TRIANGULATED CATEGORIES WITHOUT MODELS FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND Abstract.We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subc* *ategory of the homotopy category of a stable model category. Even more drastical* *ly, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories. Introduction. Triangulated categories are fundamental tools in both algebra and topology. In algebra they often arise as the stable category of a Frobenius ca* *t- egory ([Hel68, 4.4], [GM03 , IV.3 Exercise 8]). In topology they usually appear as a full triangulated subcategory of the homotopy category of a Quillen stable model category [Hov99 , 7.1]. The triangulated categories which belong, up to exact equivalence, to one of these two families will be termed algebraic and to* *po- logical, respectively. We borrow this terminology from [Kel06, 3.6] and [Sch06* *]. Algebraic triangulated categories are generally also topological, but there are* * many well-known examples of topological triangulated categories which are not algebr* *aic. In the present paper we exhibit examples of triangulated categories which are neither algebraic nor topological. As far as we know, these are the first examp* *les of this kind. Even worse (or better, depending on the perspective), our examples do not even admit non-trivial exact functors to or from algebraic or topological triangulated categories. In that sense, the new examples are completely orthogo* *nal to previously known triangulated categories. Let (R, (2)) be a commutative local ring of characteristic 4, such as R = Z=4, or more generally R = W2(k) the 2-typical Witt vectors of length 2 over a perfe* *ct field k of characteristic 2. We denote by F(R) the category of finitely genera* *ted free R-modules. Theorem 1. The category F(R) has a unique structure of a triangulated category with identity translation functor and such that the diagram R -2!R -2!R -2!R is an exact triangle. Given an object X in an algebraic triangulated category T and an exact triang* *le A 2.1A-!A -! C -! A, the equation 2 . 1C = 0 holds, compare [Kel06, 3.6] and [Sch06]. Since the ring* * R has characteristic 4, the triangulation of the category F(R) is not algebraic. * * We ____________ 1991 Mathematics Subject Classification. 18E30, 55P42. Key words and phrases. Triangulated category, stable model category. The first author was partially supported by the Spanish Ministry of Educatio* *n and Science under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellow* *ship EX2004-0616, and a Juan de la Cierva research contract. 1 2 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND cannot rule out the possibility of a topological model for F(R) as easily: the * *classical example of A = S the sphere spectrum in the stable homotopy category shows that the morphism 2 . 1C can be nonzero in this more general context. Nevertheless, F(R) is not topological either, which follows from Theorem 2. H* *ere we call an exact functor between triangulated categories trivial if it takes ev* *ery object to a zero object. Theorem 2. Every exact functor from F(R) to a topological triangulated category is trivial. Every exact functor from a topological triangulated category to F(R* *) is trivial. Acknowledgements. We are grateful to Bernhard Keller for helpful conversations on the results of this paper, and to Amnon Neeman, who suggested the possibility of constructing a triangulated structure on F(Z=4) by using Heller's theory [He* *l68]. In the original version of this note the first author alone constructed the t* *ri- angulation of the category F(Z=4) and proved that it does not admit any model. The second author joined the project later by providing a simpler and more gene* *ral proof that the triangulation is not topological. The third author's contributi* *on was an old preprint on the example considered in Remark 8, which provided some guidance for the other results. The triangulated categories. Let T be an additive category and let : T ~!T be a self-equivalence that we call translation functor. A candidate triangle (f* *, i, q) in (T, ) is a diagram (3) A -f!B -i! C -q! A, where if, qi, and ( f)q are zero morphisms. A