Minimal free resolutions that are not supported by a CW-complex.

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1 Minimal free resolutions that are not supported by a CW-complex. Mauricio Velasco Department of Mathematics, Cornell University, Ithaca, NY 4853, US Introduction The study of monomial resolutions and their connection with combinatorial topology has been an active area of research in recent years. In [] it is shown that every monomial ideal admits a simplicial resolution (Taylor s resolution) and that some minimal free resolutions are supported in simplicial complexes (Scarf ideals, monomial regular sequences). This idea is generalized in [2] where cellular resolutions are introduced. The authors show that every monomial ideal admits a resolution supported in a regular cell complex (the hull resolution) which is minimal in many cases (for example ideals in three variables) but not in general. More recently, in [3] and [6] discrete Morse theory is used to construct CWcomplexes which support the minimal free resolutions of virtually all classes of monomial ideals for which these resolutions are known. s the class of allowed topological spaces grows (simplicial complexes, regular cell complexes, CW complexes) more minimal free resolutions can be endowed with one of these geometries. s more and more positive examples are found, the following fundamental question (see e.g. [6]) became very interesting: Can every minimal monomial free resolution be supported by a CW-complex? The purpose of the present paper is to explore this question and to answer it in the negative. Our fundamental tool will be the nearly Scarf ideals, a family of ideals associated to simplicial complexes introduced in joint work with I. Peeva [9]. The minimal free resolution of a nearly Scarf ideal is closely related to the topology/combinatorics of an associated simplicial complex. s a result,

2 these resolutions have a remarkably rich structure, and we can use topological methods to produce minimal free resolutions with interesting properties. The main results in this paper are:. We find necessary and sufficient conditions, in terms of the topology of the simplicial complex, for the associated nearly Scarf ideal to have a minimal cellular resolution. 2. We show that there are minimal CW-resolutions which are not cellular. 3. We show that there are minimal free resolutions that are not supported in any CW-complex and we construct a large class of counterexamples: the nearly Scarf ideals of certain Eilenberg-Maclane spaces. cknowledgements I would like to thank Mike Stillman, Irena Peeva, llen Hatcher and Ken Brown for many helpful discussions. I would especially like to thank Irena Peeva for suggesting the question in [6] as an interesting topic and llen Hatcher, who was very generous with his time. 2 Preliminaries In this section we will briefly recall the definition of CW complexes, regular cell complexes and cellular homology and establish notation which will be used in the rest of the paper. For more detailed information see [5] and [7]. Let B n, U n and S n denote the closed unit ball, the open unit ball and the unit sphere in R n respectively. Definition. (finite) CW -complex is a topological space constructed by finitely many steps of the following inductive procedure:. Start with a finite discrete set X (). 2. For any finite collection of continuous maps φ α : S n X (n ), let X (n) = X (n ) α Bn α/ where x φ α (x) for all x in Sα n. Endow this space with the quotient topology. Definition 2. Every attaching map φ α can be naturally extended to a map Φ α : B n α X (n) which is a homeomorphism between U n α and it s image e n α. We will call e n α an n-cell and Φ α the characteristic map of e n α. Note that X is the disjoint union of it s cells. 2

