Fibrewise nullification and the cube theorem David Chataur and J'er^ome Scherer * February 26, 2003 Abstract In this paper we explain when it is possible to construct fibrewise loc* *alizations in model categories. For pointed spaces, the general idea is to decompose * *the total space of a fibration as a diagram over the category of simplices of the ba* *se and replace it by the localized diagram. This of course is not possible in an * *arbitrary category. We have thus to adapt another construction which heavily depend* *s on Mather's cube theorem. Working with model categories in which the cube the* *orem holds, we characterize completely those who admit a fibrewise nullificatio* *n. Introduction Mather's cube theorem states that the top face of a cube of spaces whose bottom* * face is a homotopy push-out and all vertical faces are homotopy pull-backs is again * *a homo- topy push-out ([Mat76 , Theorem 25]). This theorem is one of the very few occu* *rences of a situation where homotopy limits and colimits commute. It is actually rela* *ted to a theorem of Puppe about commuting fibers and push-outs ([Pup74 ]), and also to Q* *uillen's Theorem B in [Qui73 ]. Doeraene's work on J-categories has incorporated the cub* *e theo- rem as an axiom in pointed model categories and allowed him to study the L.S.-c* *ategory in an abstract setting ([Doe93 ]). Roughly speaking a J-category is a model cat* *egory in which the cube theorem holds. Such a model category is very suitable for studyi* *ng the relationship between a localization functor (constructed by means of certain ho* *motopy colimits) and fibrations. ________________________________ *The first author was partially supported by DGESIC grant PB97-0202. 1 Recall that a localization functor in a model category M is any coaugmented * *idempo- tent functor L : M ! M. The coaugmentation is a natural transformation j : Id !* * L. We will only deal with nullification functors PA. In this context the image of * *PA is charac- terized by the property that map(A, PAX) ' *. We are looking for an existence t* *heorem of fibrewise nullification, i.e. a construction which associates to any fibrati* *on F ! E ! B another fibration together with a natural transformation F _____//_E___//_B |''| || || fflffl| fflffl|fflffl| PAF ____//_~E__//_B where E ! ~Eis a PA-equivalence. This is achieved by imposing the join axiom fo* *r the object A: We require the join X * A to be killed by PA, i.e. PA(X * A) ' *, f* *or any object X. For pointed spaces, the most elegant construction of fibrewise localization * *is due to E. Dror Farjoun (in [DF96 , Theorem F.3]). His idea is to decompose the total s* *pace of a fibration as a diagram over the category of simplices of the base and replace* * it by the corresponding localized diagram. In certain particular settings, some authors u* *sed other constructions (P. May [May80 ], W. Dwyer, H. Miller, and J. Neisendorfer in [DM* *N89 ] for completions, C. Casacuberta and A. Descheemaker in [CD02 ] in the category of g* *roups), but none of these can be adapted in model categories. We prove the following: Theorem 3.3 Let M be a model category which is pointed, left proper, cellular * *and in which the cube and the join axiom hold. Then the nullification functor PA a* *dmits a fibrewise version. This condition is actually necessary and we characterize completely the mode* *l cate- gories for which fibrewise nullifications exist. This is closely related to th* *e property of preserving products: A nullification functor PA preserves (finite) products if * *PA(X xY ) ' PAX x PAY . Theorem 3.5 Let M be a model category which is pointed, left proper, cellular a* *nd in which the cube axiom holds. Then the following conditions are equivalent: (i)The nullification functor PA admits a fibrewise version. (ii)The nullification functor PA preserves finite products. 