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23-524: His father, Abel Chevalley, was a French diplomat who, jointly with his wife Marguerite Chevalley née Sabatier , wrote The Concise Oxford French Dictionary . Chevalley graduated from the École Normale Supérieure in 1929, where he studied under Émile Picard . He then spent time at the University of Hamburg , studying under Emil Artin and at the University of Marburg , studying under Helmut Hasse . In Germany, Chevalley discovered Japanese mathematics in

46-510: A dense open subset of its closure. Terminology: The definition given here is the one used by the first edition of EGA and the Stacks Project . In the second edition of EGA constructible sets (according to the definition above) are called "globally constructible" while the word "constructible" is reserved for what are called locally constructible above. A major reason for the importance of constructible sets in algebraic geometry

69-422: A class of subsets of a topological space that have a relatively "simple" structure. They are used particularly in algebraic geometry and related fields. A key result known as Chevalley's theorem in algebraic geometry shows that the image of a constructible set is constructible for an important class of mappings (more specifically morphisms ) of algebraic varieties (or more generally schemes ). In addition,

92-413: A large number of "local" geometric properties of schemes, morphisms and sheaves are (locally) constructible. Constructible sets also feature in the definition of various types of constructible sheaves in algebraic geometry and intersection cohomology . A simple definition, adequate in many situations, is that a constructible set is a finite union of locally closed sets . (A set is locally closed if it

115-400: A locally constructible subset. EGA IV § 9 covers a large number of such properties. Below are some examples (where all references point to EGA IV): One important role that these constructibility results have is that in most cases assuming the morphisms in questions are also flat it follows that the properties in question in fact hold in an open subset. A substantial number of such results

138-428: A use of L-functions and replacing it by an algebraic method. At that time use of group cohomology was implicit, cloaked by the language of central simple algebras . In the introduction to André Weil 's Basic Number Theory , Weil attributed the book's adoption of that path to an unpublished manuscript by Chevalley. Around 1950, Chevalley wrote a three-volume treatment of Lie groups . A few years later, he published

161-612: Is compact for every compact open subset U ⊂ X {\displaystyle U\subset X} . A subset of X {\displaystyle X} is constructible if it is a finite union of subsets of the form U ∩ ( X − V ) {\displaystyle U\cap (X-V)} where both U {\displaystyle U} and V {\displaystyle V} are open and retrocompact subsets of X {\displaystyle X} . A subset Z ⊂ X {\displaystyle Z\subset X}

184-510: Is locally constructible if there is a cover ( U i ) i ∈ I {\displaystyle (U_{i})_{i\in I}} of X {\displaystyle X} consisting of open subsets with the property that each Z ∩ U i {\displaystyle Z\cap U_{i}} is a constructible subset of U i {\displaystyle U_{i}} . Equivalently

207-547: Is also locally constructible in Y {\displaystyle Y} . In particular, the image of an algebraic variety need not be a variety, but is (under the assumptions) always a constructible set. For example, the map A 2 → A 2 {\displaystyle \mathbf {A} ^{2}\rightarrow \mathbf {A} ^{2}} that sends ( x , y ) {\displaystyle (x,y)} to ( x , x y ) {\displaystyle (x,xy)} has image

230-476: Is that the image of a (locally) constructible set is also (locally) constructible for a large class of maps (or "morphisms"). The key result is: Chevalley's theorem. If f : X → Y {\displaystyle f:X\to Y} is a finitely presented morphism of schemes and Z ⊂ X {\displaystyle Z\subset X} is a locally constructible subset, then f ( Z ) {\displaystyle f(Z)}

253-423: Is the intersection of an open set and closed set .) However, a modification and another slightly weaker definition are needed to have definitions that behave better with "large" spaces: Definitions: A subset Z {\displaystyle Z} of a topological space X {\displaystyle X} is called retrocompact if Z ∩ U {\displaystyle Z\cap U}

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276-652: The Séminaire Cartan–Chevalley of the academic year 1955-6, with Henri Cartan and the Séminaire Chevalley of 1956-7 and 1957-8. These dealt with topics on algebraic groups and the foundations of algebraic geometry, as well as pure abstract algebra . The Cartan–Chevalley seminar was the genesis of scheme theory , but its subsequent development in the hands of Alexander Grothendieck was so rapid, thorough and inclusive that its historical tracks can appear well covered. Grothendieck's work subsumed