3 Definition 3. The cells of a CW -complex X form a poset under the relation e n α e m whenever en α e m. If en α e m we say that en α is a face of e m. We denote this poset by F aces(x). We will allways assume that the cells of our CW-complexes are oriented (i.e. that we have chosen a generator of H i (e α i, e α i \e i α) for each α). 2. Cellular homology of CW -complexes If X is a CW complex, its singular homology can be computed via cellular homology (see [5] pag. 37). Definition 4. The augmented oriented cellular homology chain complex of X, C(X, Z) is a complex of free Z-modules. Ci (X, Z) is generated by the i-cells of X and the boundary map i is given by: The usual reduced simplicial boundary for i =,. For i 2, (e i α) = X (e i α, e i )ei. Here as in the rest of the paper X (e i α, e i ) denotes the degree of the composition: Sα i X (i ) S i. The first arrow is the attaching map φ α and the second is the quotient map collapsing X (i ) \e i to a point which we will denote by q. The reduced singular homology of X with coefficients in any field k, H (X, k) is obtained by computing the homology of C(X, Z) Z k. 2.2 Regular cell complexes Definition 5. CW -complex is a regular cell complex if all characteristic maps are homeomorphisms. This additional requirement gives a much more rigid structure to regular cell complexes, as exemplified by the following three fundamental properties which do not hold for general CW -complexes:. For any n-cell e n, e n \e n is the union of closures of (n )-cells. 2. Whenever e n α e n+2 there are exactly two (n + )-cells e n+ such that e n e n+ e n Two regular cell complexes with isomorphic face posets are homeomorphic. 3

4 Moreover, for a regular cell complex, cellular homology can be described purely combinatorially in terms of an incidence function. Definition 6. n incidence function η on a regular cell complex is a function whose domain consists of pairs of faces of consecutive dimension and which satisfies the following three properties:. η(e n, e n ) {,, } and is nonzero iff e n e n. 2. If e α and e are the two vertices which are faces of a one-cell e then η(e, e α) + η(e, e ) =. 3. Suppose e n e n+2 and e n+ α, e n+ e n e n+ e n+2. Then are the only n + -cells such that η(e n+2, e n+ α )η(e n+ α, e n ) + η(e n+2, e n+ )η(e n+, e n ) = Theorem 7.2 in [7] states that every regular cell complex X admits an incidence function η and that any such function determines an orientation of the cells for which X (e i α, eα i ) = η(e i α, eα i ). 2.3 Cellular resolutions and CW resolutions. Let R = k[x,..., x n ] be multigraded by letting deg(x i ) be the i-th basis vector in N n. We will identify the monomials in R with their degrees in N n and order them by a b iff a i b i for all i. For cellular resolutions we will use the definition proposed in [2] where the term was introduced. We will use the term CW-resolutions to refer to the generalization of cellular resolutions proposed in [3] (In section 5 we will show that CW resolutions are a strictly larger class). Definition 7. CW-complex X is N n -graded if there is a function gr : F aces(x) N n which is order preserving. Let M be a monomial ideal in R and let F M be an N n -graded free resolution of R/M. Definition 8. F M is a CW-resolution if there exists an N n -graded CW - complex X such that:. For all i (F M ) i has a basis ê i,..., ê i j (i )-dimensional cells of X. in bijection with the 4

5 2. deg(ê i α) = gr(e i α). 3. d(ê i α) = X (e i α, e i α) deg(ê i )xdeg(êi )êi. Definition 9. F M is a cellular resolution if the CW -complex X in the last definition can be chosen to be a regular cell complex. Note that, if F M is a CW-resolution (or a cellular resolution) supported on X, the complex F M k k[x,..., x n ]/(x,..., x n ) is C(X, Z) Z k. 3 Nearly Scarf ideals and their minimal free resolutions. Definition. Given a simplicial complex Ω (with more than one point) its nearly Scarf ideal J Ω is the ideal in the ring R = k[{x F : F Ω, F }] whose generators are the monomials m v = F v x F as v ranges through the vertices of Ω. Definition. For each nonempty face G Ω, let m G = F G x F and let z Ω = F Ω x F. The following easy property will be used frequently in the rest of the paper: ( ) for F, G in Ω, m F m G if and only if F G In particular if F G are faces of the same dimension, m F incomparable. and m G are We will now construct the minimal free resolution of R/J Ω in four steps:. Given Ω, consider its oriented homology chain complex L L : CD (Ω, k) D... C (Ω, k) C (Ω, k) C (Ω, k) k f D k f k f k 2. For each i, let h i = dim( H i (Ω, k)) and choose a set {q,..., q hi } of cycles whose classes in ker( i )/im( i+ ) form a basis. 3. Define φ i : k h i Ker( i ) by φ i (e j ) = q j (where the e j are the canonical basis of k h i ). 5