2 (iii)The canonical projection X x A ! X is a PA-equivalence for any X 2 M. (iv)The join axiom for A is satisfied. We show in the last part of the paper that the category of algebras over an * *admissible operad satisfies the cube axiom. Therefore the plus-construction developed in [* *CRS ] has a fibrewise analogue. Let us only say that the plus-construction performed on a* * O-algebra B kills the maximal O-perfect ideal in ß0B and preserves Quillen homology. As a* * direct consequence we get the following result which is classical for spaces. Theorem 4.4 Let O - alg be the category of algebras over an admissible operad * *O. For any O-algebra B, denote by B ! B+ the plus construction. The homotopy fib* *er AB = F ib(B ! B+ ) is then acyclic with respect to Quillen homology. Acknowledgements. We would like to thank Gustavo Granja and Sophie Reinberg for helpful comments. 1 The cube axiom We work in a model category M which is pointed, i.e. the terminal object coinci* *des with the initial one and is denoted by *. In such a category the homotopy fiber F ib* *(p) of a map p : E ! B is defined as the homotopy pull-back of the diagram * ! B E. We also assume the category is left proper, meaning that the push-out of a weak equival* *ence along a cofibration is again a weak equivalence. Finally we require M to be cellular * *as defined in [Hir, Definition 14.1.1]. Basically the small object argument applies in a c* *ellular model category, as one has I-cells which replace the usual spheres. There exists a ca* *rdinal ~ such that any morphism from an I-cell to a telescope of length ~ ~ factorizes thro* *ugh an object of this telescope. Moreover every object has a cofibrant replacement by * *an I-cell complex by [Hir, Theorem 13.3.7]. Localization functors exist in this setting,* * see [Hir, Theorem 4.1.1], but in general we do not know if it is possible to localize fib* *rewise in any (pointed, left proper, cellular) model category. We will thus work in model ca* *tegories satisfying an extra-condition. Definition 1.1 A model category M satisfies the cube axiom if for every commuta* *tive cubical diagram in M in which the bottom face is a homotopy push-out square and* * all vertical faces are homotopy pull-back squares, then the top face is a homotopy * *push-out square as well. 3 M. Mather proved the cube Theorem for spaces in [Mat76 , Theorem 25] and J.-* *P. Do- eraene introduced it as an axiom for model categories. His paper [Doe93 ] conta* *ins a very useful appendix with several examples of model categories satisfying this rathe* *r strong axiom. Example 1.2 Any stable model category satisfies the cube axiom. Indeed homot* *opy push-outs coincide with homotopy pull-backs, so that this axiom is a tautology.* * On the other hand the category of groups does not satisfy the cube axiom. Let us give * *an easy counter-example by considering the push-out of (Z * ! Z), which is a free gro* *up on two generators a and b. The pull-back along the inclusion Z < ab >,! Z < a > *Z* * < b > is obviously not a push-out diagram. However fibrewise localizations exist in t* *he category of groups as shown by the recent work of Casacuberta and Descheemaker [CD02 ]. The following proposition claims that under very special circumstances the p* *ush-out of the fibers coincides with the fiber of the push-outs. In the category of sp* *aces this is originally due to V. Puppe, see [Pup74 ]. The close link between the cube Theor* *em and Puppe's theorem was already well-known to M. Mather and M. Walker, as can be se* *en in [MW80 ]. Proposition 1.3 Let M be a pointed model category in which the cube axiom hold* *s. Consider natural transformations between push-out diagrams: F = hocolim F1 oo___F0 ____//_F2 j|| j1|| j0|| j2|| fflffl| fflffl| fflffl| fflffl| E = hocolim E1 oo___E0 ____//_E2 p|| p1|| p0|| p2|| fflffl| fflffl| fflffl| fflffl| B = hocolim B _____B_______B Assume that Fi= F ib(pi) for any 0 i 2. Then F = F ib(p). Proof. Denote by k : G ! E the homotopy fiber of p. We show that G and F are weakly equivalent. Let us construct a cube by pulling-back Ei! E along k. The b* *ottom face consists thus in the middle row of the above diagram and the top face cons* *ists in the homotopy pull-backs of Ei ! E G, which are the same as the homotopy pull-backs of Ei ! B *, i.e. Fi. The cube axiom now states that the top face is a homo* *topy push-out and we are done. 4 This result will be the main tool in constructing fiberwise localization in * *M. In his paper [Doe93 ] on L.S.-category, J.