Technique de descente et th茅or猫mes d'existence en g茅om茅trie alg茅briques. II. Le th茅or猫me d'existence en th茅orie formelle des modules

<!DOCTYPE html> <html lang="fr"> <head> <title> Technique de descente et th茅or猫mes d'existence en g茅om茅trie alg茅briques. II. Le th茅or猫me d'existence en th茅orie formelle des modules </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/> <meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1.0, minimum-scale=1.0"> <link rel="shortcut icon" type="image/x-icon" href="/static/numdam/img/numdam_favicon.ico"> <link rel="stylesheet" href="//fonts.googleapis.com/css?family=Lato"/> <link rel="stylesheet" href="/static/ptf/vendors/bootstrap/css/bootstrap.min.css"/> <link rel="stylesheet" href="/static/ptf/vendors/font-awesome/css/font-awesome.min.css"> <link rel="stylesheet" href="/static/ptf/vendors/lightbox/css/lightbox.css"> <link rel="stylesheet" href="/static/ptf/css/ptf.css" type="text/css"/> <link rel="stylesheet" href="/static/ptf/css/search-bar.css" type="text/css"/> <link rel="stylesheet" href="/static/numdam/vendors/owl-carousel/css/owl.carousel.css"> <link rel="stylesheet" href="/static/numdam/vendors/owl-carousel/css/owl.theme.css"> <link rel="stylesheet" href="/static/numdam/css/numdam.css" type="text/css"/> </head> <body id="page-" style="font-family: ;font-size: px;"> <div id="main"> <div id="top"> <div id="bandeau"> <div style="float:left"> <a href="/" class='no-focus'> <img src="/static/numdam/img/numdam_logo.gif" title='Accueil' alt='Accueil' class='logo no-text-decoration'/> </a> </div> <div class="bandeau-background"> <div style="float:left;width:calc(50% - 16px)"> </div> <div style="float:left" class="smartphone"> <div class="dropdown"> <button class="btn light-blue dropdown-toggle" type="button" data-toggle="dropdown"> <img src='/static/mersenne_cms/img/main-menu.svg' alt='main-menu'/> </button> <ul class="dropdown-menu"> <li><a href="/journals/">Revues</a></li> <li><a href="/actas/">S茅minaires</a></li> <li><a href="/books/">Livres</a></li> <li><a href="/lectures">Notes de cours</a></li> <li><a href="/thesis">Th猫ses</a></li> <li><a href="/authors">Auteurs</a></li> </ul> </div> </div> </div> <div class="onoffswitch-group"> <div id="language-switch" title="Cliquer pour afficher le site en anglais"> <form id="language-switch-form" action="/i18n/setlang/" method="post"><input type="hidden" name="csrfmiddlewaretoken" value="YC9v3oBKh046KmtKh4K5gzwtjpJVRKNIuQ4qYiW7rDVXNwFkx3EMrmrUvO38jvy0"> <div class="onoffswitch" id="lang_flags"> <input type="checkbox" name="language" value="en" class="onoffswitch-checkbox" id="lang_onoffswitch" checked> <label class="onoffswitch-label" for="lang_onoffswitch"> <span class="onoffswitch-inner" data-on="" data-off=""></span> <span class="onoffswitch-switch"></span> </label> </div> </form> </div> <div id="mathml-switch" title="Cliquer pour afficher le code source TeX" > <form action="/set-formula-display/" method="post"><input type="hidden" name="csrfmiddlewaretoken" value="YC9v3oBKh046KmtKh4K5gzwtjpJVRKNIuQ4qYiW7rDVXNwFkx3EMrmrUvO38jvy0"> <div class="onoffswitch"> <input type="checkbox" name="formula-display" value="tex" class="onoffswitch-checkbox" id="tex_onoffswitch" onclick="this.form.submit();" > <label class="onoffswitch-label" for="tex_onoffswitch"> <span class="onoffswitch-inner" data-on="TeX" data-off="TeX"></span> <div