6 4. Construct the complex F Ω which in homological degree i is given by: ( R( m F ) ) R( z Ω ) h i 2 F:dim(F)=i- and whose differential ψ i is represented by the unique map of multidegree whose matrix (with respect to the canonical basis in the above direct sum) equals ( i φ i 2 ) when we set all variables appearing in its entries to (see example below). Theorem. F Ω is the minimal free resolution of R/J Ω. For a proof see theorem 6. in [9]. Note that, ignoring the multidegrees of the generators, F Ω has the form: F Ω :... R f 2 R ˆ 2 ˆφ ˆ ˆφ R f R h R f ˆ R where f i is the number of i-dimensional faces of Ω. Example: Let Ω be a square with vertices {a, b, c, d} and edges e = (d, a), f = (a, b), g = (b, c), h = (c, d). Each vertex determines a monomial m a = bcdgh, m b = acdeh, m c = abdef, m d = abcfg and J Ω is the ideal of in k[a,..., h] generated by (m a, m b, m c, m d ). Steps (),(2),(3) yield the complex: k 4 k C C k 4 (,,,) k 6

7 Step (4) gives the minimal free resolution of R/J Ω : F Ω : R( z Ω ) e f g h C af -ae bg -bf ch -cg C -dh de R( abcdfgh) R( bcdgh) R( abcdegh) R( abcdef h) R( abcdef g) R( acdeh) R( abdef) R( abcf g) R The minimal free resolutions of nearly Scarf ideals are especially useful since we can describe the matrix form of the differential for all choices of multihomogeneous basis. Lemma. For any choice of multihomogeneous basis in F Ω the differential is represented by matrices: F Ω :... R f 2 R ˆF Ê2 Ê ˆF R f R h R f Ê R where Êi = D i ˆ i D i+ for some invertible diagonal matrices D i with entries in k. Proof. Recall that the generators of R f i have multidegrees m F as F ranges through the i-dimensional faces of Ω. By ( ) any two of these monomials are incomparable and all of them divide the multidegree z Ω strictly. s a result any matrix representing a homogeneous automorphism: R f i R h D E R f i R h i must have the above block form where E and D are invertible matrices with coefficients in k and D is diagonal. Thus any homogeneous change of coordinates leads to a commutative dia- 7

8 gram:... R f i R h D i+ i+ B i+... R f i R h Êi Ĵ i ˆF i Ĝ ˆ i ˆφi R f i R h i D i i B i R f i R h i 2... which implies that Ĵi = Ĝi = and that Êi = D i ˆ i D i+. 4 Cellular resolutions of nearly Scarf ideals In this section we will determine necessary and sufficient conditions for F Ω to be a cellular resolution. Theorem 2. F Ω is a cellular resolution over k if and only if for all i, H i (Ω, k) can be generated by cycles whose support is homeomorphic to S i. Proof. Suppose that the graded regular cell complex X supports F Ω. Then the faces of X whose degree strictly divides z Ω form a subcomplex isomorphic to Ω (since by ( ) their face posets must coincide and the fact that a regular CW- complex is determined up to homeomorphism by its face poset). Moreover, since X supports F Ω there is some choice of basis such that F Ω k k[x F : F Ω]/(x F : F Ω) is the cellular homology complex of X. By Lemma this complex must be:... k f 3 k h 3 F 2 k f 2 k 2 F F k f k h k f k where i are the simplicial boundary maps of Ω (for some orientation of the cells). Since this complex is exact (it is isomorphic to the multidegree z Ω strand of F Ω ) the F i (e i+ α ) are cycles whose classes in ker( i )/im( i+ ) form a basis. Moreover F i (e i+ α ) is supported in the image of the attaching map of e i+ α which is homeomorphic to S i since X is regular. Conversely, assume that the homology of Ω is generated by cycles a j e i j supported on spheres and let X be the regular cell complex obtained from 8

