ) = 0.
Proof.The left Kan extension Ldef=LBTK of the functor BT K : cat(Kx ) -! Top + *
*along
the functor cat (Kx ) -! c is defined by L(c) def=hocolim+_#cBT K . And hocoli*
*mc L '
hocolimcat (Kx)BT K [1]. By Lemma 2.5 and Theorem 2.2, there exists a natural t*
*rans-
formation L -! F, which induces a homotopy equivalence at each object. This pro*
*ves the
first part.
For the second part we apply the composition of functor spectral sequence (e.*
*g. see [10]
where the dual situation for colimits is discussed in detail). That is there ex*
*ists a spectral
sequence
Ei,j2def=limicoplimjc#_opeH*(BT K) =) limi+jcat(Kx)opeH*(BT K).
By Theorem 2.2 and Lemma 2.5 limjc#_opeH*(BT K) = 0 for j 1 and limc#_opeH*(B*
*T K) =
He*(F (c)). This proves part (ii).
7
Proof of Theorem 2.3. The first part follows from Proposition 2.6 and Proposi*
*tion 2.7,
and the second part from the general relation between pointed and unpointed hom*
*otopy
colimits as discussed above.
In the rest of the proof all limits are taken over cat(Kx )op. Let 1R denote *
*the constant
functor. Since a category and it's opposite category have the same geometric re*
*alization,
we have limi1R ~=Hi(K). The short exact sequence
0 -! eH*(cstK) -! H*(cstK) -! 1R -! 0
of functors establishes a long exact sequence of the higher limits. By part (i*
*), this long
exact sequence splits into a short exact sequence
0 -! eH*(c(K)) ~=lim0eH*(cstK) -! lim0H*(cstK) -! lim01R ~=H0(K) -! 0
and isomorphisms limiH*(cstK) ~=limi1R ~=Hi(K) for i 1. The short exact seque*
*nce
can be rewritten as
0 -! R(K) -! lim0R(stK) -! eH0(K) -! 0,
which proves part (iii).
Remark 2.8. For later purpose we will calculate the higher limits for particula*
*r functors.
Let M be an abelian group and _M : cat (K)op -! ab be the atomic functor given*
* by
OEM (;) def=M and OEM (oe) def=0 for oe 6= ;.Let 1M : cat (K)op -! ab denote t*
*he constant
functor which maps all objects to M and all morphisms to the identity. Then, we*
* have a
short exact sequence
0 -! OEM - ! 1M - ! _M def=1M =OEM - ! 0.
Since limcat (K)op1M ~= M ~=limcat (K)op_M , the long exact sequence for the *
*higher
limits establishes isomorphisms limicat(K)op_M ~=limi+1OEM . Since N*(cat (K)o*
*p; _M ) ~=
N*(cat (Kx )op; 1M ), we get a sequence of isomorphisms
Hi(K) ~=limicat(Kx)op1M ~= limicat(K)op_M ~= limi+1cat(K)opOEM .
By construction, this composition is natural with respect to maps between simpl*
*icial com-
plexes.
3. A vector bundle over c(K)
In [17], Theorem 2.2 was used to construct a particular nontrivial map c(K) -*
*! BU(n),
whose construction we recall next.
Let T m -! U(m) denote the maximal torus of the unitary group U(m) given by d*
*iagonal
matrices. The cohomology H*(BU(m); Z) ~=Z[c1, ..., cm ] of the classifying spac*
*e BU(m)
is a polynomial algebra generated by the Chern classes ci and H*(BT m; Z) ~= Z[*
*V ] is a
polynomial algebra as well which we identify with the polynomial algebra genera*
*ted by
the set V of vertices. The above map induces an isomorphism H*(BU(m); Z) ~=Z[V *
*] m ~=
Z[oe1, ..., oem ], where we identify m with the Weyl group of U(m) and where *
*m acts on
8
Z[V ] by permutations . Here, oej denotes the j-th elementary symmetric polynom*
*ial. We
can and will identify the Chern classes cj with the elementary symmetric polyno*
*mials oej.
Since dim K = n - 1, a monomial vo vanishes in Z(K) if ]o n + 1. Hence, t*
*he
composition Z[c1, ..., cm ] -! Z[V ] -! Z(K) factors through Z[c1, ..., cn] and*
* establishes a
commutative diagram
Z[c1, ..., cm-]--!Z[V ] ~=Z[v1, ...., vm ]
? ?
? ?
y y
_
Z[c1, ..., cn]---! Z(K)
The left vertical arrow is induced by the canonical inclusion BU(n) -! BU(m), i*
*.e. we
put cj = 0 for n + 1 j m. The following statement concerning a topological *
*realization
of _ is proven in [17].
Theorem 3.1. [17] There exists a topological realization fK : c(K) -! BU(n) of *
*_, i.e.
H*(f; Z) = _, which is unique up to homotopy. Moreover, the diagram
fK
c(K) ---! BU(n)
? ?
? ?
y y
BT m ---! BU(m)
commutes up to homotopy.
The map fK is constructed as follows. Let T n U(n) denote the maximal toru*
*s of
U(n) given by diagonal matrices. Since ]o n, for each face o 2 K, we can thin*
*k of o as
a subset of the set {1, ..., n} which we also denote by n. Such an inclusion es*
*tablishes a
monomorphism T o-! T n U(n) and, passing two classifying spaces, a map fo : BT*
* o-!
BU(n). Moreover, for a different inclusion o n, the two monomorphisms T o- !*
* T n
differ only by a permutation. Hence, they are conjugate in U(n) and the induce*
*d maps
between the associated classifying spaces are homotopic [19]. This establishes *
*a map from
the 1-skeleton of the homotopy colimit into BU(n) unique up to homotopy. There *
*exists an
obstruction theory for extending this map to a map hocolimBT K -! BU(n) as well*
* as for
the uniqueness question of such extensions [21]. The obstruction groups are hig*
*her limits of
the form limjssi(map (BT o, BU(n))foe). If we pass to completion, i.e. we repla*
*ce BU(n) by
it's p-adic completion BU(n)^p, the mapping space can be identified with (BT ox*
*BU(n\o))^p
[14]. For j = i, i+1 and target BU(n)^p, these higher limits do vanish [17], wh*
*ich is sufficient
to prove existence and homotopical uniqueness of maps fK : c(K) -! BU(n)^preali*
*zing
_ for allQprimes [1] [21]. Rationally, the map _ can be realized, since the rat*
*ionalisation
BU(n)0 ' ni=1K(Q, 2i) of BU(n) is a product of rational Eilenberg-MacLane spa*
*ces in
even degrees. An arithmetique square argument then establishes a map fK -! BU*
*(n)
and also shows that the homotopy class of this map is uniquely determined (for *
*details see
[17]).
As already mentioned in the introduction, we define XK as the homotopy fibre *
*of fK :
c(K) -! BU(n). We are particularly interested in the top degrees of H*(XK ).