morphism of candidate triangles (ff, fi, fl): (f, i, q) ! (f0, i0, q0) is a commutative diagram f i q A _____//_B____//_C____//_ A ff|| fi|| fl|| ||ff fflffl| fflffl| fflffl| fflffl| A0__f0_//B0_i0_//C0_q0_// A0 The category of candidate triangles is additive. The mapping cone of the morphi* *sm (ff, fi, fl) is the candidate triangle ` -i0' ` -q0' `- f 0' fif0 fli0 ffq0 B A0_______//C B0______// A C0______// B A0. A homotopy ( , , ) from (ff, fi, fl) to (ff0, fi0, fl0) is given by morphisms f i q A _____//_B____//_C____//_ A ___ ___|| zzz|| ff||ff0||fi||fi0||___fl|fl0|___|ff|ff0zzz fflffl|fflffl|fflffl|fflffl|""__fflffl|fflffl|""__fflffl|f* *flffl|__zz A0__f0_//B0_i0_//C0_q0_// A0 such that fi0- fi= i + f0 , fl0- fl= q + i0 , (ff0- ff)= ( f) + q0 . TRIANGULATED CATEGORIES WITHOUT MODELS 3 We say in this case that the morphisms are homotopic. The mapping cones of two homotopic morphisms are isomorphic. A contractible triangle is a candidate triangle such that the identity is homotopic to the zero morphism. A homotopy ( , , ) from 0 to 1 is called a contracting homotopy. Any morphism from or to a contractible triangle is always homotopic to zero. A triangulated category is a pair (T, ) as above together with a collection * *of candidate triangles, called distinguished or exact triangles, satisfying the fo* *llow- ing properties. The family of exact triangles is closed under isomorphisms. T* *he candidate triangle (4) A -1!A -! 0 -! A, is exact. Any morphism f :A ! B in T can be extended to an exact triangle like (3). A candidate triangle (3) is exact if and only if its translate B --i!C -q-! A --f! B, is exact. Any commutative diagram f i q A _____//_B____//_C____//_ A ff|| fi|| ||ff fflffl| fflffl| fflffl| A0__f0_//B0_i0_//C0_q0_// A0 whose rows are exact triangles can be extended to a morphism whose mapping cone is also exact. This non-standard set of axioms for triangulated categories* * is equivalent to the classical one, see [Nee01], and works better for the purposes* * of this paper. Now we are ready to prove Theorem 1. Proof of Theorem 1.Given an object X in F(R) we consider the candidate triangle X2 defined as (5) X -2! X -2! X -2! X. We are going to prove that the category F(R) has a triangulated category stru* *c- ture with identity translation functor where the exact triangles are the candid* *ate triangles isomorphic to the direct sum of a contractible triangle and a candida* *te triangle of the form (5). The family of exact triangles is closed under isomorphisms by definition. The candidate triangle (4)is contractible, and hence exact. The ring R is a quotien* *t of a discrete valuation ring with maximal ideal generated by 2, see [Coh46 , Corolla* *ry 3]; therefore any morphism f :A ! B in F(R) can be decomposed up to isomorphism as 0 1 1 0 0 f = @ 0 2 0 A : A = W X Y -! W X Z = B. 0 0 0 Then f is extended by the direct sum of (5)and the contractible triangle `10' `00' `00' 00 01 10 W Y ______//_W Z______//Y Z______//_W Y. 4 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND The translate of a contractible triangle is also contractible, and the triangle* * (5)is invariant under translation. This proves that the translate of an exact triangl* *e is exact. Translating a candidate triangle six times yields the original one, ther* *efore if a candidate triangle has an exact translate then the original candidate tria* *ngle is also exact. We say that a candidate triangle A f!B !iC !qA is a quasi-exact triangle if A -f!B -i! C -q! A -f!B, is an exact sequence of R-modules. The exact triangles are all quasi-exact. Now we are going to show that any diagram of candidate triangles (6) A __f__//_B_i__//_C_q__//_A ff|| fi|| |ff| fflffl| fflffl| fflffl| A0__f0_//B0_i0_//C0_q0_//A0 with exact rows can be completed to a morphism with exact mapping cone. Suppose that the upper row in (6)is contractible and the lower row is quasi- exact. Since f0ff = fif then f0ffq = 0; since C is projective, there exists fl0* *:C ! C0 such that q0fl0 = ffq. Let ( , , ) be a contracting homotopy for the upper ro* *w. Then fl = fl0+ (i0fi - fl0i) completes (6) to a morphism of candidate triangle* *s. If the upper row in (6) is quasi-exact and the lower row