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30-524: Corps Locaux by Jean-Pierre Serre , originally published in 1962 and translated into English as Local Fields by Marvin Jay Greenberg in 1979, is a seminal graduate-level algebraic number theory text covering local fields , ramification , group cohomology , and local class field theory . The book's end goal is to present local class field theory from the cohomological point of view. This theory concerns extensions of " local " (i.e., complete for

60-412: A discrete valuation ) fields with finite residue field . This article about a mathematical publication is a stub . You can help Misplaced Pages by expanding it . Jean-Pierre Serre Jean-Pierre Serre ( French: [sɛʁ] ; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology , algebraic geometry and algebraic number theory . He

90-467: A complex projective variety X and the category of coherent analytic sheaves on the corresponding analytic space X are equivalent. The analytic space X is obtained roughly by pulling back to X the complex structure from C through the coordinate charts. Indeed, phrasing the theorem in this manner is closer in spirit to Serre's paper, seeing how the full scheme-theoretic language that the above formal statement uses heavily had not yet been invented by

120-806: A finite étale map – are important. This acted as one important source of inspiration for Grothendieck to develop the étale topology and the corresponding theory of étale cohomology . These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique (SGA) 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures by Pierre Deligne . From 1959 onward Serre's interests turned towards group theory , number theory , in particular Galois representations and modular forms . Amongst his most original contributions were: his " Conjecture II " (still open) on Galois cohomology; his use of group actions on trees (with Hyman Bass );

150-466: A finitely generated projective module over a polynomial ring is free . This question led to a great deal of activity in commutative algebra , and was finally answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976. This result is now known as the Quillen–Suslin theorem . Serre, at twenty-seven in 1954, was and still is the youngest person ever to have been awarded

180-462: A fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures . Two major foundational papers by Serre were Faisceaux Algébriques Cohérents (FAC, 1955), on coherent cohomology , and Géométrie Algébrique et Géométrie Analytique ( GAGA , 1956). Even at an early stage in his work Serre had perceived

210-496: A need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field could not capture as much topology as singular cohomology with integer coefficients. Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties – those that become trivial after pullback by

240-453: A very young age he was an outstanding figure in the school of Henri Cartan , working on algebraic topology , several complex variables and then commutative algebra and algebraic geometry , where he introduced sheaf theory and homological algebra techniques. Serre's thesis concerned the Leray–Serre spectral sequence associated to a fibration . Together with Cartan, Serre established

270-471: Is closed (in the ordinary topological sense) is an algebraic subvariety. This can be rephrased as "any analytic subspace of complex projective space that is closed in the strong topology is closed in the Zariski topology ." This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry. Foundations for the many relations between the two theories were put in place during

300-477: Is coherent, a result known as the Oka coherence theorem , and also, it was proved in “Faisceaux Algebriques Coherents” that the structure sheaf of the algebraic variety O X {\displaystyle {\mathcal {O}}_{X}} is coherent. Another important statement is as follows: For any coherent sheaf F {\displaystyle {\mathcal {F}}} on an algebraic variety X

330-501: Is easy to prove that the analytic functions from the Riemann sphere to itself are either the rational functions or the identically infinity function (an extension of Liouville's theorem ). For if such a function f is nonconstant, then since the set of z where f(z) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f(z) equal to infinity. Consider

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360-896: The Fields Medal . He went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000, and was the first recipient of the Abel Prize in 2003. He has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre (Centre National de la Recherche Scientifique, CNRS). He is a foreign member of several scientific Academies (US, Norway, Sweden, Russia,

390-521: The Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C , which by Liouville's theorem is constant. Thus f is a rational function. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere . There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in

420-631: The Lefschetz principle , named for Solomon Lefschetz , was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field K of characteristic 0, by treating K as if it were the complex number field. An elementary form of it asserts that true statements of the first order theory of fields about C are true for any algebraically closed field K of characteristic zero. A precise principle and its proof are due to Alfred Tarski and are based in mathematical logic . This principle permits

