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GROTHENDIECK-SERRE CORRESPONDENCE Classification Math´Ematique Par Sujets (2000)

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GROTHENDIECK-SERRE CORRESPONDENCE Classification Math´Ematique Par Sujets (2000) GROTHENDIECK-SERRE CORRESPONDENCE Classification math´ematique par sujets (2000). | 11, 14. Mots clefs. | Sheaf cohomology, schemes, Riemann-Roch, fundamental group, existence theorems, motives. GROTHENDIECK-SERRE CORRESPONDENCE R´esum´e. | This volume contains a large part of the mathematical correspon- dence between A.Grothendieck and J-P.Serre. This correspondence forms a vivid introduction to the algebraic geometry of the years 55-65 (a rich period if ever there was one). The readers will discover, for instance, the genesis of some of Grothendieck's ideas: Sheaf cohomology (T^ohoku),Schemes, Riemann-Roch, Fundamental Group, Existence Theorems and Motives... They also will get an idea of the mathematical atmosphere of this time (Bourbaki, seminars, Paris, Harvard, Princeton, Algeria war, ...). Abstract (Correspondance Grothendieck-Serre) Ce volume contient une grande partie de la correspondance math´ematique entre A.Grothendieck et J-P.Serre. Cette correspondance constitue une intro- duction particuli`erement vivante `ala g´eom´etrie alg´ebriquedes ann´ees1955{1965 (p´eriode faste s'il en fut). Le lecteur y d´ecouvrira,en particulier, la gen`ese de certaines des id´eesde Grothendieck: cohomologie des faisceaux (T^ohoku), sch´emas,Riemann-Roch, groupe fondamental, th´eor`emesd'existence, motifs... Il se fera aussi une id´eede l'atmosph`eremath´ematiquede cette ´epoque (Bour- baki, Paris, Harvard, Princeton, guerre d'Alg´erie,...). TABLE DES MATIERES` Foreword .............................................................. vii Correspondence ...................................................... 1 Bibliographie ..........................................................281 FOREWORD A large part of my correspondence with Grothendieck between 1955 and 1970 is reproduced in the present volume of the series \Mathematical Documents". I have chosen those of our letters which seemed to me most likely to be of interest to the reader, either from a purely mathematical point of view or from a historical point of view. Of course, this choice is far from representing all the questions I discussed with Grothendieck over these fifteen years: when we were both in Paris, our usual means of communication was the telephone. The letters reflect rather the periods when we were separated: one of us being in Princeton or Harvard and the other being in Paris. I have also included several of the letters that we exchanged between 1984 and 1987, at the moment when R´ecoltes et Semailles was being written. The only changes which have been made to the original texts are the following: | suppression of abbreviations, such as \com." for \commutative" g; | correction of both spelling mistakes and obvious mis-typings. | suppression of certain personal passages (1) which have been replaced by suspension points between brackets [...]. On the other hand, wrong statements have been preserved as they were. They are corrected (insofar as it was possible) in the Notes placed at the end of the volume. These notes also give some more recent results. They are indicated in the text by reference numbers in the margin. (1)For curiosity's sake, here is one of the suppressed passages, on the subject of a mathematician who I will call X: \... the horrible prose of X, grand master wizard of the den of horrors, who of course is careful to keep from writing a clear little paper, in which the helpful claims would appear in a usable form..." It will be understood why I wished to spare X from reading this passage. viii FOREWORD Jean-Pierre Serre, Paris, December 2000 This volume would never have seen the day without the help of a large number of people. Jean Malgoire authorized publication of this correspondence in Grothendieck's name. A (small) army took on the work of typing the letters in TeX which in certain cases was no light task. Once the typing was finished, the readers pinpointed a certain number of points deserving commentary, which contributed to filling out the notes written by Serre. When the volume was reasonably advanced, we had to think about publica- tion. Since none of the S.M.F.'s existing series had room for a volume of this kind, it was decided to create a new series, \Mathematical Documents", whose raison d'^etrewould be the publication of historico-mathematical texts. I would like to thank all the people mentioned above for the enthusiasm with which they participated in this undertaking, and I hope the reader will have as much pleasure consulting this correspondence as I had editing it. Pierre Colmez, Paris, January 2001 222 scale=.70]Page2.eps Facsimile of Grothendieck's letter of 1.28.55, page 1 CORRESPONDENCE January 28, 1955 Alexandre Grothendieck J.