class="onoffswitch-switch"> <div class="onoffswitch-switch-label"> OFF</div> </div> </label> </div> </form> </div> </div> </div> <nav class="navbar navbar-default desktop"> <div class="container-fluid"> <ul class="nav navbar-nav"> <li ><a href="/journals/">Revues</a></li> <li class="active"><a href="/actas/">S茅minaires</a></li> <li ><a href="/books/">Livres</a></li> <li ><a href="/lectures">Notes de cours</a></li> <li ><a href="/thesis">Th猫ses</a></li> <li ><a href="/authors">Auteurs</a></li> </ul> </div> </nav> <div id="bandeau3"> <div class="container" id="search-bar"> <div> <form class="form-horizontal" method="post" action="/search"><input type="hidden" name="csrfmiddlewaretoken" value="YC9v3oBKh046KmtKh4K5gzwtjpJVRKNIuQ4qYiW7rDVXNwFkx3EMrmrUvO38jvy0"> <div class="form-group"> <div class="col-sm-offset-1 col-xs-8" id="line-container"> <div class="input-group search-group"> <div class="search-group-inner"> <input class="searchinput form-control" type="text" name="q0" value="" placeholder="Rechercher des articles, des livres..."/> <span class="searchclear glyphicon glyphicon-remove"></span> </div> <div class="input-group-btn" id="search-group-btn"> <input type="hidden" name="qt0" value="all"> <button type="button" class="btn btn-default dropdown-toggle" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false"> Tout <span class="caret"></span> </button> <ul class="dropdown-menu"> <li><a href="#">Tout</a></li> <li><a href="#">Auteur</a></li> <li><a href="#">Titre</a></li> <li><a href="#">Bibliographie</a></li> <li><a href="#">Mots cl茅s</a></li> <li><a href="#">Plein texte</a></li> </ul> <button id="add-search-field" type="button" class="btn btn-default"> <span class="glyphicon glyphicon-plus"></span> </button> </div> </div> </div> <div class="col-xs-2 search-button-div"> <button type="submit" class="btn search-button" name="Submit"> <span class="glyphicon glyphicon-search"></span> <span id="search-button-text">Rechercher</span> </button> <a href="/help/"><img src="/static/mersenne_cms/img/question.png"></a> </div> </div> </form> <div id="template-search-form" style="display:none"> <div class="input-group search-group search-group-added"> <div class="input-group-addon"> <input class="bt-checkbox" type="checkbox" name="not##i##" /> <span style="line-height:20px">NOT</span> </div> <div class="search-group-inner"> <input class="searchinput form-control" type="text" name="q##i##"/> <span class="searchclear glyphicon glyphicon-remove"></span> </div> <div style="display:none" class="bt-input-with-text">   <span class="bt-input-text-between">Entre</span> <input class="form-control-in" type="text" name="q-f-##i##" placeholder="Ann茅e" /> <span class="bt-input-text">et</span> <input class="form-control-in" type="text" name="q-l-##i##" placeholder="Ann茅e" />   </div> <span class="input-group-btn"> <input type="hidden" name="qt##i##" value="author"> <button type="button" class="btn btn-default dropdown-toggle" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false"> Auteur <span class="caret"></span> </button> <ul class="dropdown-menu"> <li><a href="#">Tout</a></li> <li><a href="#">Auteur</a></li> <li><a href="#">Titre</a></li> <li><a href="#">Date</a></li> <li><a href="#">Bibliographie</a></li> <li><a href="#">Mots cl茅s</a></li> <li><a href="#">Plein texte</a></li> </ul> <button type="button" class="btn btn-default remove-search-field"> <span