is contractible then (6) can also be completed to a morphism. This can be shown directly, but it also follows from the previous case since we have a duality functor Hom R(-, R): F(R) -~!F(R)op, which preserves contractible triangles and quasi-exact triangles. Here we use t* *hat R is injective as an R-module, see [Lam99 , Example 3.12]. If the upper and the lower rows in (6) are X2 and Y2, respectively, then fl =* * fi+2ffi extends (6) to a morphism of candidate triangles for any ffi :X ! Y . This proves that any diagram like (6) with exact rows can be completed to a morphism ' = (ff, fi, fl). Now we have to check that the completion can be done in such a way that the mapping cone is exact. Suppose that the upper and the lower rows are X2 T and Y2 T 0, respectively, with T and T 0contractible. T* *he morphism ' is given by a matrix of candidate triangle morphisms ` ' ' = '11' '12 :X2 T -! Y2 T 0, 21'22 where 'ij= (ffij, fiij, flij). Here '12, '21and '22are homotopic to 0 since eit* *her the source or the target is contractible, therefore the mapping cone of ' is isomor* *phic to the mapping cone of ` ' _ = '110 00 :X2 T -! Y2 T 0, which is the direct sum of the mapping cone of '11 and two contractible triangl* *es, T 0and the translate of T . TRIANGULATED CATEGORIES WITHOUT MODELS 5 We can suppose that 0 1 1 0 0 ff11= @ 0 2 0 A : X = L M N -! L M P = Y. 0 0 0 Moreover, as we have seen above we can take fl11= fi11+ 2ffi for 0 1 0 0 0 ffi= @ 0 1 0 A :X = L M N -! L M P = Y. 0 0 0 We have 2fi11 = 2ff11, therefore fi11 = ff11+ 2 for some : X ! Y . This shows that (0, , 0) is a homotopy from '11 to i = (ff11, ff11, ff11+ 2ffi), so the m* *apping cone of '11 is isomorphic to the mapping cone of i. The mapping cone of i is the direct sum of four candidate triangles, namely N* *2, P2, the mapping cone of the identity 1: L2 ! L2, which is contractible, and the mapping cone of (2, 2, 0): M2 ! M2, which is the upper row of `20' `2 0' `20' 22 0 2 22 M M ______//_M M_______//M M_______//M M ` ' ` ' ` ' ` ' 1011~=|| 1001|~=| 1001~=|| 1101|~=| fflffl| fflffl| fflffl| fflffl| M M _`20'_//_M M_`2_0'_//M M__`20'_//M M 02 0 2 02 Here the lower row is (M M)2, which is exact. Therefore the mapping cone of i is exact, and also the mapping cone of '11, _ and '. It remains to show the uniqueness claim in Theorem 1. In any triangulation, a* *ll contractible candidate triangles are exact. The triangle X2 is a finite direct * *sum of copies of R2. Hence every triangulation of (F(R), Id) which contains R2 contains all the exact triangles which we considered above. Two triangulations with the same translation functor necessarily agree if one class of triangles is contain* *ed in the other, so there is only one triangulation in which R2 is exact. This comple* *tes the proof. Remark 7. The exact triangles in F(R) can be characterized more intrinsically as follows. Let T be a quasi-exact triangle, which we can regard as a Z=3-graded c* *hain complex of free R-modules with H*(T ) = 0. As T is free we have a short exact sequence 2 2T ,! T i 2T, and the resulting long exact sequence in homology reduces to an isomorphism oe :H*(2T ) ! H*-1(2T ). As the grading is 3-periodic we can regard oe3 as an automorphism of H*(2T ). We claim that T is exact if and only if oe3 = 1. One direction is straightforward: if T is contractible then H*(2T ) = 0, and if T =* * X2 then Hi(2T ) = 2X for all i and oe is the identity. The converse is more fiddly* * and we will not go through the details. It would be nice to give a proof of Theorem* * 1 based directly on this definition of exactness, but we do not know how to do so. Remark 8. Let k be a field of characteristic 2. The same arguments as in the pr* *oof of Theorem 1 show that the category F(k["]="2) of finitely generated free modul* *es 6 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND over the algebra k["]="2 of dual numbers admits a triangulation with the identi* *ty translation functor and such that the diagram (9) k["]="2 -"!k["]="2 -"!k["]="2 -"!k["]="2 is an exact triangle. However, this triangulated category is both algebraic and topological, and hence, from our current perspective, less interesting. We exhibit a