9 Ω by attaching a new cell on each of these. We must show that the incidence function η in Ω can be extended so that, for each new cell e n+ α, (eα n+ ) = a j e n j since in that case X supports F Ω. Let η(e n+ α, e n j ) = a j. If τ is any (n ) face of e n+ α and e n j and en k are the cells in the support of e n+ α containing τ we have: η(e n+ α, e n j )η(e n j, τ) + η(e n+ α, e n k )η(en k, τ) = a jη(e n j, τ) + a k η(e n k, τ) = where the last equality follows since it is the τ component of the boundary of the cycle a j e n j. Thus η is an incidence function in X and (en+ α ) = η(e n+ α, e n j )en j = a j e n j. Corollary. Let Ω be any finite graph with at least two vertices. Then F Ω is a cellular resolution for all fields k. Proof. Choose a point p i in each connected component of Ω. Then the cycles p j p : j >, form a basis for H (Ω, k) by cycles supported in -spheres. Now, assume that Ω is connected (otherwise repeat the following procedure in each connected component) and let T be any spanning tree. For each e T consider the cycle obtained by concatenating e with the shortest path in T which joins it s endpoints; these cycles form a basis for H (Ω, k) by subcomplexes homeomorphic to circles (since collapsing T to a point induces a homotopy equivalence between Ω and a wedge of circles, one for each edge e T ). By theorem 2, F Ω is a cellular resolution. The next example shows that cellularity can depend on the characteristic: Example: Let Ω be a simplicial complex homeomorphic to RP 2. When char(k) 2, Ω is acyclic so F Ω is a simplicial (hence cellular) resolution. If char(k) = 2, H2 (Ω, k) = k and all generators are multiples of the sum of all simplices of Ω. Since Ω is not homeomorphic to S 2 theorem 2 shows that F Ω is not cellular. 5 The need for CW resolutions. It is legitimate to wonder whether the notion of CW-resolution is really necessary. That is, whether every resolution which can be supported in a CW complex can also be supported in a regular cell complex. 9

10 There is an obvious restriction: every resolution supported by a regular cell complex must (for some choice of basis) have all coefficients in its differential matrices equal to either or. In section 5 of [8] the authors provide an example of a Stanley-Reisner ideal which requires a coefficient of 2 in its minimal free resolution (for every choice of basis); thus this resolution cannot be cellular. The next example shows that CW resolutions are needed even for minimal resolutions whose only coefficients are, (in particular this example is not a cellular resolution in any characteristic). Example: Figure : Let Ω be any triangulation of the torus (S S ). For any field k, the generators of H 2 (Ω, k) = k are supported on Ω and thus, by theorem 2, F Ω is not cellular. On the other hand, F Ω is always a CW resolution. We will construct a graded CW complex X that supports it by attaching the following cells to Ω: pair of 2-cells e 2, e2 2 identifying their boundary circles to the usual generators of H (Ω, k) = k 2. single 3-cell e 3, attaching its boundary S2 to X(2) via the map shown in figure (note that this map is not injective and hence not a homeomorphism).