-P. Doeraene used the cube axiom in a very s* *imilar fashion to study fiberwise joins. Indeed Ganea's characterization of the L.S.-* *category uses iterated fibers of push-outs over a fixed base space. The same ideas have * *also been used in [DT95 ]. Lemma 1.4 Let M be a model category in which the cube axiom holds. Let D be* * the homotopy push-out in M of the diagram A B ! C. Then, for any object X 2 M, X x D is the homotopy push-out of the diagram X x A X x B ! X x C. Proof. It suffices to consider the cube obtained by pulling back the mentionned* * push-out square along the canonical projection X x D ! D. 2 The join We check here that we can use all the classical facts about the join in any mod* *el category and introduce the join axiom. Most proofs here are not new, but probably folklo* *re. Recall p1 * * p2 that the join A*B of two objects A, B 2 M is the homotopy push-out of A - AxB* * -! B. First notice that the induced maps A ! A * B and B ! A * B are trivial. Inde* *ed the p1 map A ! A * B can be seen as the composite A -i1!A x B -! A ! A * B which by p2 definition coincides with the obviously trivial map A -i1!A x B -! B ! A * B. Lemma 2.1 For any objects A, B 2 M, we have A * B ' (A ^ B). Proof. We use a "classical" Fubini argument (homotopy colimit commute with itse* *lf, cf. for example [CS02 , Theorem 24.9]). Let P be the homotopy push-out of A A _ B* * ! A x B and consider first the commutative diagram A oo___A _ B _______//BOO || || | || || | || || | A oo___A _ B _____//A x B || | || || | || || fflffl| || A oo___A x B ______A x B Its homotopy colimit can be computed in two different ways. By taking first ve* *rtical homotopy push-outs and next the resulting horizontal homotopy push-out one gets* * A * B. 5 By taking first horizontal homotopy push-outs one gets the homotopy cofiber of * *P ! A. Consider finally the commutative diagram *________*__________*OOOOOO | | | | | | | | | A oo___A _ B _____//A x B || | || || | || || fflffl| || A oo___A x B ______A x B The same process as above shows that Cof(P ! A) is homotopy equivalent to (A ^* * B). Lemma 2.2 For any objects A, B 2 M, we have A ^ B ' (A ^ B). Proof. Apply again the Fubini commutation rule to the following diagram *________*________*OOOOOO | | | | | | | | | B oo__A__ B ____//_B || | || || | || || fflffl| || B oo__A_x B ____//_B where one uses Lemma 1.4 to identify the push-out of the bottom line. For a fibration F ! E!! B, the holonomy action is the map m : B xF ! F indu* *ced on the pull-backs by the natural transformation from B ! * F to P B!! B E. Corollary 2.3 For any fibration F ! E!! B, the homotopy push-out of B B x F -m! F is weakly equivalent to B * F . Proof. Copy the proof above to compare this homotopy push-out to ( B ^ F ). When working with a nullification functor PA for some object A 2 M, we say t* *hat X is A-acyclic or killed by A if PAX ' *. By universality this is equivalent to m* *ap(X, Z) ' * for any A-local object Z, or even better to the fact that any morphism X ! Z to* * an A-local object is homotopically trivial. Definition 2.4 A cellular model category M satisfies the join axiom for the nul* *lification functor PA if the join of A with any I-cell is A-acyclic. 6 Example 2.5 Any stable model category satisfies trivially the join axiom, as * *push-outs coincide with pull-backs. In such a category the join is always trivial. The * *category of groups satisfies the join axiom for a similar reason (but we saw in Example 1.2* * that the cube axiom does not hold). Proposition 2.6 Let M be a cellular model category in which the join axiom and* * the cube axiom hold. Then iA * Z is A-acyclic for any i 0 and any object Z. Proof. The join is a homotopy colimit and thus commutes with other homotopy col* *imits. Since any object in M has a cofibrant approximation which can be constructed as* * a telescope by attaching I-cells, the lemma will be proven if we show that iA * * *Z is acyclic for any I-cell Z. By assumption we know that A * Z is acyclic and we conclude * *by Lemma 2.2 since iA * Z ' i(A * Z) is PA-acyclic. Remark 2.7 Given a family S of I-cells, we say M satisfies the restricted joi* *n axiom if the join of A with any I-cell in S is A-acyclic. One refines then the above pr* *oposition