differential k-algebra A such that F(k["]="2) is exact equivalent to the category of compact objects in the derived category D(A) of differential A-modules. We let A = k=I be the free k-algebra generated by two non- commuting variables a and u modulo the two-sided ideal I generated by a2 and au + ua + 1. A differential d : A ! A is determined by d(u) = 0, d(a) = u2 and the Leibniz rule. The hypothesis that k has characteristic 2 guarantees that t* *he differential is well-defined on A. The homology algebra H(A) = ker(d)=im(d) is isomorphic to the dual numbers k["]="2, where " = [u] is the homology class of * *the cycle u. We claim that the homology functor H : Dc(A) ! F(k["]="2) is an equivalence of categories, where the left hand side is the derived catego* *ry of those (ungraded) differential A-modules whose homology is finitely generated ov* *er k["]="2. Let M be any differential A-module and let [x] 2 H(M) be a homology class with " . [x] = 0. We choose a representing cycle x and an element y with d(y) =* * ux; then the element z = uy + ax is a cycle with x = uz + d(ay), so [x] = " . [z] in homology. So every homology class which is annihilated by " is also divisible b* *y ", which proves that H(M) is a free k["]="2-module. The universal case of this is the free module M = A{x, y} with d(x) = 0 and d(y) = ux, which is just the mapping cone of the chain map A -u! A. Here the cycle z = uy + ax 2 M gives a quasiisomorphism A ! M. Using this, we obtain an exact triangle A -u!A -u!A -u! A = A in Dc(A) which maps to the exact triangle (9). The rest of the proof that H is * *an exact equivalence from Dc(A) to F(k["]="2) is relatively straightforward, and we omit it. It is more traditional to work with differential graded algebras and differen* *tial graded modules, and to get into that context we could introduce the dg algebra A* = A k k[v, v-1] where v is a central unit of dimension 1. The differential "don A*, which now decreases degree, is determined by d"(u) = d"(v) = 0 and "d(a) = u2 v-1. The category of (ungraded) differential A-modules is equivale* *nt to the category of dg A*-modules, via the functor which takes M to M kk[v, v-1] with differential "d(x vn) = dx vn-1. We still owe the proof that the triangulated category F(R) does not admit non- trivial exact functors to or from a topological triangulated category. For this* * pur- pose we introduce two intrinsic properties that an object A of a triangulated c* *ate- gory may have. TRIANGULATED CATEGORIES WITHOUT MODELS 7 A Hopf map for an object A is a morphism j : A ! A which satisfies 2j = 0 and such that for some (hence any) exact triangle (10) A -2!A -i!C -q! A we have ijq = 2 . 1C . An object which admits a Hopf map will be termed hopfian. We note that the class of hopfian objects is closed under isomorphism, suspensi* *on and desuspension. If F is an exact functor with natural isomorphism o : F ~=F and j : A ! A a Hopf map for A, then the composite F (j)o : F (A) -! F (A) is a Hopf map for F (A). We call an object E exotic if there exists an exact triangle (11) E -2! E -2! E -h! E for some morphism h : E ! E. We note that the class of exotic objects is closed under isomorphism, suspension and desuspension. Every exact functor takes exotic objects to exotic objects. Every object of the triangulated category F(R* *) of Theorem 1 is exotic. We remark without proof that the morphism h which makes (11)exact is unique and natural for morphisms between exotic objects. We show below that h is of the form h = 2_ for an isomorphism _ : E ! E. Remark 12. The integer 2 plays a special role in the definition of exotic objec* *ts, which ultimately comes from the sign which arises in the rotation of a triangle* *. In more detail, suppose that there is an exact triangle (13) E -n! E -n! E -h! E for some integer n. We claim that if E is nonzero, then n 2 mod 4 and 4.1E =* * 0, so that the triangle (13)equals the `exotic' triangle (11)with n = 2. Indeed, we can find a morphism _ : E ! E which makes the diagram E _n__//_E_n__//_E_h__//___ E || || _____ || || || _____ || || || _fflffl__|| E _n__//_E_h_//_ E_-n_//_ E commute, and _ is an isomorphism. We have n_ = h = -n_ which gives 2n_ = 0. Since _ is an isomorphism, this forces 2n . 