class="glyphicon glyphicon-minus"></span> </button> </span> </div> </div> </div> </div> </div> </div> <div class="container"> </div> <div id="content" class="wrapper-float"> <div id="left"> <div id="content_left"> <div class="cover-thumbnail"> <a href="/icon/SB/SB_1958-1960__5_/couv1.jpg" class="lightbox" title=""> <img src="/icon/SB/SB_1958-1960__5_/couv1_small.jpg" width="80px"> </a> </div> </div> </div> <div id="center"> <div class="row breadcrumb-row"> <div class="col-xs-12"> <div class="breadcrumb-lg"> <div style="float:left"> <ul class="cms-breadcrumb"> <li class="breadcrumb-previous"> <a href="/item/SB_1958-1960__5__359_0/">Pr茅c茅dent</a> </li> </ul> </div> <div style="float:left"> <ul class="cms-breadcrumb"> <li class=""> <a href="/actas/SB/">S茅minaire Bourbaki</a> </li> <li class=""> <a href="/item/SB_1958-1960__5_/">no. 5</a> </li> <li class="active"> <a href="/item/SB_1958-1960__5__369_0/">Expos茅 no. 195</a> </li> </ul> </div> <div style="float:left"> <ul class="cms-breadcrumb"> <li class="breadcrumb-next"> <a href="/item/SB_1958-1960__5__390_1/">Suivant</a> </li> </ul> </div> </div> <div class="breadcrumb-sm"> <a href="/item/SB_1958-1960__5__359_0/" class="btn btn-default"><i class="glyphicon glyphicon-chevron-left"></i></a> <a href="/item/SB_1958-1960__5__390_1/" class="btn btn-default"><i class="glyphicon glyphicon-chevron-right"></i></a> </div> <div class="clearfix"> </div> </div> </div> <div id="article-div" class="article-div" style="margin-left:-15px"> <div class="row " style="margin:0px"> <div class="float-minus-100"> <div style="font-size:18px;margin-bottom:10px"> <span class="common-title article-title"> Technique de descente et th茅or猫mes d'existence en g茅om茅trie alg茅briques. II. Le th茅or猫me d'existence en th茅orie formelle des modules </span> </div> <div class="article-author" style="display:inline-block"> <div style="margin-bottom:5px; font-size: 16px"> <a class="author_link" href="/search/%22Grothendieck,%20Alexander%22-c/">Grothendieck, Alexander</a> </div> </div> <div style="margin-bottom:10px"> <span style="display:inline-block;">S茅minaire Bourbaki : ann茅es 1958/59 - 1959/60, expos茅s 169-204, </span> S茅minaire Bourbaki, no. 5 (1960), Expos茅 no. 195, 22 p. </div> <div style="margin-top:20px;margin-bottom:10px"> <div class="wrapper-float"> <div style="float:left;margin-top:0px"> <a href="/item/SB_1958-1960__5__369_0.pdf" target="_blank"> <img src="/static/ptf/img/pdf.gif"> </a> <a href="/item/SB_1958-1960__5__369_0.djvu"> <img src="/static/ptf/img/djvu.png"> </a> </div> </div> </div> </div> <div style="float:left;margin-top:20px;width:98px;padding-right:10px;padding-left:15px;"> <a href="https://smf.emath.fr"><img src="/static/mersenne_cms/img/logo-societe-mathematique-de-france.jpg" width="80px"></a> </div> </div> <div class="row" style="margin:10px"> <div style="margin-bottom:10px; margin-top:20px" class="relations-div col-xs-12"> <i class="fa fa-info-circle" aria-hidden="true"></i> corrig茅 par <a href="/item/SB_1961-1962__7__300_1/"> <span class="common-title article-title"> Erratum 脿 l'expos茅 n掳195 </span> </a> </div> </div> <div class="row" style="margin:0px"> <div class="col-xs-12"> <div style="margin-top:10px;margin-bottom:10px"> <ul class="nav nav-tabs article-tab"> <li class="nav-item active"><a class="nav-link active" data-bs-toggle="tab" data-toggle="tab" data-bs-target="#info-tab" href="#info-tab">D茅tail</a></li> <li