9
Proposition 3.2.
(i) XK has the homotopy type of a finite CW -complex of dimension n2 + n.
(ii) ae
0 fori > n2 + n
Hi(XK ) ~= Hn-1(K) fori = n2 + n
2+n-1
(iii) If Hei(K) = 0 for i < n - 1, then Hn (Xk) = 0 and there exists a shor*
*t exact
sequence
2+n-2 Y n-2 n-1
0 -! Hn (XK ) -! eH (linkK({i})) -! H (K) -! 0
i2V
Proof.In the proof cohomology is always taken with coefficients in R. The comp*
*osition
fK
BT oe-! c(K) -! BU(n) is natural with respect to maps in cat (K). Interpreti*
*ng
this map as the classifying map of a U(n)-principal bundle establishes a diagra*
*m of U(n)-
principal bundles Y (oe) -! BT oeover cat (K). By construction, Y (oe) ' U(n)=T*
* oe. Since
U(n) acts freely on Y (oe), the Borel construction Y (oe)hU(n)def=Y (oe) xU(n)E*
*U(n) projects
to Y (oe)=U(n) = BT oeby a homotopy equivalence. These projections are natural *
*with re-
spect to morphisms in cat(K). Since Borel constructions commute with taking hom*
*otopy
colimits [6] we get a commutative diagram of fibrations
hocolimcat(K)Y (-) ---! hocolimcat (K)BT K - --! BU(n)
? ? fl
'?y '?y flfl
XK ---! c(K) - --! BU(n)
We can calculate H*(XK ) with the help of the Bousfield-Kan spectral sequence. *
* Since
the normalised cochain complex Ni(cat (K)op, OE) vanishes for i > n for any fun*
*ctor OE, we
have limicat(K)OE = 0 for i > n. Moreover, Hi(U(n)=T oe) = 0 for i > n2 = dim *
*U(n).
Since limiHj(Y (-)) is always a finitely generated abelian group, this shows th*
*at H*(XK )
vanishes in degrees > n2 + n and is a finitely generated abelian group in each *
*degree. By
construction, XK is simply connected, and is therefore homotopy equivalent to *
*a finite
CW -complex of dimension n2 + n.
2 n n2
The above argument also shows, that the group En,n2~= limcat (K)H (Y (-)) is*
* the
only possibly non vanishing term of total degree n2 + n and survives in the spe*
*ctral se-
2 n2+n
quence. In particular, En,n2~= H (XK ). On the other hand, for any oe 6= ;*
* we have
2
dim U(n)=T oe< n2. Hence, the functor Hn (Y (-)) has it's only non vanishing v*
*alue for
2 n2 n,n2 n-1
oe = ; and Hn (Y (;)) ~=H (U(n)) ~=R. Hence, by Remark 2.8, E2 ~=He (K). T*
*his
proves the second part of the claim.
2 i,n2
In fact, Remark 2.8 shows that Hei-1(K) ~= limicat(K)opHn (Y (-)) ~= E2 . H*
*ence,
if Hej(K) vanishes for j 6= n - 1, the only term of total degree n2 + n - 1 is *
*given by
2-1 def n2-1 o 2
limnHn (Y (-)). Let OE = H (Y (-)). Since dim U(n)=T n - ]o, the fu*
*nctor
OE vanishes for all faces o of order 2. Moreover, for each vertex i 2 V the*
* projection
10
2-1 {i} ~= n2-1
U(n) -! U(n)=T {i}induces an isomorphism Hn (U(n)=T ) - ! H (U(n)). This
follows from an analysis of the Serre spectral sequence of the fibration T {i}-*
*! U(n) -!
U(n)=T {i}.
For r = 0, 1 we define functors OEr by OEr(oe) def=OE(oe) if ]oe = r and OEr(*
*oe) def=0 otherwise.
We get short exact sequences of functors
0 -! OE0 -! OE -! OE1 -! 0
and
0 -! OE -! 1R -! _ def=1R=OE -! 0
where 1R denotes the constant functor. The functor _ is non trivial only for fa*
*ces of order
2.
In [16] higher limits of functors defined on cat(K) are discussed in detail. *
*Those results
show, that limj1R = 0 for j 1, that R = lim01R ~= lim0OE, that limj_ ~= limj+*
*1OE for
j 1 and that limj_ = 0 for j n - 1. In particular, 0 = limn-1_ = limnOE. Th*
*is proves
2+n-1
that Hn (XK ) = 0.
The first of the above sequences establishes an exact sequence
0 = limn-1OE0 -! limn-1OE -! limn-1OE1 -! limnOE0 -! limnOE = 0
ByQpart (i) the fourth term can be identified with eHn-1(K), by [16] the third *
*term with
n-2 n2+n-2
i2VHe (link({i})), and the second term with H (XK ). This finishes the*
* proof of
the third part.
Corollary 3.3. Let L K be a subcomplex of the same dimension. Then, the compo*
*sition
2+n n2+n n-1
Hn-1(K) ~=Hn (XK ) -! H (XL) ~=H (L)
is the map induced in cohomology by the inclusion.
Proof.This follows from the above proof and Remark 2.8
We are also interested in the top degree of H*(Xst({i})).
Lemma 3.4. Xst({i})has the homotopy type of a finite CW -complex of dimension *
* n2+
2+n-2 n-2
n - 2 and Hn (Xst({i})) ~=H (link({i})).
Proof.Since c(st({i})) ' BT {i}xc(link({i})) we have a commutative diagram of f*
*ibrations
Xlink({i})---!c(link({i})) x BT {i}---!BU(n - 1) x BT {i}
? ? ?
? ? ?
y ' y y
Xst({i})---! c(st({i})) ---! BU(n)
Since the homotopy fibre of the right vertical arrow is homotopy equivalent to *
*the n - 1-
dimensional complex projective space P(n - 1), this establishes a fibration Xli*
*nk({i})-!
Xst({i})-! P(n - 1). This shows that Xst({i})is simply connected and has the ho*
*motopy
type of a finite CW -complex of dimension (n - 1)2 + (n - 1) + 2(n - 1) = n2 *
*+ n - 2
11
2+n-2 (n-1)2+(n-1)
and that Hn (Xst({i})) ~=H (link({i})). The last equation follows *
*from an
analysis of the Serre spectral sequence of the above fibration.
For later purpose we need the following lemma.
Lemma 3.5. R(K) is a finitely generated R[c1, ..., cn]-module.
Proof.Since R[V ] -! R(K) is a surjection and since R[V ] is a finitely generat*
*ed R[c1, ..., cm ]-
module, the same holds for R(K) as R[c1, ..., cm ]-module. By Theorem 3.1 the *
*map
R[c1, ..., cm ] -! R(K) factors through R[c1, ..., cn], which implies the state*
*ment.