1E = 0. Exactness of (13)lets us ch* *oose a morphism f : E ! E with 2 . 1E = n . f. But then 4 . 1E = n2f2 = 0. So if n is divisible by 4, then E = 0. If n is odd, then E is anhihilated by 4 and the * *odd number n2, so also E = 0. Hopf maps are incompatible with the property of being exotic in the sense that these two classes of objects are orthogonal. Proposition 14. Let T be a triangulated category, A a hopfian object and E an exotic object. Then the morphism groups T(A, E) and T(E, A) are trivial. In par- ticular, every exotic and hopfian object is a zero object. Proof.Let j : A ! A be a Hopf map. Given any morphism f : E ! A there exists g :E ! C such that (f, f, g) is a morphism from (11)to (10), and hence if = 2g = ijqg = ij( f)h = ij( f)2_ = 0. Here we use the notation of Remark 12 for n = 2 and the fact that 2j = 0. Moreover, (10)is exact, so f = 2f0 for s* *ome 8 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND f0:E ! A. This equation follows for any morphism f :E ! A, hence f is divisible by any power of 2, but 4 . 1E = 0, so f = 0. The proof of T(A, E) = 0 is similar. Alternatively, we can reduce this statem* *ent to the previous one by observing that the properties of being exotic and hopfian are self-dual. In other words, an object E is exotic in a triangulated category* * T if and only if E is exotic as an object of the opposite category Top with the oppo* *site triangulation, and similarly for Hopf maps. Proposition 15. Every object of a topological triangulated category is hopfian. Proof.We can assume that the topological triangulated category is Ho M for a stable model category M. We use that for every object A of Ho M there exists an exact functor F : Ho Sp ! Ho M from the stable homotopy category which takes the sphere spectrum to an object isomorphic to A. Here Sp is the category of `sequential spectra' of simplicial sets with the stable model structure of Bous* *field and Friedlander [BF78 , Sec. 2]. To construct F we let X be a cofibrant-fibrant object of the model category M which is isomorphic to A in the homotopy category Ho M. The universal property of the model category of spectra [SS02, Thm. 5.1 (1)] provides a Quillen adjoint functor pair ___X^____// Sp oo_______ M Hom(X,-) whose left adjoint X^ takes the sphere spectrum S to X, up to isomorphism. The left derived functor of the left Quillen functor X ^ - : Sp ! M is exact and can serve as the required functor F . Since exact functors preserve Hopf maps it thus suffices to treat the `univer* *sal example', i.e., to exhibit a Hopf map for the sphere spectrum as an object of t* *he stable homotopy category. The stable homotopy class j : S ! S of the Hopf map from the 3-sphere to the 2-sphere precisely has this property, hence the name. * *In more detail, we have an exact triangle S 2.1S-!S -i! S=2 -q! S in the stable homotopy category, where S=2 is the mod-2 Moore spectrum; then the morphism 2 . 1S=2factors as ijq, and moreover 2j = 0. In topological triangulated categories, something a little stronger than Prop* *osi- tion 15 is true in that Hopf maps can be chosen naturally for all objects. Howe* *ver, we don't need this and so we omit the details. Now we can give the Proof of Theorem 2.Every object of the triangulated category F(R) is exotic and every object of a topological triangulated category is hopfian. So an exact fun* *ctor from one type of triangulated category to the other hits objects which are both exotic and hopfian. But such objects are trivial by Proposition 14. Remark 16. The only special thing we use in the proof of Theorem 2 about topolo* *g- ical triangulated categories is that therein every object has a Hopf map. Hopf * *maps can also be obtained from other kinds of structure that were proposed by differ* *ent authors in order to `enrich' or `enhance' the notion of a triangulated category* *. So our argument also proves that the triangulated category F(R) of Theorem 1 does not admit such kinds of enrichments, and every exact functors to or from such TRIANGULATED CATEGORIES WITHOUT MODELS 9 enriched triangulated categories is trivial. For example, if T is an algebraic * *trian- gulated category, then for some (hence any) exact triangle (10)we have 2 . 