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. References [1] A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localisation* *s, SLNM 304, Springer Verlag. [2] W. Bruns and H.J. Herzog. Cohen-Macaulay rings, Cambridge Studies in Advan* *ced Mathematics 39, Cambridge University Press, 1998. [3] Victor M Buchstaber and Taras E Panov. Torus Actions and Their Application* *s in Topology and Combinatorics, volume 24 of University Lecture Series, American Mathematic* *al Society, 2002. [4] M.W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and t* *orus actions, Duke Mathematical Journal 62 (1991), 417-451. [5] F.-X. Dehon and J. Lannes, Sur les espaces fonctionnels dont la source est* * le clasifiant d'un group de Lie compact commutatif, Inst. Hautes 'Etudes Sci. Publ. 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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 26