We finish this section with the following observation:
fK
Remark 3.6. The composition c(st(o)) - ! c(K) -! BU(n) makes R(st(o)) into a
R[c1, ..., cn] ~=H*(BU(n); R)-module and, with respect to this structure, all d*
*ifferentials
of the normalised chain complex N*(cat (K); H*(c(stK); R) become H*(BU(n); R)-l*
*inear.
Hence, limiH*((cstK); R) = limiR(stK) is an H*(BU(n); R)-module. The proof of T*
*heo-
rem 2.3 shows that part (iii) can be refined. There exists a short exact sequen*
*ce
0 -! R(K) ~=H*(c(K); R) -! lim0H*(cstK); R) ~=lim0R(stK) -! eH0(K; R) -! 0
of H*(BU(n); R)-modules. Here, H*(BU(n); R) acts on limiR(stK) ~=Hi(K; R) as w*
*ell
as on eH0(K; R) via the augmentation H*(BU(n); R) -! R.
4. Cohen-Macaulay and Gorenstein conditions
In this section we assume that R = F is a field and cohomology is always take*
*n with
coefficients in F. In particular, H*(-)def=H*(-; F).
Let A* be a non negatively graded commutative algebra over F. We say that A**
* is
connected if A0 ~=F and F-finite if Aj = 0 for j large and Ajis a finitely gene*
*rated F-module
in each degree. We call a finite sequence of elements a1, .., ar 2 A a homogene*
*ous system of
parameters, a hsop for short, if they are homogeneous and algebraically indepen*
*dent and
if the quotient A*=(a1, ..., ar) is F-finite. We say that the sequence is a reg*
*ular sequence,
if, for all i, ai+1is not a zero divisor in A=(a1, ..., ai).
We call A* Cohen-Macaulay, if there exists a sequence a1, ..., an which is bo*
*th, a hsop
and regular. If A* is Cohen-Macaulay, then it is known, that every hsop is also*
* a regular
sequence [2].
A Noetherian local ring S is called Gorenstein if S considered as a module ov*
*er itself
has a finite injective resolution. If A* is a commutative connected non negativ*
*ely graded
algebra, we can use the following equivalent definition [20, Theorem I.12.4]. *
*That is,
A* is Gorenstein, if A* is Cohen-Macaulay and if for any hsop a1, ...an of A*, *
*we have
soc(A*=a1, ..., an) ~=F. The socle soc(B*) of a non negatively graded algebra B*
** is defined
as soc(B*) def={b 2 B* : B+ b = 0} ~= Hom B*(F, B*) where B+ denotes the elemen*
*ts of
positive degree.
We call A* a Poincar'e duality algebra, or PD-algebra for short, if there exi*
*sts a class
[A]
[A] 2 Hom F(A*, F) such that the composition A* A* -! A* -! F is a non degen*
*erate
12
bilinear form. In particular, every PD-algebra is connected. As a straight forw*
*ard argument
shows, the above condition is equivalent to the fact that socA*def=HomA*(F, A*)*
* ~=F. That
is, a F-finite non negatively graded algebra is a P D-algebra if and only if it*
* is Gorenstein.
If A* ~=H*(X; F) for a topological space, then we call X a Poincar'e duality sp*
*ace over
F, a F-PD-space for short, if A* is a PD-algebra. In this case [X] def=[A] 2 H**
*(X) is the
fundamental class of X.
We want to describe Cohen-Macaulay and Gorenstein properties of the face ring*
* F(K)
in terms of the map fK : c(K) -! BU(n) described in Theorem 3.1 and it's homoto*
*py
fibre XK . The next result contains part (i) of Theorem 1.3.
Theorem 4.1. The following conditions are equivalent:
(i) F(K) is Cohen-Macaulay.
(ii) The map c(K) - ! BU(n) makes F(K) into a finitely generated free H*(BU(n))-
module.
(iii) The cohomology ring H*(XK ) is concentrated in even degrees.
(iv) TorjH*(BU(n))(F(K), F) = 0 for j -1.
For the proof we need the following lemma. Again we denote by oej the j-th el*
*ementar
symmetric polynomial in the generators of F(K) and identify H*(BU(n)) with F[oe*
*1, ..., oen],
that is with image of the map H*(BU(n)) -! H*(c(K)).
Lemma 4.2. The sequence oe1, .., oen 2 F(K) is a hsop for F(K).
Proof.By Lemma 3.5 F(K)=(oe1, ..., oen) is F-finite. We have only to show that *
*the elements
oe1, ..., oen are algebraic independent in F(K).
Let ~ be a maximal face of K, that is ]~ = n. The composition BT ~- ! c(K) -!
BU(n) is induced by a maximal torus inclusion T ~-! U(n) (see Section 3). The i*
*mages
of oe1, ..., oen in H*(BT ~) ~=F[~] are given by the elementary symmetric polyn*
*omials and
therefore are algebraic independent as well as oe1, ...oe1 in F(K).
Proof of Theorem 4.1: If F(K) is Cohen-Macaulay, then every hsop is given by *
*a regular
sequence. And if we have a hsop of F(K) given by a regular sequence, then F(K)*
* is
Cohen-Macaulay [2]. In the light of Lemma 4.2 this shows that the first two con*
*ditions are
equivalent.
If H*(XK ; Z) is concentrated in even degrees (part (iii)), then, by degree r*
*easons, the
Serre spectral sequence for the fibration XK - ! c(K) -! BU(n) collapses at the*
* E2 page
and H*(c(K)) ~=H*(XK ) F H*(BU(n)) as H*(BU(n))-module. This shows that F(K) is
a finitely generated free H*(BU(n))-module and therefore Cohen-Macaulay, which *
*is part
(i).
If F(K) is Cohen-Macaulay, then, by (ii), it is a finitely generated free mod*
*ule over
H*(BU(n)). In particular, TorjH*(BU(n))(F(K), F) = 0 for j -1. This is part (*
*iv).
If condition (iv) is satisfied, the Eilenberg-Moore spectral sequence for cal*
*culating
H*(XK ; F) collapses at the E2-page and shows that H*(XK ; F) ~=F(K) H*(BU(n);*
*F)F and
that H*(XK ; F) is concentrated in even degrees, which is condition (iii). This*
* proves the
equivalence of (i), (iii) and (iv).
13
Proof of Theorem 1.3 (ii): By our definition of Gorenstein, F(K) is Gorenstei*
*n if and
only if F(K) is Cohen-Macaulay and F(K) F[oe1,...,oen]F ~=F(K)=(oe1, ..., oen) *
*is a PD-algebra.
Hence the equivalence of the two conditions follows from the first part of the *
*Theorem.
5.Homotopy fixed point sets
For an action of a group G on a space X we can think of the fixed point set a*
*s the
mapping space XG = map G(*, X) of G-equivariant maps from a point into X. The n*
*otion
is not flexible enough for doing homotopy theory, since a homotopy equivalence *
*between
G-spaces, which happens to be G-equivariant in addition, does not induce an equ*
*ivalence
between the fixed-point sets in general. We therefore are interested in homoto*
*py fixed
point sets which do have this property. They are defined as the equivariant map*
*ping space
XhG def=mapG(EG, X) where EG is a contractible G-CW-complex with a free G-actio*
*n.