1C =* * 0; so the zero map is a Hopf map. Another example of such extra structure is the notion of a triangulated deriv* *ator, due to Grothendieck [Gro90], and the closely related notions of a stable homoto* *py theory in the sense of Heller [Hel88, Hel97] or a system of triangulated diagram categories in the sense of Franke [Fra96]. In each of these settings, the stab* *le homotopy category is the underlying category of the free example on one gener- ator (the sphere spectrum). We do not know a precise reference of this fact for triangulated derivators, but we refer to [Cis02, Cor. 4.19] for the `unstable' * *(i.e., non-triangulated) analog. In Franke's setting the universal property is formula* *ted as Theorem 4 of [Fra96]. These respective universal properties in the enhanced context provide, for every object A, an exact functor (Ho Sp)cp ! T which takes the sphere spectrum S to A, up to isomorphism. This functors sends the classical Hopf map for the sphere spectrum to a Hopf map for A. Another kind of structure which underlies many triangulated categories is tha* *t of a stable infinity category as investigated by Lurie in [Lur06]. The appropriate* * uni- versal property of the infinity category of spectra is established in [Lur06, C* *or. 17.6], so again every object of the homotopy category of any stable, presentable infin* *ity category has a Hopf map. References [BF78]A. K. Bousfield and E. M. Friedlander, Homotopy theory of -spaces, spect* *ra, and bisim- plicial sets, Geometric applications of homotopy theory (Proc. Conf., Eva* *nston, Ill., 1977), II. Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. * *80-130. [Cis02]D.-C. Cisinski, Propri'et'es universelles et extensions de Kan d'eriv'ee* *s, Preprint (2002). [Coh46]I. S. Cohen, On the structure and ideal theory of complete local rings, * *Trans. Amer. Math. Soc. 59 (1946), 54-106. [Fra96]J. Franke, Uniqueness theorems for certain triangulated categories posse* *ssing an Adams spectral sequence, K-theory Preprint Archives #139 (1996). http://www.math.uiuc.edu/K-theory/ [GM03]S. I. Gelfand and Y. I. Manin, Methods of homological algebra, second ed.* *, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. [Gro90]A. Grothendieck, D'erivateurs, manuscript, around 1990, partially av* *ailable from http://www.math.jussieu.fr/~maltsin/groth/Derivateurs.html [Hel68]A. Heller, Stable homotopy categories, Bull. Amer. Math. Soc. 74 (1968),* * 28-63. [Hel88]A. Heller, Homotopy theories, Mem. Amer. Math. Soc. 71 (1988), no. 383, * *vi+78 pp. [Hel97]A. Heller, Stable homotopy theories and stabilization, J. Pure Appl. Alg* *ebra 115 (1997), 113-130. [Hov99]M. Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63* *, American Mathematical Society, Providence, RI, 1999. [Kel06]B. Keller, On differential graded categories, Proceedings of the Interna* *tional Congress of Mathematicians, Madrid, Spain, 2006, vol. II, European Mathematical So* *ciety, 2006, pp. 151-190. [Lam99]T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics,* * 189. Springer-Verlag, New-York, 1999. [Lur06]J. Lurie, Derived algebraic geometry I: Stable infinity categories. math* *.CT/0608228 [Nee01]A. Neeman, Triangulated Categories, Annals of Mathematics Studies, vol. * *148, Princeton University Press, Princeton, NJ, 2001. [Sch06]S. Schwede, Algebraic versus topological triangulated categories, Extend* *ed notes of a talk given at the ICM 2006 Satellite Workshop on Triangulated Categories,* * Leeds, UK, http://www.math.uni-bonn.de/people/schwede/leeds.pdf, 2006. 10 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND [SS02]S. Schwede and B. Shipley, A uniqueness theorem for stable homotopy theor* *y, Math. Z. 239 (2002), 803-828. Universitat de Barcelona, Departament d'`Algebra i Geometria, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain E-mail address: fmuro@ub.edu Mathematisches Institut, Universit"at Bonn, Beringstr. 1, 53115 Bonn, Germany E-mail address: schwede@math.uni-bonn.de Department of Pure Mathematics, University of Sheffield, Hicks Building, Houn* *sfield Road, Sheffield S3 7RH, UK E-mail address: n.p.strickland@sheffield.ac.uk