-P. Serre : Grothendieck was in Lawrence (Kansas), where he had been invited (by N.Aronszajn, I believe) because of his work on Topological Vector Spaces. He had decided to change subjects and move towards topology and functions of complex variables, which had led him to algebraic geometry (as it had led me, a year or two earlier). Dear Serre, A really annoying thing has happened: as I could not carry all my papers on the plane, I sent part of them in two packages, which were posted together the day I left. But only one of them has arrived, a week ago, and by now it is unlikely that the other will ever arrive. It contained (among other things) those lectures from the Schwartz, Cartan and Cartier seminars that existed at the time, your paper on algebraic sheaves and the Kodaira- Spencer papers. I am very upset about this, as it will probably not be possible to replace the last two for some time, and I had not even had time to glance at Kodaira-Spencer. In any case, send me the complete set of all the lectures from all three seminars, if there are any left. I hope Barros remembers to bring you the Schwartz lectures. Can you also tell Spencer I have not received any of the papers he wanted to send me from Princeton? Here I have practically all my time to myself; I am giving a few talks on Malgrange's thesis, but this hardly takes any time. I will probably bother you shortly with various technical questions; I have started reading Thom's \varieties" J.-P. Serre : \Thom's `varieties' ". This is a reference to: R. Thom, Quelques propri´et´esglobales des vari´et´esdiff´erentiables, Comment. Math. Helv. 28 (1954), 17{86., which is horribly badly written, but probably 2 CORRESPONDENCE useful to look at, and your article on H(Π; q; Z2) J.-P. Serre : \your article on H(Π; q; Z2)" : [Se53b]. which has pleased me so far. I have also gone thoroughly over the basic principles of ho- and coho-mology; I needed to do so. I came across several questions of a standard type, to which you may have an 0 0 answer. Let X be a space with a “Φ family", X ⊂ X a closed subset, U = {X , and F a sheaf on X. If B is a subset of X, then let Φ \ B be the set of all A\B (A 2 Φ), and let ΦB be the set of all A 2 Φ which are in B. O denotes an arbitrary open neighborhood of X0. There are canonical homomorphisms, (taking the singular (co)homology point of view, for the sake of argument) ( Hp (X mod X0; F) Hp (U; F) bijective if X and X0 are HLC, for example. Φ ΦU Φ 0 Φ\U Hp (X mod X ; F) ! Hp (U; F) when is this bijective? J.-P. Serre : (HLC) means \homologically locally connected". A space X has this property if for any x 2 X , any integer p ≥ 0 and any neighborhood V of x, there is a neighborhood U of x in X contained in V such that the homomorphism Hp(U; Z) ! Hp(V; Z) is trivial. (Here, Hp denotes the p-th group of singular homology if p > 0; when p = 0, this is replaced with the corresponding \reduced group", i.e. the kernel of H0 ! Z.) Every locally contractible space (any simplicial complex, for example) is (HLC). In the Cartan seminar of 1948/1949, Cartan showed that if X is a locally compact (HLC) space, then the singular cohomology of X can be identified with its sheaf cohomology. ( Φ −! Φ\O Hp (X mod U; F) ! limO Hp (O; F) bijective if X has a normal neighborhood − Hp (X mod U; F) lim Hp (O; F) when is this bijective? Φ O ΦO ( p 0 −! p HΦ\X0 (X ; F) limO HΦ\O(O; F) always bijective. 0 − Φ\X 0 ΦO Hp (X ; F) ! limO Hp (O; F) when is this bijective? −! Homomorphisms which are dual to each other are bracketed together. lim = − inductive limit, lim = projective limit; in any case it is always clear what I mean. The first two pairs of homomorphisms arise when one seeks to eliminate \relative" groups by interpreting them in terms of \absolute groups" . The questions marked (?) are those to which I have only managed to give very unsatisfactory answers by duality arguments: I am forced to assume that F is at least locally constant, and that Φ is the family of all closed sets (co), respectively, of all compact sets (ho). A reasonable conjecture would be that everything works if X and X0 are both HLC. Here is an analogous question: is CORRESPONDENCE 3 the homomorphism p − p H (X; F) ! limK H (K; F) (where K runs over all compact sets of X, which is assumed to be locally compact and paracompact) bijective, at least in certain cases? Once again, I have a partial answer by duality if F is locally constant, but this is scarcely satisfactory. This question arises in justifying the \passage to the limit" in Cartan's theorems A and B once the compact case is established, for example.
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