class="nav-item"><a class="nav-link" data-bs-toggle="tab" data-toggle="tab" data-bs-target="#citation-tab" href="#citation-tab">Export</a></li> <li class="nav-item "> <a class="nav-link " data-bs-toggle="tab" data-toggle="tab" data-bs-target="#how-to-cite-tab" href="#how-to-cite-tab">Comment citer</a> </li> </ul> <div class="tab-content" style="margin-top:10px"> <div id="info-tab" class="tab-pane fade in active">  <div style="margin-top:5px"> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1603480" class="badge badge-secondary ref-badge extid-badge"> MR </a>   <a href="https://zbmath.org/?q=an:0234.14007" class="badge badge-secondary ref-badge extid-badge"> Zbl </a> | <a href="/articles/SB_1958-1960__5__369_0/citations/"> 22 citations dans Numdam</a> </div> <div style="margin-top:5px"> </div> </div> <div id="citation-tab" class="tab-pane fade"> <div class="upper"> <ul class="nav nav-pills"> <li role="presentation" class="active"><a data-toggle="pill" href="#bibtex">BibTeX</a></li> <li role="presentation"><a data-toggle="pill" href="#ris">RIS</a></li> <li role="presentation"><a data-toggle="pill" href="#enw">EndNote</a></li> </ul> <div class="tab-content"> <div id="bibtex" class="tab-pane fade in active"> <pre class="upper-right-content" id="bibtex-div">@incollection{SB_1958-1960__5__369_0, author = {Grothendieck, Alexander}, title = {Technique de descente et th\'eor\`emes d'existence en g\'eom\'etrie alg\'ebriques. {II.} {Le} th\'eor\`eme d'existence en th\'eorie formelle des modules}, booktitle = {S\'eminaire Bourbaki : ann\'ees 1958/59 - 1959/60, expos\'es 169-204}, series = {S\'eminaire Bourbaki}, note = {talk:195}, pages = {369--390}, publisher = {Soci\'et\'e math\'ematique de France}, number = {5}, year = {1960}, mrnumber = {1603480}, zbl = {0234.14007}, language = {fr}, url = {http://www.numdam.org/item/SB_1958-1960__5__369_0/} }</pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#bibtex-div" aria-hidden="true" title="Copier dans le presse-papier"></button> <a class="fa fa-download export-button btn btn-primary" title="T茅l茅charger" href="/export_citation/SB_1958-1960__5__369_0/bib/"></a> </div> <div id="ris" class="tab-pane fade"> <pre class="upper-right-content" id="ris-div">TY - CHAP AU - Grothendieck, Alexander TI - Technique de descente et th茅or猫mes d'existence en g茅om茅trie alg茅briques. II. Le th茅or猫me d'existence en th茅orie formelle des modules BT - S茅minaire Bourbaki : ann茅es 1958/59 - 1959/60, expos茅s 169-204 AU - Collectif T3 - S茅minaire Bourbaki N1 - talk:195 PY - 1960 SP - 369 EP - 390 IS - 5 PB - Soci茅t茅 math茅matique de France UR - http://www.numdam.org/item/SB_1958-1960__5__369_0/ LA - fr ID - SB_1958-1960__5__369_0 ER - </pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#ris-div" aria-hidden="true" title="Copier dans le presse-papier"></button> <a class="fa fa-download export-button btn btn-primary" title="T茅l茅charger" href="/export_citation/SB_1958-1960__5__369_0/ris/"></a> </div> <div id="enw" class="tab-pane fade"> <pre class="upper-right-content" id="enw-div">%0 Book Section %A Grothendieck, Alexander %T Technique de descente et th茅or猫mes d'existence en g茅om茅trie alg茅briques. II. Le th茅or猫me d'existence en th茅orie formelle des modules %B S茅minaire Bourbaki : ann茅es 1958/59 - 1959/60, expos茅s 169-204 %A Collectif %S S茅minaire Bourbaki %Z