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Grothendieck and J. Dieudonné appeared, an encyclopedic work on the foundations of Grothendieck’s algebraic geometry. We sketch some of the most important concepts developed there, comparing it to the classical language, and mention a few results in algebraic and arithmetic geometry which have since been proved using the new framework.","datePublished":"2018-06-29T00:00:00Z","dateModified":"2018-06-29T00:00:00Z","pageStart":"235","pageEnd":"290","sameAs":"https://doi.org/10.1365/s13291-018-0181-1","keywords":["Éléments de Géométrie Algébrique","Algebraic Geometry","Schemes","Mathematics","general"],"image":["https://media.springernature.com/lw1200/springer-static/image/art%3A10.1365%2Fs13291-018-0181-1/MediaObjects/13291_2018_181_Figa_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1365%2Fs13291-018-0181-1/MediaObjects/13291_2018_181_Figb_HTML.gif"],"isPartOf":{"name":"Jahresbericht der Deutschen Mathematiker-Vereinigung","issn":["1869-7135","0012-0456"],"volumeNumber":"120","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Springer Berlin Heidelberg","logo":{"url":"https://www.springernature.com/app-sn/public/images/logo-springernature.png","@type":"ImageObject"},"@type":"Organization"},"author":[{"name":"Ulrich Görtz","affiliation":[{"name":"University of Duisburg-Essen, Fakultät für Mathematik","address":{"name":"University of Duisburg-Essen, Fakultät für Mathematik, Essen, Germany","@type":"PostalAddress"},"@type":"Organization"}],"email":"ulrich.goertz@uni-due.de","@type":"Person"}],"isAccessibleForFree":false,"hasPart":{"isAccessibleForFree":false,"cssSelector":".main-content","@type":"WebPageElement"},"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> </head> <body class="" >  <noscript> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>   <noscript data-test="gtm-body"> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" 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article" data-track-category="article body" data-track-label="link">Cite this article</a> </li> </ul> <div class="app-article-masthead__buttons" data-test="download-article-link-wrapper" data-track-context="masthead"> </div> </div> <div class="app-article-masthead__brand"> <a href="/journal/13291" class="app-article-masthead__journal-link" data-track="click_journal_home" data-track-action="journal homepage" data-track-context="article page" data-track-label="link"> <picture> <source type="image/webp" media="(min-width: 768px)" width="120" height="159" srcset="https://media.springernature.com/w120/springer-static/cover-hires/journal/13291?as=webp, https://media.springernature.com/w316/springer-static/cover-hires/journal/13291?as=webp 2x"> <img width="72" height="95" src="https://media.springernature.com/w72/springer-static/cover-hires/journal/13291?as=webp" srcset="https://media.springernature.com/w144/springer-static/cover-hires/journal/13291?as=webp 2x" alt=""> </picture> <span class="app-article-masthead__journal-title">Jahresbericht der Deutschen Mathematiker-Vereinigung</span> </a> <a href="https://link.springer.com/journal/13291/aims-and-scope" class="app-article-masthead__submission-link" data-track="click_aims_and_scope" data-track-action="aims and scope" data-track-context="article page" data-track-label="link"> Aims and scope <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> <a href="mailto:alexander.zimmermann@u-picardie.fr" class="app-article-masthead__submission-link" data-track="click_submit_manuscript" data-track-context="article masthead on springerlink article page" data-track-action="submit manuscript" data-track-label="link"> Submit manuscript <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> </div> </div> </div> </section> <div class="c-article-main u-container u-mt-24 u-mb-32 l-with-sidebar" id="main-content" data-component="article-container"> <main class="u-serif js-main-column" data-track-component="article body"> <div class="c-article-header"> <header> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Ulrich-G_rtz-Aff1" data-author-popup="auth-Ulrich-G_rtz-Aff1" data-author-search="Görtz, Ulrich" data-corresp-id="c1">Ulrich Görtz<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff1">1</a></sup> </li></ul> <div data-test="article-metrics"> <ul class="app-article-metrics-bar u-list-reset"> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>1139 <span class="app-article-metrics-bar__label">Accesses</span></p> </li> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-citations-medium"></use> </svg>2 <span class="app-article-metrics-bar__label">Citations</span></p> </li> <li class="app-article-metrics-bar__item app-article-metrics-bar__item--metrics"> <p class="app-article-metrics-bar__details"><a href="/article/10.1365/s13291-018-0181-1/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>About 50 years ago, <i>Éléments de Géométrie Algébrique</i> (EGA) by A. Grothendieck and J. Dieudonné appeared, an encyclopedic work on the foundations of Grothendieck’s algebraic geometry. We sketch some of the most important concepts developed there, comparing it to the classical language, and mention a few results in algebraic and arithmetic geometry which have since been proved using the new framework.</p></div></div></section> <div class="c-notes"> <p class="c-notes__text c-status-message--info"> <svg width="24" height="24" focusable="false" role="img" aria-hidden="true" class="c-status-message__icon"> <use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-info-filled-medium"></use> </svg> This is a preview of subscription content, <a id="test-login-banner-link" href="//wayf.springernature.com?redirect_uri=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1365%2Fs13291-018-0181-1%3Ferror%3Dcookies_not_supported%26code%3D10ac4296-b412-4404-a603-9eb69f511285" data-track="click" data-track-action="login" data-track-label="link" class="c-preview-message__link">log in via an institution</a> <svg width="16" height="16" focusable="false" role="img" 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The language of schemes will allow us to deal with this point more elegantly.