450-403: The fundamental group of the complement of the ramification points . Since the Riemann surface property is local, such coverings are quite easily seen to be coverings in the complex-analytic sense. It is then possible to conclude that they come from covering maps of algebraic curves—that is, such coverings all come from finite extensions of the function field . In the twentieth century,

480-550: The Borel–Serre compactification; results on the number of points of curves over finite fields; Galois representations in ℓ-adic cohomology and the proof that these representations have often a "large" image; the concept of p-adic modular form ; and the Serre conjecture (now a theorem) on mod- p representations that made Fermat's Last Theorem a connected part of mainstream arithmetic geometry . In his paper FAC, Serre asked whether

510-455: The Legion of Honour (Grand Croix de la Légion d'Honneur) and Grand Cross of the Legion of Merit (Grand Croix de l'Ordre National du Mérite). A list of corrections , and updating, of these books can be found on his home page at Collège de France. GAGA Let X be a projective complex algebraic variety . Because X is a complex variety, its set of complex points X ( C ) can be given

540-670: The Royal Society, Royal Netherlands Academy of Arts and Sciences (1978), American Academy of Arts and Sciences , National Academy of Sciences , the American Philosophical Society ) and has received many honorary degrees (from Cambridge, Oxford, Harvard, Oslo and others). In 2012 he became a fellow of the American Mathematical Society . Serre has been awarded the highest honors in France as Grand Cross of

570-470: The algebraic variety X is equivalent to the category of analytic coherent sheaves on the analytic variety X , and the equivalence is given on objects by mapping F {\displaystyle {\mathcal {F}}} to F an {\displaystyle {\mathcal {F}}^{\text{an}}} . (Note in particular that O X an {\displaystyle {\mathcal {O}}_{X}^{\text{an}}} itself

600-411: The carrying over of some results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0. (e.g. Kodaira type vanishing theorem . ) Chow (1949) , proved by Wei-Liang Chow , is an example of the most immediately useful kind of comparison available. It states that an analytic subspace of complex projective space that

630-430: The common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions , algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way. For example, it

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660-455: The comparison of categories of sheaves. Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings. In slightly lesser generality, the GAGA theorem asserts that the category of coherent algebraic sheaves on

690-508: The early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory . The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Jean-Pierre Serre , now usually referred to as GAGA . It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these to

720-419: The homomorphisms are isomorphisms for all q' s. This means that the q -th cohomology group on X is isomorphic to the cohomology group on X . The theorem applies much more generally than stated above (see the formal statement below). It and its proof have many consequences, such as Chow's theorem , the Lefschetz principle and Kodaira vanishing theorem . Algebraic varieties are locally defined as

750-408: The natural homomorphism: is an isomorphism. Here O X {\displaystyle {\mathcal {O}}_{X}} is the structure sheaf of the algebraic variety X and O X an {\displaystyle {\mathcal {O}}_{X}^{\text{an}}} is the structure sheaf of the analytic variety X . More precisely, the category of coherent sheaves on

780-485: The nineteenth century. Some of the more important advances are listed here in chronological order. Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an (smooth projective) algebraic curve . Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of

810-640: The structure of a compact complex analytic space . This analytic space is denoted X . Similarly, if F {\displaystyle {\mathcal {F}}} is a sheaf on X , then there is a corresponding sheaf F an {\displaystyle {\mathcal {F}}^{\text{an}}} on X . This association of an analytic object to an algebraic one is a functor . The prototypical theorem relating X and X says that for any two coherent sheaves F {\displaystyle {\mathcal {F}}} and G {\displaystyle {\mathcal {G}}} on X ,

840-473: The technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres , which at that time was one of the major problems in topology. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, and also made the point that the award was for the first time awarded to a non-analyst. Serre subsequently changed his research focus. In the 1950s and 1960s,

870-712: Was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003. Born in Bages , Pyrénées-Orientales , to pharmacist parents, Serre was educated at the Lycée de Nîmes. Then he studied at the École Normale Supérieure in Paris from 1945 to 1948. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris . In 1956 he

900-629: Was elected professor at the Collège de France , a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil . The French mathematician Denis Serre is his nephew. He practices skiing, table tennis, and rock climbing (in Fontainebleau ). From

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