talk:195 %D 1960 %P 369-390 %N 5 %I Soci茅t茅 math茅matique de France %U http://www.numdam.org/item/SB_1958-1960__5__369_0/ %G fr %F SB_1958-1960__5__369_0</pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#enw-div" aria-hidden="true" title="Copier dans le presse-papier"></button> <a class="fa fa-download export-button btn btn-primary" title="T茅l茅charger" href="/export_citation/SB_1958-1960__5__369_0/enw/"></a> </div> </div> </div> </div> <div id="how-to-cite-tab" class="tab-pane fade "> <div class="upper-right"> <pre class="upper-right-content" id="how-to-cite-div">Grothendieck, Alexander. Technique de descente et th茅or猫mes d'existence en g茅om茅trie alg茅briques. II. Le th茅or猫me d'existence en th茅orie formelle des modules, dans <em>S茅minaire Bourbaki : ann茅es 1958/59 - 1959/60, expos茅s 169-204</em>, S茅minaire Bourbaki, no. 5 (1960), Expos茅 no. 195, 22 p. http://www.numdam.org/item/SB_1958-1960__5__369_0/</pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#how-to-cite-div" aria-hidden="true" title="Copier dans le presse-papier"></button> </div> </div> </div> </div> </div> </div> <div class="row" style="margin:0px"> <div class="col-xs-12"> <div style="margin-top:30px;margin-bottom:10px"> <ul class="nav nav-tabs article-tab" > <li class="nav-item active" id="main-citations-li" > <a class="nav-link" data-bs-toggle="tab" data-toggle="tab" data-bs-target="#references-tab" href="#references-tab">Bibliographie</a> </li> <li class="nav-item" id="citations-html" style="display:none"> <a class="nav-link" data-bs-toggle="tab" data-toggle="tab" data-bs-target="#citations-tab" href="#citations-tab">Cit茅 par</a> </li> </ul> <div class="tab-content" style="margin-top:10px"> <div id="references-tab" class="tab-pane fade in active"> <div style="margin-top:20px;margin-bottom: 0px"> <p class="bibitemcls " > [1] <span class="citation-author">Grothendieck (Alexander)</span>. - <span class="citation-document-title">Sur quelques points d'alg猫bre homologique</span>, <span class="citation-publication-title">Tohoku math. J.</span>, t. <span class="citation-volume">9</span>, 1957, p. 119-221. | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=102537" class="badge badge-secondary ref-badge "> MR </a> | <a href="https://zbmath.org/?q=an:0118.26104" class="badge badge-secondary ref-badge "> Zbl </a> </p> <p class="bibitemcls " > [2] <span class="citation-author">Grothendieck (Alexander)</span>. - <span class="citation-document-title">G茅om茅trie formelle et g茅om茅trie alg茅brique</span>, <span class="citation-publication-title">S茅minaire Bourbaki</span>, t. <span class="citation-volume">11</span>, 1958/59, fasc. 3, n掳 182. | <a href="/item/SB_1958-1960__5__193_0" class="badge badge-secondary ref-badge">Numdam</a> | <a href="https://zbmath.org/?q=an:0229.14005" class="badge badge-secondary ref-badge "> Zbl </a> </p> <p class="bibitemcls " > [3] <span class="citation-author">Grothendieck (Alexander)</span>. - <span class="citation-document-title">Technique de descente et th茅or猫mes d'existence en g茅om茅trie alg茅brique, I : G茅n茅ralit茅s, Descente par morphismes fid猫lement plats</span>, <span class="citation-publication-title">S茅minaire Bourbaki</span>, t. <span class="citation-volume">12</span>, 1959/60, fasc. 1, n掳 190. | <a href="/item/SB_1958-1960__5__299_0" class="badge badge-secondary ref-badge">Numdam</a> | <a href="https://zbmath.org/?q=an:0229.14007" class="badge