The projection EG -! * induces a map XG - ! XhG .
Applying the Borel construction establishes a fibration
X -! XhG def=X xG EG -ss!BG.
A straight forward argument shows that we can equivalently define the homotopy *
*fixed
point set as the space (XhG ! BG) of sections of this fibration. The latter de*
*finition also
allows to define homotopy fixed point sets in more general situations. A proxy *
*G-action
on X is a fibration X -! E - ss!BG, where we think of E as the Borel constructi*
*on of
this action. We define XhG def= (E ! BG). This establishes a fibration
XhG -! map (BG, E){id}-! map (BG, BG)id
Here, the middle term consist of all lifts of the identity id of BG up to homot*
*opy, i.e.
of all maps g : BG -! E such that ssf ' id. If G is a finite abelian group the*
* base
space is homotopy equivalent to BG [12], and composition of maps yields an equi*
*valence
BG x XhG -'! map (BG, E){id}. This equivalence fits into a commutative diagra*
*m of
fibrations
XhG - --! BG x XhG - --! BG
? ? fl
? ? fl
y y fl
X - --! E - --! BG
where the vertical arrows are induced by evaluation at a basepoint.
Typical examples of proxy actions arise from pull back constructions. Let
X -! E - ss!B be a fibration and f : BG -! B be a map. The pull back constructi*
*on
establishes a commutative diagram
0
E0 --ss-!BG
? ?
? ?
y fy
E --ss-! B
14
and applying the mapping space functor yields a pull back diagram
ss0*
map (BG, E0){id}---! map (BG, BG)id
? ?
? ?
y fy
map (BG, E){f} --ss*-!map (BG, B)f.
The fibre of both horizontal arrows is given by the homotopy fixed point set Xh*
*G . Here, the
left mapping space in the bottom row consists of all maps BG -! E which are hom*
*otopic
to a lift of f. If G is finite and abelian the composition BG ' map (BG, BG)id*
* -!
map (BG, B)f -ev!B equals the map f : BG -! B.
The main goal of this section is to show that, in favourable cases, H*(XhG ; *
*Fp) is con-
centrated in even degrees or a PD-algebra, if H*(X; Fp) satisfies these propert*
*ies. For our
method of proof we have to make some restrictions. According to our situation, *
*we have
the spaces XK in mind, we will always assume that:
(1) X is Fp-finite and p-complete.
(2) H*(X; Fp) is concentrated in even degrees.
In particular, H*(X; Fp) is a graded algebra, commutative in the non graded sen*
*se. Spaces
satisfying both conditions will be called special.
We also restrict ourselves to particular proxy actions. We say that a proxy *
*action
X -! E -! BG is orientable if ss1(BG) acts trivially on H*(X; Z^p).
Remark 5.1. If G is an elementary abelian p-group, and the G-action extends to *
*a torus
action of T r, i.e. the proxy action is induced by a pull back of a fibration X*
* -! E0 -!
BT r, the group G always acts trivially on the cohomology of X and the proxy ac*
*tion is
orientable.
Moreover, if the proxy action X - ! E - ! BG is orientable, the Serre spectra*
*l se-
quence for H*(-; Z^p) collapses at the E2-page by degree reasons. And the same *
*holds for
H*(-; Fp). In particular, H*(E; Fp) ~=H*(BG; Fp) H*(X; Fp) as H*(BG; Fp)-modu*
*le [5,
Corollary 2.5] .
The first part of the next theorem is due to Dehon and Lannes [5, Corollary 2*
*.10].
Theorem 5.2. Let G be an elementary abelian p-group and X -! E -! BG an orienta*
*ble
proxy G-action on a space X.
(i) If X is special, then so is XhG .
(ii) If X is special and a Fp-PD-space, then so is each component of XhG .
The proof of the second part needs some preparation. For the rest of this sec*
*tion we make
the following assumptions. H*(-) denotes H*(-; Fp), G is an elementary abelian *
*p-group
and HG def=H*(BG). Moreover, X is special and the proxy G-action X -! E -! BG is
orientable. In particular, H*(E) ~=HG H*(X) as HG -module (see Remark 5.1).
For such actions H*(E) is not concentrated in even degrees and hence not comm*
*utative in
the non graded sense. To avoid technical difficulties we will use the following*
* construction.
We denote by J HG the ideal generated by all classes of degree 1. And for an *
*graded
HG -module M we denote by M~ the quotient M=JM. If G ~= (Z=p)r, the composition
15
H*(BT r) - ! HG -! H~G is an isomorphism, where the first map is induced by the
canonical inclusion G ~=(Z=p)r T r. In particular, ~HG is a polynomial algebr*
*a generated
by elements of degree 2. In fact, we can think of it as the polynomial part an*
*d as a
subalgebra of HG . Hence, every HG -module is naturally an ~HG-module.
Lemma 5.3. H*(X) is a PD-algebra if and only if H~*(E) is Gorenstein.
Proof.By assumption, H*(E) ~= H*(X) HG as HG -module. And hence, H~*(E) ~=
H*(X) ~HGas ~HG-module. Since ~HG is a polynomial algebra, this implies that *
*~H*(E) is
Cohen-Macaulay. And hence, ~H*(E) is Gorenstein if and only if H*(X) ~=H~*(E) *
*H~GFp
is a PD-algebra.
Let S HG denote the multiplicative subset generated by all Bockstein image*
*s in
degree 2 of nontrivial elements of degree 1. That is the subset of all images o*
*f non trivial
elements of H*(BT r) of strictly positive degree. For any HG -module M we deno*
*te by
S-1M the localised module over S-1HG . Let X -! E -! BG be a G-proxy action on *
*Y .
Following [8] (Corollary 1.2 and the following remark), there exists a map S-1H*
**(E) -!
S-1(HG H*(Y hG)) between the localised modules. Under favourable circumstance*
*s this
is an isomorphism, which, as shown in [5, Section 2], always hold if X is speci*
*al and if the
G-action is orientable. We collect this into the following theorem.
Theorem 5.4. ([8] [5]) Let X -! E -! BG be an orientable G-proxy action on a sp*
*ecial
space X. Then there exist isomorphisms
S-1H*(E) ~=S-1(HG H*(XhG )) ~=(S-1HG ) H*(XhG ).
Since the multiplicative subset S HG consist of elements of even degree, *
*it also
is a multiplicative subset of H~G. We can think of the localised module S-1H~**
*(E) ~=
H~*(E) H~GS-1H~G in a different way. Since H~G -! H~*(E) is a monomorphisms, *
*the
set S gives rise to a multiplicative subset S0 H~*(E). Then multiplication i*
*nduces an
isomorphism ~H*(E) H~GS-1H~G ~=(S0)-1H~*(E), where the latter localisation is *
*obtained
by localising the algebra ~H*(E) with respect to the subset S0.