11 Note that X (e 3, e2 i ) = for i =, 2 since on the new cells the attaching map of the figure is either not surjective or fills the cell twice, once with each orientation. Let gr(f ) = m F for F in Ω and gr(f ) = z Ω for all the new cells. Note that the boundaries of the new cells are cycles which generate the homology of Ω and thus that X supports F Ω. 6 Minimal non CW monomial resolutions. We will begin with a special case of our main theorem. The proof will serve as motivation for the constructions in the remainder of the section. Throughout the rest of the paper the term homology sphere will denote any d-dimensional CW -complex with the homology of S d. The results in this section involve 2-dimensional homology spheres with contractible covering spaces. Their existence is not at all obvious and we will construct one at the end of the paper. Lemma 2. Let Ω be a 2-dimensional homology sphere with a contractible cover Ω. Then F Ω cannot be supported in any CW complex which contains Ω. Proof. Suppose the contrary and let Z be a CW complex which supports F Ω. By Lemma the resolution is (ignoring the grading) of the form F Ω : R d 3 R f 2 d 2 R f d R f (m v,... ) R Since Z supports F Ω and contains Ω we see that: Z is obtained from Ω by attaching a single 3-cell e 3 α (via some map φ α : S 2 α Ω). d 3 (ê 3 α) = xdeg(ê3 α ) deg(ê2 ) Z (e 3 α, e 2 )ê2 Now, S 2 α is simply connected so φ α admits a lift ˆφ α to any covering space (in particular to Ω) and the following diagram is commutative (for all ): ˆφ α Ω π Sα 2 Ω φ q α S 2

12 pplying H 2 (, k) we see that (q φ α ) = (this homomorphism factors through H 2 ( Ω, k) which is since Ω is contractible). Hence Z (e 3 α, e 2 ) = for all and d 3 =, which contradicts the fact that F Ω is exact. Note that the above argument can be carried out whenever the 2-skeleton of the supporting complex Z (in this case Ω) admits a covering space with trivial H 2. We will show that such covering spaces exist for every possible supporting complex Z, that is, we will remove the condition that Z contains Ω. For the rest of this section we will make the following assumptions: Ω is a compact two dimensional simplicial homology sphere with a contractible (universal) cover π : Ω Ω. Ω is endowed with the simplicial complex structure lifted from Ω via π : Ω Ω. So in particular the restriction π : Ω() Ω () is a (normal) covering space. For every cell e n α in Ω, its closure e n α is evenly covered (that is π (e n α) is a disjoint union of components mapped homeomorphically via π to e n α). In particular the boundaries of the simplices of Ω are evenly covered. Note that the two last assumptions are made without loss of generality since they follow from the unique lifting property of covering spaces (see proposition.34 in [5]). Lemma 3. Suppose Z supports F Ω and let X = Z (2). Then the following statements hold:. X and Ω have isomorphic chain complexes over k. 2. X and Ω have isomorphic face posets. In particular their -skeletons are identical. Proof. () The standard basis in F Ω and the choice of basis for which X supports F Ω are related by a homogeneous change of coordinates, that is 2

13 (according to Lemma ) a commutative diagram of the form: R ψ R f 2 Ω 2 R f Ω R f (m v,... ) R R ψ R f 2 D 2 X 2 R f D X R f D (m v,... ) R Tensoring with k[x]/(x, x 2,... ) (that is, setting all variables in the matrices to and replacing R by k) we see that the reduced homology chain complexes of X and Ω over k (the first 4 modules in each row after tensoring) are related by a chain map of invertible diagonal matrices and hence isomorphic. (2) Consider the mapping ρ : F aces(x) F aces(ω) given by ρ(f ) = The unique face F of Ω with m F = gr(f ) (recall that m F is the product of all faces not containing F ). This map is a bijection (since Z supports F Ω ) and is order preserving (since Z is a graded CW complex). To see that ρ is order preserving assume F is a codimension face of G in Ω, then Ω (F, G ) iff Ω (F, G ) iff X (F, G) iff X (F, G) so F is a face of G. The face poset of a CW complex does not determine its homotopy type (for example a disc and RP 2 constructed using one cell in each dimension have the same face poset). In fact, the face poset does not even determine the image of the attaching map of a cell (the image must contain all the faces of this cell but it may be strictly larger). We can get rid of this nuisance via the following Lemma 4. ssume that X is a 2-dimensional CW complex such that the closure of every 2-cell contains at least one vertex. Then X is homotopy equivalent to a CW complex X with the same face poset and the same cellular homology complex with the additional property that the image of the attaching map of every cell in X equals the union of its faces. Proof. Let e 2 α be any 2-cell attached via φ α : S X (), let α be the union of faces of e 2 α and note that im(φ α ) deformation retracts onto α (since every edge h with h h im(φ α ) must have at least one endpoint in α by connectedness of im(φ α )). Composing φ α with this deformation retraction we obtain a homotopic map ψ α with image α. Let X be the CW complex obtained from X () by attaching cells via the 3