to cellular model categories in which the restricted join axiom holds. Here i* *A * Z is A-acyclic for any i 0 and any S-cellular object Z, i.e. any object weakly equ* *ivalent to one which can be built by attaching only I-cells in S. 3 Fibrewise nullification Let A be any object in M. Recall that it is always possible to construct mappin* *g spaces up to homotopy in M eventhough we do not assume M is a simplicial model category (see [CS02 ]). Thus we can define an object Z 2 M to be A-local if there is a * *weak equivalences map(A, Z) ' *. A map g : X ! Y is a PA-equivalence if it induces a* * weak equivalences on mapping spaces g* : map(Y, Z) ! map(X, Z) for any A-local objec* *t Z. Hirschhorn shows that there exists a coaugmented functor PA : M ! M such that t* *he coaugmentation j : X ! PAX is a PA-equivalence to an A-local object. This funct* *or is called nullification or periodization. The nullification X ! PAX can be constructed up to homotopy by imitating the topological construction 2.8 in [Bou94 ]. One must iterate (possibly transfini* *tely, for a cardinal given by the smallness of any cofibrant object in M, see [Hir, Theorem* * 14.4.4]) the process of gluing A-cells, i.e. take the homotopy cofiber of a map iA ! X* *. We 7 assume throughout this section that the model category M satisfies both the joi* *n axiom and the cube axiom. Let us explain now how to adapt the fibrewise construction [DF96 , F.7] in a* * model category. The following lemma is the step we will iterate on and on so as to co* *nstruct the space ~E(in Theorem 3.3). Proposition 3.1 Consider a commutative diagram "'' F Ø___//PAF_____//F1 j|| || |j2| |fflfflØ fflffl|"fflffl|' E _____//E0_____//E1 | | | p| p0| |p1 |fflfflfflffl|fflffl|fflffl|fflffl| B _______B_______B where the left column is a fibration sequence, the upper left square is a homot* *opy push-out square, p0: E0! B is the unique map extending p such that the composite PAF ! E* *0! B is trivial, p1 is a fibration, and F1 is the homotopy fiber of p1. Then the co* *mposites E ! E0! E1 and F ! PAF ! F1 are both PA-equivalences. Proof. We can assume that the map j : F ,! PAF is a cofibration as indicated in* * the diagram, so that E0is obtained as a push-out, not only a homotopy push-out. Sin* *ce j is a PA-equivalence, so is its push-out along j by left properness (see [Hir, Propos* *ition 3.5.4]). To prove that F ! F1 is a PA-equivalence, it suffices to analyze the map PAF ! * *F1. We use Puppe's Proposition 1.3 to compute F1 as homotopy push-out of the homoto* *py fibers of PAF F ! E over the fixed base B. This yields the diagram PAF x B F x B ! F whose homotopy push-out is F1. We investigate more closely the map F ! F1 by decomposing the map F ! PAF into several steps obtained by gluing A-c* *ells. f Consider a cofibration of the form iA -! F ! Cf. Let Ef be the homotopy pu* *sh- out of Cf F ! E and compute as above the homotopy fiber Ff of Ef ! B. It is weakly equivalent to the homotopy push-out of Cf x B F x B ! F . Hence Ff is also weakly equivalent to the homotopy push-out of B iA x B ! F , using the definition of Cf. Decompose this push-out as follows iA x B ______//_ iA______//_F | | | | | | fflffl| fflffl| fflffl| B _______//_ iA * B___//_Ff 8 The right-hand square must be a homotopy push-out square as well. But both iA * *and iA * B are A-acyclic (by Proposition 2.6), so that the map iA ! iA * B is a PA-equivalence. Thus so is F ! Ff by left properness. Iterating this process of* * gluing A-cells shows that F ! F1 is a telescope of PA-equivalences, hence a PA-equival* *ence. Remark 3.2 In the category of spaces it is of course true that B x F ! B x * *PAF is a PA-equivalence, because localization commutes with finite products. In genera* *l we will see in Theorem 3.5 that the join axiom is actually equivalent to the commutatio* *n of PA with products. With the restricted join axiom we would have to impose the addi* *tional restriction on B that B be S-cellular. Theorem 3.3 Let M be a model category which is pointed, left proper, cellular* * and in which the cube axiom and the join axiom hold. Let PA : M ! M be a nullification functor. Then there exists a fibrewise nullification, i.e. a construction whi* *ch