Proof of Theorem 5.2 (ii): Let us assume that X is an Fp-PD-space. Then, H~*(*
*E) is
Gorenstein (Lemma 5.3) as well as S-1H~*(E) [2, Proposition 3.1.19]. Since
M
S-1H~*(E) ~=H*(XhG ) S-1H~G ~= H*(XhGg) S-1H~G
g
as algebras, each of the summands is Gorenstein. Here, the direct sum is taken*
* over
components of XhG . Let XhGgdenotes the component associated to a section g : B*
*G -! E
of ss : E -! BG. Now we give the algebra H*(XhGg) S-1H~G a different grading *
*induced
by the grading of the first factor. That is elements of S-1HG get degree 0. Sin*
*ce H~G ~=
Fp[t1, ..., tn], the sequence {t1- 1, ...tn - 1} H*(XhGg) S-1H~G is regular*
*, homogeneous
and consists of elements of degree 0. Hence H*(XhGg) ~=H*(XhGg) S-1H~G=(t1-1, .*
*.., tn-1)
is a connected Fp-finite Gorenstein algebra and therefore a PD-algebra (see Sec*
*tion 4).
16
6. Cohen-Macaulay and Gorenstein properties for links of faces
In this section we want to prove that Cohen-Macaulay or Gorenstein properties*
* are
inherited to the Stanley Reisner algebras of links of faces. We call a simplic*
*ial complex
pure, if all maximal faces have the same dimension. We first consider the case *
*of algebras
over Fp.
Theorem 6.1. If Fp(K) is Cohen-Macaulay respectively Gorenstein, then K is pure*
* and,
for each face o 2 K the algebra Fp(linkK(o)) is also Cohen-Macaulay respectivel*
*y Goren-
stein.
For the proof we will use the space XK given by the fibration XK - ! c(K) -! *
*BU(n)
described in Section 3. Since BU(n) is simply connected, the p-adic completion *
*maintains
the fibration [1]. Since we are working with Fp as coefficients and since for s*
*imply connected
spaces completion induces an isomorphism in mod-p cohomology, we can and will a*
*ssume
that all spaces are completed. To simplify notation, we will always drop the no*
*tation for
completion. The proof also relies on the interpretation of Fp(link(o)) as a cer*
*tain mapping
space given in [16], which we recall next.
For each face o 2 K we denote by Go T othe maximal elementary abelian subgr*
*oup
of the torus T o. Let go : BGo -! c(K) denote the composition BGo -! BT o- !
go fK
hocolimcat (K)BT K ' c(K). The composition BGo - ! c(K) - ! BU(n) establishes
a proxy Go-action XK -! E -! BGo. Applying the mapping space functor we get a
fibration
o o (fK)* o
XhG -! map (BG , c(K)){fKgo}- ! map (BG , BU(n))fKgo
(see Section 5). The composition fK go is induced from a coordinate-wise inclus*
*ion Go
T o T n U(n) into the set of diagonal matrices (see Section 3). The centralis*
*er of this
image equals T ox U(n \ o) where we again think of n as the set {1, ..., n} and*
* o becomes a
subset of the set n via the coordinate-wise inclusion. Then, by construction, f*
*K go factors
through a map idx const: BT o- ! BT ox BU(n \ o) where the fist coordinate is t*
*he
identity and the second the constant map. There also exists a map BT ox c(link*
*(o)) =
c(st(o)) -! c(K) as constructed in Section 2 and the map go factors through i x*
* const:
BGo -! BT ox c(link(o)). Moreover all these maps fit into a diagram
idxflink(o)
Xlink(o)---! BT ox c(link(o)) ------! BT ox BU(n \ o)
? ? ?
? ? ?
y y y
fK
XK ---! c(K) ---! BU(n).
Applying the mapping space functor, there exit the following equivalences (of p*
*-completed)
spaces; BT o- '! map (BGo, BT o)i, BT ox BU(n \ o) -'! map(BGo, BU(n)fKgo [9] *
*and
BT ox c(link(o)) - '! map (BGo, c(K))go [16]. Putting all this information tog*
*ether we
17
have a commutative diagram
flink(o))
Xlink(o))---! BT ox c(link(o)) ----! BU(n \ o)
? ? ?
? ? ?
y 'y 'y
o o o
(XK )hGgo---! map (BG , c(K))go ---! map (BG , BU(n))fKgo
*
* o
of horizontal fibrations. Since BT ox BU(n \ o) is simply connected, the map Xh*
*GK -!
map (BGo, c(K)){fKgo}induces a bijection between the components of both spaces.*
* Hence,
the bottom left space is connected. This proves the following proposition.
o
Proposition 6.2. Xlink(o)' (XK )hGgo.
Now we are in the position to prove Theorem 6.1.
Proof of Theorem 6.1 If Fp(K) is Cohen-Macaulay, the fibre XK is special (Th*
*eorem
1.3(i)). Since the map fK go : BGo -! BU(n) factors through BT o, the proxy Go *
*action
extends to a torus action andois orientable (see Remark 5.1). We can apply The*
*orem
5.2. That is that (XK )hGgo' Xlink(o)is special and that Fp(link(o)) is Cohen-*
*Macaulay
(Theorem 1.3(i)). If Fp(K) is Gorenstein, we use Theorem 1.3(ii) instead of the*
* first part.
Finally we have to show that K is pure. The above argument shows that, if Fp(*
*K) is
Cohen-Macaulay, then Fp(link(o)) is a free H*(BU(n \ o))-module. If ~ 2 K is a *
*maximal
simplex, then link(~) is the empty complex and Fp(link(~)) ~=Fp. This implies t*
*hat ~ has
order n and that K is pure.
We finally give up the restriction on the coefficients.
Corollary 6.3. If F(K) is Cohen-Macaulay respectively Gorenstein, then K is pur*
*e and,
for every face o 2 K, the Stanley-Reisner algebra F(linkK(o)) is also Cohen-Mac*
*aulay
respectively Gorenstein.
Proof.We will use Theorem 6.1 and Theorem 1.3. If F is a field of characteristi*
*que p > 0,
then H*(XK ) ~=H*(XK ; Fp) FpF. This shows that F(K) is Cohen-Macaulay if and *
*only
if Fp(K) is so. The same holds for links. This covers the Cohen-Macaulay case a*
*s well as
the Gorenstein case and also shows that K is pure.