badge-secondary ref-badge "> Zbl </a> </p> <p class="bibitemcls " > [4] <span class="citation-author">Serre (Jean-Pierre)</span>. - <span class="citation-document-title">Corps locaux et isog茅nies</span>, <span class="citation-publication-title">S茅minaire Bourbaki</span>, t. <span class="citation-volume">11</span>, 1958/59, fasc. 3, n掳 185. | <a href="/item/SB_1958-1960__5__239_0" class="badge badge-secondary ref-badge">Numdam</a> | <a href="https://zbmath.org/?q=an:0137.02501" class="badge badge-secondary ref-badge "> Zbl </a> </p> </div> </div> </div> </div> </div> </div> </div> </div> <div id="right"> </div> </div> </div> <div id="footer"> <div class="horizontal-menu menu-footer"> <ul> <li><a href="/about/">脌 propos</a></li> <li><a href="/help/">Aide</a></li> <li><a href="/legal/">Mentions l茅gales</a></li> <li><a href="/contact/">Contact</a></li> </ul> </div> <div class="logo-footer"> <div class="logo-footer-left" style="float:left;"> </div> <div style="float:left"> <div class="lg-desktop"> <p>D茅velopp茅 par</p> </div> <div style="display:inline-block"> <a href="http://www.mathdoc.fr"><img class="logo-auto" src="/static/numdam/img/logo-mathdoc.png"/></a> </div> </div> <div style="float:left;width:5%"> </div> <div style="float:left"> <div class="lg-desktop"> <p>Soutenu par</p> </div> <div style="display:inline-block"> <a href="http://www.cnrs.fr/insmi"><img class="logo-auto" src="/static/numdam/img/logo-insmi.png"/></a>  <a href="http://www.univ-grenoble-alpes.fr/"><img class="logo-auto" src="/static/numdam/img/logo-uga.svg"/></a> </div> </div> <div style="float:left;width:5%"> </div> <div style="float:left"> <div class="lg-desktop"> <p>Partenaire de</p> </div> <div style="display:inline-block"> <a href="http://www.eudml.eu"><img class="logo-auto" src="/static/numdam/img/logo-eudml.png"/></a>  <a href="http://portail.math.cnrs.fr"><img class="logo-auto" src="/static/numdam/img/logo-portail.png"/></a> </div> </div> <div style="float:right; margin-right:20px"> <div id="logo-collex"> <a href="https://www.collexpersee.eu/"><img class="logo-auto" src="/static/numdam/img/logo-collex.png"/></a>  </div> <div id="logo-collex-lg"> <a href="https://www.collexpersee.eu/"><img class="logo-auto" src="/static/numdam/img/logo-collex.png"/></a>  </div> </div> </div> </div> <script type="text/javascript" src="/static/ptf/vendors/jquery/js/jquery-3.5.1.min.js"></script> <script type="text/javascript" src="/static/ptf/vendors/clipboard/js/clipboard.min.js"></script> <script type="text/javascript" src="/static/ptf/vendors/bootstrap/js/bootstrap.min.js"></script> <script type="text/javascript" src="/static/ptf/vendors/lightbox/js/lightbox.js"></script> <script type="text/javascript" src="/static/ptf/js/ptf.js"></script> <script type="text/javascript" src="/static/ptf/js/search-bar.js"></script> <script type="text/javascript" src="/static/numdam/vendors/owl-carousel/js/owl.carousel.min.js"></script> <script type="text/javascript" src="/static/numdam/js/scripts.js"></script> <script src="/static/ptf/js/mathjax-config.js" defer></script> <script type="text/javascript" id="MathJax-script" defer src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"> </script> </body> </html>

CINXE.COM

Technique de descente et th茅or猫mes d'existence en g茅om茅trie alg茅briques. II. Le th茅or猫me d'existence en th茅orie formelle des modules