associates to any fibration F ! E ! B another fibration together with a natural transforma* *tion F _____//_E___//_B |''| || || fflffl| fflffl|fflffl| PAF ____//_~E__//_B where E ! ~Eis a PA-equivalence. Proof. We construct first by the method provided in Lemma 3.1 a natural transf* *or- mation to the fibration F1 ! E1 ! B. We iterate then this construction and get a fibration F~ ! ~E! B where F~ = hocolim(F ! F1 ! F2 ! . .).and E~ = hocolim(E ! E1 ! E2 ! . .).. All maps in these telescopes are PA-equivalences * *by the lemma, hence so are E ! ~Eand F ! ~F. Moreover any map Fn ! Fn+1 factorizes as Fn ! PAFn ' PAF ! Fn+1 so that ~F' PAF . We obtain thus the desired fibration PAF ! ~E! B. Define ~PAX = F ib(X ! PAX), the fiber of the nullification. As in the case * *of spaces we get: Corollary 3.4 For any object X in M we have PAP~AX ' *. Proof. Apply the fiberwise localization to the fibration ~PAX ! X ! PAX. This y* *ields a fibration PAP~AX ! ~X! PAX in which the base and the fiber are A-local. There* *fore X~ is A-local as well. But then X~' PAX and so PAP~AX ' *. 9 We end this section with a complete characterization of the model categories* * which admit fibrewise nullifications. Theorem 3.5 Let M be a model category which is pointed, left proper, cellular* * and in which the cube axiom holds. Then the following conditions are equivalent: (i)The nullification functor PA admits a fibrewise version. (ii)The nullification functor PA preserves finite products. (iii)The canonical projection X x A ! X is a PA-equivalence for any X 2 M. (iv)The join axiom for A is satisfied. Proof. We prove first that (i) implies (ii). Consider the trivial fibration X !* * XxY ! Y and apply the fibrewise nullification to get a new fibration PAX ! E ! Y . The * *inclusion of the fiber admits a retraction E ! PAX, i.e. E ' PAX x Y . Applying once agai* *n the fibrewise nullification to Y ! Y xPAX ! PAX, we see that the map XxY ! PAXxPAY is a PA-equivalence. As a product of local objects is local, this means precis* *ely that PA(X x Y ) ' PAX x PAY . Property (iii) is a particular case of (ii). We show now that (iii) implies * *(iv). If the canonical projection X x A ! X is a PA-equivalence, the push-out of it along th* *e other projection yields another PA-equivalence, namely A ! X * A. Therefore the join * *X * A is PA-acyclic. Finally (iv) implies (i) as shown in Theorem 3.3. The construction we propose for fibrewise nullification does not translate t* *o the setting of general localization functors. We do not know if the cube and join axioms ar* *e sufficient conditions for the existence of fibrewise localizations. 4 Algebras over an operad In this section we provide the motivating example for which this theory has bee* *n devel- opped. For a fixed field k, we work with Z-graded differential k-vector spaces* * (k-dgm) and consider the category of algebras in k-dgm over an admissible operad. This * *is indeed a pointed, left proper and cellular category. Weak equivalences are quasi-isom* *orphisms and fibrations are epimorphisms. 10 We do not know if the join axiom holds in full generality for any object A. * * It does so however when A is acyclic with respect to Quillen homology, which is the cas* *e we are most interested in, or when A is a free algebra. We check that the cube ax* *iom always holds, following the strategy of [Doe93 , Proposition A.15], which guara* *ntees the existence of fibrewise versions of the plus-construction and Postnikov sections* *. In the case of N-graded O-algebras (the case O = As is treated by Doeraene) one has to rest* *rict to a particular set of fibrations (the so-called J-maps), because they must be surje* *ctive in each degree in order to compute pull-backs. In our context all fibrations are epimor* *phisms, so that the cube axiom holds in full generality. Theorem 4.1 The cube axiom holds in the category of O-algebras. Proof. Let us briefly recall the key steps in Doeraene's strategy. We consider * *a push-out ` square of O-algebras (along a generic cofibration B ,! B O(V )): " ` B Ø___//_C = B O(V ) | | | | | fflffl|Ø " fflffl|` A ___//_D = A O(V ) We need to compute the pull-back of this square along a fibration p : E!! D (wh* *ich