14 maps ψ α. Theorem.8 in [5] shows that X and X are homotopy equivalent. Lemma 5. Suppose Z supports F Ω and let X = Z (2). covering space π : X X with H 2 ( X, k) =. Then there is a Proof. By Lemmas 3 and 4 the -skeletons of X and Ω are identical. Thus X and Ω are obtained from Ω () by attaching 2-cells via maps {φ α } and {ψ α } respectively. By Lemma 4 we can assume that im(φ α ) = im(ψ α ); we will denote this triangle in Ω () by T α. Recall that π : Ω () Ω () is a covering space and that π (T α) is a disjoint union of triangles g G T αg, each mapped homeomorphically via the restriction π : T αg T α (which we will denote by by p αg ). Moreover, recall that Ω has the simplicial complex structure lifted from Ω via π. Thus, Ω is obtained from Ω () by attaching cells u 2 αg via the maps ˆψ αg = p αg ψ α for all α and g. By analogy we define X as the space obtained from Ω () by attaching one 2-cell e 2 αg along each lift ˆφ αg = p αg φ α. Define q : X X by q(x) = π (x) for x in Ω () and by q(x) = x in e 2 α if x e 2 αg for some g G. Note that each deck transformation of Ω () extends to a homeomorphism X X which commutes with q. By construction X is the quotient of X by the group of these extensions (recall that π : Ω () Ω () is normal) so q : X X is a covering space. Now we will show that H 2 ( X, k) is trivial. For that, let ˆv be any edge of Ω () and let v = π (ˆv). By construction ˆv is a face of e 2 αg in X iff it is a face of u 2 αg in Ω. Moreover, since q maps cells and their closures homeomorphically, the degrees of cells in X agree with those of their projections in X and f X (e 2 αg, ˆv) = X (e 2 α, v) = D (v) Ω (u 2 α, v)d 2 (e 2 α) = D (v) eω (u 2 αg, ˆv)D 2 (e 2 α) (the second equality follows from Lemma 3). Thus, defining ˆD 2 (u 2 αg) = D 2 (u 2 α)e 2 αg and ˆD (ˆv) = D (v)ˆv we obtain a commutative diagram: kû 2 Ω e 2 αg ˆv ˆD 2 kê 2 αg f X 2 ˆD ˆv 4