Let Z(p)denote the localisation of Z at the prime p. Then, for F = Q we can *
*argue
as follows. Since XK is of the homotopy type of a finite CW -complex (Propositi*
*on 3.2),
H*(XK ; Z) has only finitely many torsion primes. Hence, Q(K) is Cohen-Macaulay*
* if and
only if H*(XK ; Q) is concentrated in even degrees if and only if, for almost a*
*ll primes,
H*(XK ; Z(p)) is concentrated in even degrees and torsion free if and only if, *
*for almost all
primes, H*(XK ; Fp) is concentrated in even degrees if and only if, for almost *
*all primes,
Fp(K) is Cohen-Macaulay. Now, Theorem 6.1 and and the above chain of equivalent
statements applied in the case of link(o) proves the claim for Cohen-Macaulay a*
*lgebras
over F = Q. In the Gorenstein case, we only have to notice that H*(XK ; Q) is *
*a P D-
algebra if and only H*(XK ; Fp) is a P D-algebra for almost all primes.
For a general field of characteristique 0 we deduce the claim from the case F*
* = Q in the
same manner as for fields of characteristique p > 0 from F = Fp.
18
7. Proof of Theorem 1.1
Theorem 1.1 follows easily from the following statement by induction.
Theorem 7.1. F(K) is Cohen-Macaulay if and only if F(linkK(o)) is Cohen-Macaula*
*y for
all faces o 6= ; of K and eHr(K; F) = 0 for 0 r < n - 1.
Proof of Theorem 1.1: If dim K = 0 then F(K) is always Cohen-Macaulay. In fac*
*t, in
this case c(K) is the m-fold wedge product of BS1's and Xk the (m-1)-fold wedge*
* product
of S2's, whose cohomology is concentrated in even degrees. On the other hand th*
*e set of
conditions on the cohomology of the links of K is an empty set. This proves the*
* statement
in this case.
The general case follows by induction over the dimension of K and the above t*
*heorem.
In the following cohomology is always with F-coefficients. We define H*(-)def*
*=H*(-; F)
for the rest of this section. Also, for simplification, we set P def=H*(BU(n)) *
*~=F[oe1, ..., oen].
The proof of Theorem 7.1 is based on the homological analysis of particular d*
*ouble
complexes. To fix notation, we will recall the general concept next. For detail*
*s see [13]
Let A be a F-algebra. A differential graded A-module (C*, dC) is a cochain c*
*omplex
of A-modules such that dC is A-linear. A double complex or differential bigrade*
*d module
(M*,*, dh, dv) over A is a bigraded A- module M*,*with two A-linear maps dh : M*
**,*-!
M*,*and dv : M*,*-! M*,*of bidegree (1, 0) and (0, 1) such that dhdh = 0 = dvdv*
* and
dhdv + dvdh = 0. We think of dh as the horizontal and of dv as the vertical di*
*fferential.
To each double complex M*,*we associate a total complex T ot*(M) which is a dif*
*ferential
graded module over A defined by T otn(M) def= i+j=nMi,jwith differential D def=*
*dh + dv.
Examples are given by the tensor products of two differential graded modules (B*
**, dB ) and
(C*, dC). If we set Mi,jdef=Bi A Cj, dhdef=dB 1 and dvdef=(-1)j1 dC, we get*
* a double
complex such that T ot*(M) = B* A C*.
For a double complex (M*,*, dh, dv), we can take horizontal or vertical cohom*
*ology groups
denoted by H*h(M*,*) and H*v(M*,*). The boundary maps dh and dv induce again bo*
*und-
ary maps on these cohomology groups. We can consider cohomology groups of the f*
*orm
H*h(H*v(M*,*)) and H*v(H*h(M*,*)).
If (M*,*, dh, dv) is bounded below, that is Mi,j= 0 if i or j is small enough*
*, there exist
two spectral sequences converging towards H*(T ot(M), D). In one case, we have*
* Ei,j2=
Hih(Hjv(M)) and in the other case Ei,j2def=Hjv(Hih(M)). In the first case the d*
*ifferential have
degree (r, 1 - r, ) and in the second case degree (1 - r, r).
Let N* def=N*(cat (Kx )op, H*(cstK)) denote the normalised cochain complex fo*
*r the
functor H*(cstK) : cat(Kx )op- ! ab considered in Section 2. Actually, the comp*
*lex N*
is a bigraded object. It inherits an internal degree from the grading of H*(cst*
*K), which we
will not consider in most cases.
We collect the main properties of N* in the next proposition.
19
Proposition 7.2.
(i) Ni = 0 for i < 0 or i n.
(ii) (N*, dN ) is a differential graded P -module as well as H*(N*, dN ) ~=lim**
*cat(Kx)op, F(stK)
(iii) There exist an isomorphism Hi(N*, dN ) ~=eHi(K) for i 1 of P -modules a*
*nd a short
exact sequence
0 -! F(K) -! H0(N*, dL) ~=limcat(Kx)op0F(stK) -! eH0(K) -! 0
of P -modules.
Proof.The first part follows from the fact that Ns(cat (Kx )op, H*(cstk)) = 0 f*
*or s n,
the second and the third part from Remark 3.6.
We can also look at the projective resolution of the trivial P -module F give*
*n by the
Koszul complex Q*def= * F P . According to our convention we make this into a *
*cochain
complex and give the generators of the exterior algebra *def= *(x1, ..., xn) t*
*he degree -1.
As usual the differential dQ is defined by dQ(xi) def=oei and dQ(y) = 0 for y 2*
* P and has
degree 1. Again, Q* is a differential graded P -module, bounded below and above*
* by Qj = 0
for j > 0 or j < -n. The differential bigraded P -module N* P Q* is then bound*
*ed. In
particular, both above mentioned spectral sequences converge towards H*(Tot(N* *
*P Q*)).
Proof of Theorem 7.1: If K is the empty complex, there is nothing to show. If*
* K is a
0-dimensional complex, then F(K) is always Cohen-Macaulay as discussed in the p*
*roof of
Theorem 1.1 and Theorem 1.2.
Now we assume that dim K 1. i.e. n 2. We start with the assumption that F*
*(K)
is Cohen-Macaulay. By Theorem 6.3, we know that K is pure and that for all o 2*
* K,
the algebra F(link(o)) is also Cohen-Macaulay as well as F(st(o)). In particul*
*ar, since
dim st(o) = dim K, the algebra R(stK(o)) is a finitely generated free module ov*
*er P . We
have to show that eHi(K) = 0 for i < n - 1.
We look at the above constructed double complex N* P Q*. All modules Ni and *
*Qj
are free P -modules. Hence, the functors N* P - and - P Q* are exact. We get
ae
0 forj 6= 0
Hih(Hjv(N* P Q*)) ~=Hih(N* P Hjv(Q*)) ~= Hi(N*
P F) forj = 0
In particular, the E2-term is concentrated in one horizontal line given by j = *
*0, the spectral
sequence collapses and Hi(Tot(N* P Q*)) ~=Hi(N* P F) = 0 for i < 0 or i n.
Considering the second spectral sequence we get
Hjv(Hih(N*8 P Q*)) ~=Hjv(Hih(N*) P Q*)
>> TorjP(Hei(K), F) ~=TorjP(F, F) F eHi(K)fori > 0
< * 0 e0
~= H (XK ) TorP(F, F) H (K) fori = 0 and j = 0
>> Tor0 (F, F) eH0(K) fori = 0 and j 6= 0
: P
0 otherwise
For i = 0 this follows from Proposition 7.2 and the fact that Tor0P(F(K), F) ~=*
*H*(XK ).