</p></div></li><li class="c-article-footnote--listed__item" id="Fn3" data-counter="3."><div class="c-article-footnote--listed__content"><p>If the characteristic of <span class="mathjax-tex">$k$</span> is 2 or 3, then one must consider a slightly more general form of equation in order to obtain all elliptic curves over <span class="mathjax-tex">$k$</span>. Cf. Def <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1365/s13291-018-0181-1#FPar49">5.3</a>.</p></div></li><li class="c-article-footnote--listed__item" id="Fn4" data-counter="4."><div class="c-article-footnote--listed__content"><p>In [EGA I], the topology thus defined is called the <i>spectral topology</i> or <i>Zariski topology</i>. It appears that Zariski has only considered this topology on the sets of maximal ideals of (certain) rings. On the other hand, Jacobson [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="
					Jacobson, N.: A topology for the set of primitive ideals in an arbitrary ring. Proc. Natl. Acad. Sci. USA 31, 333–338 (1945)
				" href="/article/10.1365/s13291-018-0181-1#ref-CR35" id="ref-link-section-d12700e16322">35</a>] had defined this topology on the set of prime ideals several years earlier.</p></div></li><li class="c-article-footnote--listed__item" id="Fn5" data-counter="5."><div class="c-article-footnote--listed__content"><p>The term <i>scheme</i> was used earlier by Chevalley [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="
					Chevalley, C.: Les schémas. Séminaire Henri Cartan 8(5), 1–6 (1955–1956)
				" href="/article/10.1365/s13291-018-0181-1#ref-CR13" id="ref-link-section-d12700e26692">13</a>] in a more restrictive sense. See also [EGA I] 8.3.</p></div></li><li class="c-article-footnote--listed__item" id="Fn6" data-counter="6."><div class="c-article-footnote--listed__content"><p>What we call a <i>scheme</i> here was called a <i>prescheme</i> in EGA I–IV. A <i>scheme</i> was by definition a <i>separated prescheme</i>. In the new edition [EGA I<sub>n</sub>] of EGA I, the terminology was changed, and the definition given here is the one which is universally used nowadays.</p></div></li><li class="c-article-footnote--listed__item" id="Fn7" data-counter="7."><div class="c-article-footnote--listed__content"><p>The original [EGA I] Déf 3.2.1 only has the definition specialized to the category of schemes.</p></div></li><li class="c-article-footnote--listed__item" id="Fn8" data-counter="8."><div class="c-article-footnote--listed__content"><p>If <span class="mathjax-tex">$S$</span> is not necessarily affine, we can still define projective space over <span class="mathjax-tex">$S$</span> as the fiber product <span class="mathjax-tex">$\mathbb {P}^{n}_{S}:= \mathbb {P}^{n}_{\operatorname {Spec}\mathbb {Z}}\times _{\operatorname {Spec}\mathbb {Z}} S$</span>. Then there is a similar notion of projective <span class="mathjax-tex">$S$</span>-scheme, but the situation is a little more subtle and one finds slightly differing variants of this notion in the literature.</p></div></li><li class="c-article-footnote--listed__item" id="Fn9" data-counter="9."><div class="c-article-footnote--listed__content"><p>Although all the ingredients are in the quoted book, at least over ℂ, it seems that the result in the form given here is not explicitly stated. See the discussion [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title="
					Overflow, M.: Existence of fine moduli space for curves and elliptic curves. 
 https://mathoverflow.net/a/11282
 
 
				" href="/article/10.1365/s13291-018-0181-1#ref-CR50" id="ref-link-section-d12700e50796">50</a>] for a sketch of how to conclude.</p></div></li><li class="c-article-footnote--listed__item" id="Fn10" data-counter="10."><div class="c-article-footnote--listed__content"><p>The category of sheaves of abelian groups on a topological space is abelian, so kernels and images exist and we obtain a notion of exact sequence. More explicitly, a sequence of sheaves on a space <span class="mathjax-tex">$X$</span> is exact if and only if for every <span class="mathjax-tex">$x\in X$</span> the induced sequence of stalks is exact.</p></div></li><li class="c-article-footnote--listed__item" id="Fn11" data-counter="11."><div class="c-article-footnote--listed__content"><p>This is not to say that the proof is given in EGA—examples and applications of the theoretical machinery are almost non-existent there.</p></div></li><li class="c-article-footnote--listed__item" id="Fn12" data-counter="12."><div class="c-article-footnote--listed__content"><p>The “incomplete” refers to the fact that we omit the primes where <span class="mathjax-tex">$\mathscr {E}_{p}$</span> is not an elliptic curve from the discussion; incorporating them in a suitable way leads to a more elegant theory. Even for the incomplete <span class="mathjax-tex">$L$</span>-function, our discussion is a little over-simplified: To be consistent with the usual terminology, one should be a little more careful with the choice of the equation for ℰ over ℤ. Nevertheless, our definition is good enough for the needs of the following discussion. Generalizations of this construction are available for elliptic curves over general number fields.</p></div></li></ol></div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1"> Aholt, C., Sturmfels, B., Thomas, R.: A Hilbert scheme in computer vision. Can. J. 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Görtz, U. Classics Revisited: Éléments de Géométrie Algébrique. <i>Jahresber. Dtsch. Math. 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