is hence an epimorphism of chain complexes). We have thus the following isomorphis* *m of chain complexes: a E ~=A O(V ) ker(p). This allows to compute the successive pull-backs A xD E, C xD E, and B xD E. In* * order to construct the homotopy push-out P of these pull-backs (which must coincide w* *ith E) we factorize the morphism B xD E ! C xD E as a ~ B xD E ,! (B kerp) O(V W ) i C xD E ` Thus P is identified with (A ker(p)) O(V W ), which allows us to build fi* *nally a quasi-isomorphism to E. The plus-construction for an O-algebra is a nullification with respect to a * *universal acyclic algebra U. We refer to [CRS ] for an explicit construction and nice app* *lications. Proposition 4.2 The join axiom holds for any acyclic O-algebra A. It holds in* * par- ticular for the universal acyclic algebra U constructed in [CRS ], so that the * *fibrewise plus-construction exists. 11 Proof. The join A * X is weakly equivalent to A ^ X by Lemmas 2.1 and 2.2. Sin* *ce A is 0-connected and acyclic, it is trivial by the Hurewicz Theorem [CRS , The* *orem 1.1]. Thus A * X ' * is always PA-acyclic. We consider next the case of Postnikov sections PO(x), where x is a generato* *r of arbitrary degree n 2 Z. Because [O(x), X] ~=ßnX for any O-algebra X, the nullif* *ication functor PO(x)is really a Postnikov section, i.e. PO(x)X ' X[n - 1]. Let us also* * recall that ßn(X x Y ) ~=ßnX x ßnY . Proposition 4.3 The join axiom holds for any free O-algebra O(x) on one genera* *tor of degree n 2 Z. Therefore fibrewise Postnikov sections exist. Proof. By Theorem 3.5 we might as well check that the map X x O(x) ! X is a PO(x)-equivalence for any O-algebra X. Clearly the n-th Postnikov section of th* *e product X x O(x) is equivalent to X[n - 1] and we are done. Our final result is a particular case of Corollary 3.4. A direct proof (with* *out fibrewise techniques) seems out of reach. Theorem 4.4 Let O - alg be the category of algebras over an admissible operad* * O. For any O-algebra B, denote by B ! B+ the plus construction. The homotopy fib* *er AB = F ib(B ! B+ ) is then acyclic with respect to Quillen homology. References [Bou94] A. K. Bousfield. Localization and periodicity in unstable homotopy th* *eory. J. Amer. Math. Soc., 7(4):831-873, 1994. [CD02] Carles Casacuberta and An Descheemaker. Relative group compel* *tions. preprint, 2002. [CRS] David Chataur, Jos'e Rodr'iguez, and J'er^ome Scherer. Plus-construct* *ion of alge- bras over an operad, hochschild and cyclic homologies up to homotopy.* * preprint. [CS02] Wojciech Chach'olski and J'er^ome Scherer. Homotopy theory of diagram* *s. Mem. Amer. Math. Soc., 155(736):ix+90, 2002. 12 [DF96] E. Dror Farjoun. Cellular spaces, null spaces and homotopy localizati* *on, volume 1622 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1996. [DMN89] William Dwyer, Haynes Miller, and Joseph Neisendorfer. Fibrewise comp* *letion and unstable Adams spectral sequences. Israel J. Math., 66(1-3):160-1* *78, 1989. [Doe93] Jean-Paul Doeraene. L.S.-category in a model category. J. Pure Appl. * *Algebra, 84(3):215-261, 1993. [DT95] J.-P. Doeraene and D. Tanr'e. Axiome du cube et foncteurs de Quillen.* * Ann. Inst. Fourier (Grenoble), 45(4):1061-1077, 1995. [Hir] P. S. Hirschhorn. Localization of model categories. Unpublished, avai* *lable at P. Hirschhorn's homepage: http://www-math.mit.edu/~psh/. [Mat76] Michael Mather. Pull-backs in homotopy theory. Canad. J. Math., 28(2)* *:225- 263, 1976. [May80] J. P. May. Fibrewise localization and completion. Trans. Amer. Math* *. Soc., 258(1):127-146, 1980. [MW80] Michael Mather and Marshall Walker. Commuting homotopy limits and col* *im- its. Math. Z., 175(1):77-80, 1980. [Pup74] Volker Puppe. A remark on öh motopy fibrations". Manuscripta Math., 1* *2:113- 120, 1974. [Qui73] Daniel Quillen. Higher algebraic K-theory. I. In Algebraic K-theory, * *I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 197* *2), pages 85-147. Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973. David Chataur Centre de Recerca Matem`atica, E-08193 Bellaterra email: chataur@crm.es J'er^ome Scherer Departament de Matem`atiques, Universitat Aut'onoma de Barcelona, E-08193 Bella* *terra e-mail: jscherer@mat.uab.es 13