By degree reasons there is no differential ending at or starting from H-nv(H0h(*
*N* P ) ~=
20
n(x1, ..., xn) eH0(K) ~=eH0(K). Since n 2 this group has total degree -n +*
* 1 < 0 and
must therefore vanish. Hence, the whole column given by i = 0 and j -1 vanish*
*es and
E0,02~=H*(XK ). By induction we can apply this argument successively for i = 1,*
* ..., n - 2.
In these cases the whole columns Ei,j2vanish. For i = n - 1 there might be a no*
*n trivial
differential En-1,n2-! E0,02and we cannot conclude anymore that En-1,j2= 0. Thi*
*s shows
that eHi(K) = 0 for 0 i n - 2 and proves one direction of the claim.
Now we assume that, for ; 6= o 2 K, all algebras F(link(o)) are Cohen-Macaul*
*ay
and that Hej(K) = 0 for j < dim K = n - 1 1 In particular, K is connected. T*
*his
implies that for any pair ~, ~0 of maximal faces of K, there exists a chain of *
*maximal
faces ~ = ~1, ..., ~s = ~0 such that the intersection ~j \ ~j+1 is non empty. *
*Since linki
is pure for all vertices {i} 2 K (Theorem 6.3), this implies that all maximal s*
*implices of
K have the same order, that K is pure, that for all faces o 2 K the dimension o*
*f link(o)
equals n - 1 - ]o and that H(BT ox c(link(o))) ~=F(stK(o)) is a finitely genera*
*ted free
module over P . Now we consider again both spectral sequences. In this case, we*
* get for
Hih(Hjv(N* P Q*)) the same result as above. And, since eH0(K) = 0, Proposition *
*7.2 shows
that
8 j
< TorP(F(K), F) for i = 0
Hjv(Hih(N* P Q*)) ~=Hjv(Hih(N*) P Q*) ~= Torj(Hen-1(K), F) for i = n
: P
0 otherwise
Hence the E2-page is concentrated in two vertical lines given by i = 0, n. Cons*
*idering again
total degrees shows that TorjP(F(K), F) = 0 for j 6= 0 and that F(K) is a finit*
*ely generated
free P -module (Theorem 4.1). This proves the other implication of the claim.
We can draw the following consequence from the above proof.
Corollary 7.3. If F(K) is Cohen-Macaulay, then there exists a short exact seque*
*nce
0 -! eHn-1(K) -! H*(XK ) -! limH*(Xst(-)) -! n-1(n) eHn-1(K) -! 0
8. Proof of Theorem 1.2
The proof of Theorem 1.2 is an easy consequence of the following statement.
Theorem 8.1. F(K) is Gorenstein* if and only if F(linkK(o)) is Gorenstein* for *
*all faces
o 6= ; of K and ae
eHi(K; F) ~= F fori = dim K
0 fori 6= dim K
Proof of Theorem 1.2: We argue as in the proof of Theorem 1.1. If dim K = 0,*
* then
K is always Cohen-Macaulay. And K is Gorenstein* if and only if Xk ' S2 if and *
*only if
m = 2 if and only if eH0(K; F) = F.
The general case follows by induction over the dimension of K and the above t*
*heorem.
The proof of Theorem 8.1 needs some preparation.
21
Remark 8.2. We call a simplicial complex reduced, if for every vertex i, the in*
*clusion
st({i}) K is proper. If st({i}) = K then K = {i} * link({i}). Hence, any comp*
*lex K
can be written as a joint product * L of a full simplex and a reduced compl*
*ex L.
Moreover, K is reduced if and only if the union of all minimal missing faces eq*
*uals the set
V of vertices.
We call a simplicial complex K F-spherical if it satisfies the geometric cond*
*ition of the
Gorenstein property, i.e. if
ae
eHi(link(o); F) ~= F fori = dim link(o)
0 fori 6= dim link(o)
for every face o 2 K in including ;.
Proposition 8.3. Let F(K) be Gorenstein. Then the following holds:
(i) K is reduced if and only if eHn-1(K) 6= 0. In fact, if this is the case, th*
*en eHn-1(K) ~=F.
(ii) If K is reduced, then, for every vertex i 2 V , the link link({i}) is also*
* reduced.
Proof.By Corollary 7.3 we have an exact sequence
Y
0 -! eHn-1(K) -! H*(XK ) -! H*(Xst({i}))
i2V
Let d def=dimXK . If Hen-1(K) = 0 then the second map becomes a monomorphism a*
*nd
there exists an i 2 V such that F ~=Hd(XK ) -! Hd(Xst({i})) is a monomorphisms.*
* On
the other hand, if L K is a subcomplex of the same dimension as K such that F*
*(K) and
F(L) are Cohen-Macaulay, then the map H*(XK ) -! H*(XL) is an epimorphism. Henc*
*e,
Hd(XK ) ~=Hd(Xst({i})) ~=F for some vertex i 2 K. Since both algebras are PD-al*
*gebras
(Theorem 6.3), this implies that H*(XK ) ~= H*(Xst({i})). Comparing the two fi*
*brations
defining XK and Xst({i})shows that F(st({i})) ~=F(K), that st({i}) = K, and tha*
*t K is
not reduced.
If K is not reduced, then, as the cone of a subcomplex, |K| is contractible a*
*nd eHn-1(K) =
0. This proves the equivalence in part (i). If one of the conditions is satisfi*
*ed, then, since
2+n n-1
H*(XK ) is P D-algebra, Lemma 3.2 shows that F ~=Hn (XK ) ~=eH (K).
Since Gorenstein algebras are Cohen-Macaulay, Theorem 7.1 and Proposition 3.2*
* (ii) tell
us that there exists a short exact sequence
2+n-2 Y n-2 n-1
0 -! Hn (XK ) -! eH (link({i}) -! eH (K) ~=F -! 0.
i2V
Since XK is a P D-space of dimension n2 + n , the first term in the above sequ*
*ence is
isomorphic to H2(XK ) ~=Fm-1 . Hence, the middle term must be isomorphic to Fm *
*. Since
for each vertex i 2 V , dimF eHn-2(link({i})) 1, this shows that eHn-2(link({*
*i})) ~=F and
that link({i}) is reduced.
For o V we denote by Ko K the full subcomplex which consist of all faces *
*ae 2 K
such that ae V \ o. If o = {i} is a vertex, we denote this complex by Ki.
22
Lemma 8.4. If K is F-spherical, then, for each vertex i 2 K, the complex Ki is *
*Cohen-
Macaulay and eHn-1(Ki) = 0.