15 Since H 2 ( Ω, k) = and the vertical maps are isomorphisms, X 2 is injective and H 2 ( X, k) = as we wanted to prove. With these Lemmas at hand we can now prove our main theorem: Theorem 3. Let Ω be a 2-dimensional homology sphere with a contractible cover. Then F Ω is not a CW resolution over any field k. Proof. Suppose the contrary and let Z be a CW complex which supports F Ω. F Ω : R d 3 R f 2 d 2 R f d R f (m v,... ) R Let X = Z (2) and note that Z is obtained from X by attaching a single 3-cell e 3 α via a map φ α : S 2 α X and that d 3 (ê 3 α) = xdeg(ê3 α) deg(ê 2 ) Z (e 3 α, e 2 )ê2. Now, let X be the CW complex constructed from X using Lemma 4 and let g : X X be a homotopy equivalence. By Lemma 5, X has a covering space with H 2 ( X, k) = so, reasoning as in Lemma 2, we see that (g φ α ) : H 2 (S 2 α, k) H 2 (X, k) is the homomorphism and that φ α = (since g is an isomorphism). In particular, this implies that Z (e 3 α, e 2 ) = for all so d 3 = which contradicts the exactness of F Ω. The above theorem gives us (via nearly Scarf ideals) a non CW minimal resolution for each Ω. Hence we would like to produce explicit examples of such spaces (and of the F Ω they determine). However one might ask whether spaces Ω with the above conditions exist at all (for example, it is easy to see that no manifold satisfies them); in the remainder of this section we will construct one such example. The main idea is the observation that the universal cover of Ω is contractible iff Ω is the K(G, ) space of some group G. s a result our question becomes one about groups and we can use the tools of geometric group theory to address it. Definition 2. Given a presentation of a group by generators and relations G = g α r it s presentation complex is obtained from α S by attaching 2-cells e 2 along the loops specified by the words r. Our building block for Ω is the presentation complex of Higman s Group (defined by Higman as an example of a finitely generated infinite simple group). This space has trivial homology groups and trivial higher homotopy groups, intuitively we can think about it as a disc. By gluing two discs together we will construct the desired homology sphere. The following Lemma is proven in [4] (pag ): 5

16 Lemma 6. For the following presentation of Higman s Group: H = a, b, c, d : a[b, c], b[c, d], c[d, a], d[a, b] the presentation complex D is a K(H, ). Using H we will now construct Ω Figure 2: The simplicial complex Ω Lemma 7. Consider the group T = a,..., d, a 2,..., d 2 : a i [b i, c i ],..., d i [a i, b i ], a a 2, i =, 2 and let Ω be its presentation complex. Then. T is the amalgamated product of two copies of Higman s group along a free subgroup. 2. K(T, ) = Ω so Ω has a contractible (universal) cover. 6

17 3. H (Ω, Z) = H (Ω, Z) = and H 2 (Ω, Z) = Z. Proof. () Since K(H, ) is finite dimensional, no element of H has finite order. Let H and H 2 be two copies of Higman s group with generators a,..., d and a 2,..., d 2 respectively, then T = H Z H 2 where Z corresponds to the subgroup generated by a i in H i. (2) Follows from () and theorem W in [4]. (3) Since Ω is connected (it has just one vertex) H (Ω, Z) =. π (Ω) = T so H (Ω, Z) = (the abelianization of T is trivial). Finally a cellular homology computation shows that H 2 (Ω, Z) = Z so Ω is a homology sphere. triangulation of Ω (with 23 vertices,4 edges and 2 faces) is schematically drawn in figure 2.b, it consists of eight pentagons (triangulated as in figure 2.a) whose edges are identified according to the labels (all vertices of all pentagons in figure 2.b are identified to a single vertex V ). The corresponding nearly Scarf Ideal J Ω lies in the ring k[x,..., x 284 ] and is minimally generated by 23 monomials. By theorem 3 its minimal free resolution is not CW. References [] D. Bayer, I. Peeva, and B. Sturmfels, Monomial Resolutions, Math. Research Letters 5 (995), [2] D. Bayer and B. Sturmfels, Cellular resolutions of monomial modules, J. Reine ngew. Math. 53 (998), [3] E. Batzies and V Welker, Discrete Morse theory for cellular resolutions, J. Reine ngew. Math. 543 (22), [4] E. Dyer and.t. Vasquez, Some small aspherical spaces, J. ustral. Math. Soc. (973), [5]. Hatcher, lgebraic Topology, Cambridge Univ. Press., 22. [6] M. Jollenbeck and V. Welker, Minimal resolutions via algebraic discrete morse theory., Submitted. [7] W Massey, Singular Homology theory., Springer-Verlag, 98. [8] V. Reiner and V Welker, Linear syzygies of Stanley-Reisner ideals, Mathematica Scandinavica 89 (2), [9] I. Peeva and M. Velasco, Frames and degenerations of monomial ideals, Submitted. 7