299-584: The Boolean algebra generated by retrocompact open subsets. In a locally noetherian topological space , all subsets are retrocompact, and so for such spaces the simplified definition given first above is equivalent to the more elaborate one. Most of the commonly met schemes in algebraic geometry (including all algebraic varieties ) are locally Noetherian, but there are important constructions that lead to more general schemes. In any (not necessarily noetherian ) topological space, every constructible set contains

322-591: The U.S., Chevalley became an American citizen and wrote a substantial part of his lifetime's output in English. When Chevalley applied for a chair at the Sorbonne , the difficulties he encountered were the subject of a polemical piece by his friend and fellow Bourbakiste André Weil , titled "Science Française?" and published in the Nouvelle Revue Française . Chevalley was the "professeur B" of

345-427: The co-editor of Chevalley's collected works attests to these interests: "Chevalley was a member of various avant-garde groups, both in politics and in the arts... Mathematics was the most important part of his life, but he did not draw any boundary between his mathematics and the rest of his life." In his PhD thesis, Chevalley made an important contribution to the technical development of class field theory , removing

368-454: The constructible subsets of a topological space X {\displaystyle X} are the smallest collection C {\displaystyle {\mathfrak {C}}} of subsets of X {\displaystyle X} that (i) contains all open retrocompact subsets and (ii) contains all complements and finite unions (and hence also finite intersections) of sets in it. In other words, constructible sets are precisely

391-618: The more specialised contribution of Serre , Chevalley, Gorō Shimura and others such as Erich Kähler and Masayoshi Nagata . Louis Auguste Sabatier Too Many Requests If you report this error to the Wikimedia System Administrators, please include the details below. Request from 172.68.168.226 via cp1108 cp1108, Varnish XID 250145715 Upstream caches: cp1108 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 10:57:23 GMT Constructible set (topology) In topology , constructible sets are

414-503: The person of Shokichi Iyanaga . Chevalley was awarded a doctorate in 1933 from the University of Paris for a thesis on class field theory . When World War II broke out, Chevalley was at Princeton University . After reporting to the French Embassy, he stayed in the U.S., first at Princeton and then (after 1947) at Columbia University . His American students included Leon Ehrenpreis and Gerhard Hochschild . During his time in

437-572: The piece, as confirmed in the endnote to the reprint in Weil's collected works, Oeuvres Scientifiques, tome II . Chevalley eventually did obtain a position in 1957 at the faculty of sciences of the University of Paris and after 1970 at the Université de Paris VII . Chevalley had artistic and political interests, and was a minor member of the French non-conformists of the 1930s . The following quote by

460-512: The set { x ≠ 0 } ∪ { x = y = 0 } {\displaystyle \{x\neq 0\}\cup \{x=y=0\}} , which is not a variety, but is constructible. Chevalley's theorem in the generality stated above would fail if the simplified definition of constructible sets (without restricting to retrocompact open sets in the definition) were used. A large number of "local" properties of morphisms of schemes and quasicoherent sheaves on schemes hold true over

483-566: The solubility of equations over a finite field. Another theorem of his concerns the constructible sets in algebraic geometry , i.e. those in the Boolean algebra generated by the Zariski-open and Zariski-closed sets. It states that the image of such a set by a morphism of algebraic varieties is of the same type. Logicians call this an elimination of quantifiers . In the 1950s, Chevalley led some Paris seminars of major importance:

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506-473: The work for which he is best remembered, his investigation into what are now called Chevalley groups . Chevalley groups make up 9 of the 18 families of finite simple groups . Chevalley's accurate discussion of integrality conditions in the Lie algebras of semisimple groups enabled abstracting their theory from the real and complex fields . As a consequence, analogues over finite fields could be defined. This

529-574: Was an essential stage in the evolving classification of finite simple groups . After Chevalley's work, the distinction between "classical groups" falling into the Dynkin diagram classification, and sporadic groups which did not, became sharp enough to be useful. What are called 'twisted' groups of the classical families could be fitted into the picture. "Chevalley's theorem" (also called the Chevalley–Warning theorem ) usually refers to his result on

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