For the proof we need some preparation. The inclusions Ki K and st({i}) K
induce epimorphisms F(K) -! F(Ki) and fii : F(K) -! F(st({i})) of P -modules. *
*The
kernel of the first map is the ideal viF(K) generated by vi 2 F(K). And the se*
*cond
epimorphism induces an isomorphism viF(K) ~=viF(st({i})). This follows from th*
*e fact
that for any face o 2 K the monomial vivo = 0 in F(st({i})) if and only if o [ *
*{i} 62 K.
Moreover, since st({i}) = {i} * link({i}), multiplication by vi induces an isom*
*orphism
F(st({i})) -! viF(st({i})). In fact, all the above maps are F(K)-linear, where *
*F(K) acts
on F(Ki) and F(st({i})) via the above projections. Moreover they fit together t*
*o a short
exact sequence
0 -! F(st({i})) -! F(K) -! F(Ki) -! 0
of F(K)-modules. The first map is given by q 7! viq0where fii(q0) = q. Applying*
* the functor
- P establishes an epimorphism ffi: H*(XK ) -! H*(Xst({i})) and an H*(XK )-line*
*ar map
_i: H*(Xst({i})) -! viH*(XK )
given by _i(a) = via0where ffi(a0) = a. In fact, we will show that this map is *
*an isomor-
phism (see Corollary 8.5).
Proof of Lemma 8.4: Since for two vertices i, j 2 K we have linkK({j})iequals*
* linkK({j})
if the simplex {i, j} 62 K and equals linkKi({j}) if {i, j} 2 K, we only have t*
*o prove that
Her(Ki) = 0 for r n - 1. And this claim we prove via an induction over the di*
*mension of
K.
For n = 1, F-spherical implies that K consists only of two vertices. And for *
*n = 2, F-
spherical implies that K is a triangulation of S1. In both cases, the claim is *
*straightforward.
Now let us assume that n 3. By excision, H*(K, Ki) ~=H*(stK({i}), linkK({i*
*})) ~=
H*( linkK({i})). Here linkK({i}) denotes the suspension of linkK({i}), actua*
*lly of the
geometric realization of linkK({i}). Moreover, the map H*(K, Ki) - ! H*(K) can*
* be
identified with H*( linkK({i})) -! H*(K) induced by the last arrow in the cofi*
*bration
sequence linkK({i}) -! Ki- ! K -! linkK({i}). Since linkK({i}) is F-spheric*
*al, it
suffices to show that Hn-1(Ki) = 0.
Let j 2 V such that o def={i, j} 2 K. In the following, K will also denote th*
*e geometric
realization of K. Because of the identities linkK(o) = linklinkK({i})({j}) = li*
*nklinkK({j})({i}),
linkKi({j}) = linkK({j})i and linkKj({i}) = linkK({i})j all rows and columns in*
* the
homotopy commutative diagram
linkK(o) ---! linkKj({i})---! linkK({i})
? ? ?
? ? ?
y y y
linkKi({j})---! Ko - --! Ki
? ? ?
? ? ?
y y y
linkK({j}) ---! Kj - --! K
23
consist of cofibrations. Passing to suspensions we can extend the diagram to th*
*e right and
to the bottom yielding a homotopy commutative 4x4-diagram, whose bottom right s*
*quare
looks like
K ---! linkK({j})
? ?
? ?
y y
linkK({i})---! 2linkK(o).
In the induced diagram in cohomology in degree n - 1
Hn-1( 2linkK(o)) ---! Hn-1( linkK({j}))
? ?
? ?
y y
Hn-1( linkK({i})) ---! Hn-1(K)
the left vertical and top horizontal arrows are isomorphisms by induction hypot*
*hesis.
Therefore, the other two arrow are either both isomomorphisms or both trivial. *
* And
both are isomorphisms if and only if Hn-1(Ki) = Hn-1(Kj) = 0.
For n 2, the complex K is connected and we can connect each pair of vertice*
*s by
1-dimensional faces. The above argument now shows that the maps Hn-1(K, Ki) - !
Hn-1(K) are either isomorphisms for all vertices or trivial for all vertices. *
*Hence it is
sufficient to show this map is at least nontrivial for at least one vertex, or *
*equivalently,
that Hn-1(Ki) = 0 for at least one vertex.
2+n
Since H*(XK ) is generated by classes of degree 2, a generator of Hn (XK ) *
*~=F can
be represented by a monomial a which can be written as via0for a suitable verte*
*x i 2 K.
We fix this vertex. By the above considerations we have an exact sequence
2+n-2 n2+n n2+n
Hn (Xst({i})) -! H (XK ) -! H (XKi).
The first map is given by multiplication with viand therefore an isomorphism. B*
*y Corollary
3.3, the latter map can be identified with the map Hn-1(K) -! Hn-1(Ki), which, *
*since
2+n n-1
dim linkK ({i}) = n - 2, is an epimorphism. Hence, we have 0 = Hn (XKi) ~=H *
* (Ki),
which completes the proof.
Corollary 8.5. IF K is F-spherical, then, for each vertex i 2 K, multiplication*
* by vi
induces an isomorphism H*(Xst({i})) -! viH*(XK ).
Proof.All terms of the exact sequence
0 -! F(st({i})) -! F(K) -! F(Ki) -! 0
are Cohen-Macaulay (Corollary 6.3, Theorem 1.1, Lemma 8.4. Hence, applying the *
*functor
PF establishes a short exact sequences
0 -! H*(Xst({i})) -! H*(XK ) -! H*(XKi) -! 0.
By construction, the first map is given by multiplication by vi.
24
Proof of Theorem 8.1: If K is Gorenstein*, then Proposition 8.3 shows that Hn*
*-1(K) ~=
F and, together with Corollary 6.3, that for each face o 2 K the algebra F(link*
*(o)) is
Gorenstein*. 2
For the opposite conclusion it suffices to show that soc(H*(XK ) ~= Hn +n(XK *
*) ~= F
(Theorem 1.3(ii)). By induction we can assume that K is F-spherical.
For each vertex i 2 V the map H*(XK ) -! H*(Xst({i})) is an epimorphism. He*
*nce,
2+n-2
this map maps socH*(XK ) to socH*(Xst({i})) ~=QHn (Xst({i})). Moreover, we*
* have
an exact sequence 0 -! Hn-1(K) -! H*(XK ) -! iH*(Xst({i})) (Corollary 7.3), a*
*nd
2+n-2
hence all elements socH*(XK ) have degree n2 + n - 2. Let a 2 Hn (XK ). *
* This
2+n-2
class maps to 0 6= b 2 Hn (Xst({i})) ~=F for some vertex i 2 V . And, by Le*
*mma 8.5,
2+n * n2+n n-1
0 6= vib = via 2 Hn (XK ). This shows that socH (XK ) ~=H (XK ) ~=H (K) *
*~=F
and finishes the proof.
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Department of Mathematics, University of Leicester, University Road, Leiceste*
*r LE1 7RH,
England
E-mail address: dn8@mcs.le.ac.uk
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