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Géométrie algébrique et géométrie analytique
Open accessAbstract
Toute variété algébrique X sur le corps des nombres complexes peut être munie, de façon canonique, d’une structure d’espace analytique ; tout faisceau algébrique cohérent sur X détermine un faisceau analytique cohérent. Lorsque X est une variété projective, nous montrons que, réciproquement, tout faisceau analytique cohérent sur X peut être obtenu ainsi, et de façon unique ; de plus, cette correspondance préserve les groupes de cohomologie. Ces résultats contiennent comme cas particuliers des théorèmes classiques de Chow et Lefschetz, et permettent d’aborder la comparaison entre espaces fibrés algébriques et espaces fibrés analytiques de base une variété algébrique projective.
Overview
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TL;DR
Serre proves that for a projective complex algebraic variety, coherent algebraic sheaves and coherent analytic sheaves correspond bijectively and have isomorphic cohomology. This GAGA equivalence yields algebraic-versus-analytic comparison results, including Chow’s theorem and algebraicity criteria for certain principal bundles.
Key findings
- For projective \(X\) over \(\mathbb C\) and any coherent algebraic sheaf \(\mathcal F\), the natural map \(H^q(X,\mathcal F)\to H^q(X^{\mathrm{an}},\mathcal F^{\mathrm{an}})\) is bijective for every \(q\ge 0\) (Theorem 1). Theorems 2 and 3 add that analytic morphisms between associated sheaves come from algebraic morphisms, and every coherent analytic sheaf comes from a coherent algebraic sheaf, uniquely up to isomorphism.
- Applications include invariance of Betti numbers of nonsingular varieties under automorphisms of \(\mathbb C\) (Proposition 12), algebraicity of compact analytic subsets of algebraic varieties (Proposition 14), regularity of holomorphic maps from compact algebraic varieties to algebraic varieties (Proposition 15), and uniqueness of an algebraic structure on a compact analytic space (corollary).
- For principal bundles, the algebraic-to-analytic comparison map is injective for compact \(X\) (Proposition 16) and bijective for the additive group \(\mathbb C\) (Proposition 17), for \(\mathrm{GL}_n(\mathbb C)\) (Proposition 18), and for groups satisfying a rational-section condition (R), including solvable groups, \(\mathrm{SL}_n(\mathbb C)\), and \(\mathrm{Sp}_{2m}(\mathbb C)\) (Proposition 20 examples).
- Locally, the algebraic local ring and the ring of holomorphic germs at a point have the same completion and form a flat couple; consequently they have the same dimension (Proposition 3, Corollaries 1–2).
Methods
- The paper is a mathematical proof study: it defines analytic spaces following H. Cartan, constructs the analytic space \(X^{\mathrm{an}}\) associated to a complex algebraic variety, and compares local rings and sheaves on \(X\) and \(X^{\mathrm{an}}\).
- The main arguments use coherent algebraic sheaf theory, Cartan’s Theorems A and B, exact sequences of sheaves, the five lemma, induction on cohomological degree and dimension, and Dolbeault theory for Betti-number applications.
- There is no empirical sample or dataset; the analysis is local and global sheaf cohomology together with commutative algebra of local rings, completions, and flat couples.
Limitations
- The author states that the fiber-space applications are incomplete: among semisimple groups, only the unimodular linear group and the symplectic group are treated, and applications to automorphic functions are left aside.
- The author notes uncertainty about whether \(X\) and its transform \(X^\sigma\) are always homeomorphic; a genus-1 curve example shows they are not always analytically isomorphic. The rational-section condition (R) fails for \(\mathrm{O}_n(\mathbb C)\) when \(n\ge 3\), and bijectivity in that case is unknown.
- Evident from the text: the main equivalence theorems require projectivity and fail even for affine varieties; several applications also assume nonsingularity or compactness.
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In the paper
“COROLLAIRE 1. -Le couple £ anneaux (Q^ 3€^) est un couple plat”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
What the sources say
- Not matchedcites Annexe, déf.4
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In the paper
“LEMME 3. -Soit A un anneau commutatif, dans lequel Vidéal 0 soit intersection Sun nombre fini d^idéaux premiers minimaux distincts p;; soit K^ le corps des fractions de A/p^, et soit S V ensemble des éléments de A qui ^appartiennent à aucun des pc U anneau de fractions As est alors isomorphe au composé direct des K,.”
§ ANNALES DE L'INSTITUT FOURIER > As c As.
What the sources say
- Not matchedcites [15] [15] P. Samuel. Commutative Algebra (Notes by D. Herzig). Cornell Univ., 1953.
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In the paper
“La topologie de X^ est plus fine que la topologie de X”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
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In the paper
“b) 5i Y est un sous-espace analytique fermé de X, le faisceau .^)(Y)est un faisceau analytique cohérent”
§ ANNALES DE L'INSTITUT FOURIER > 3. Faisceaux analytiques.
What the sources say
- Not matchedcites [1] [1] H. Cartan. Idéaux et modules de fonctions analytiques de variables complexes. Bull. Soc. Math. France, 78, 1950, pp. 29-64.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [2] [2] H. Cartan. Séminaire E. N. S., 1951-1952.
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In the paper
“COROLLAIRE 3. -Si X est une variété algébrique irréductible de dimension r, l'espace analytique X^ est de dimension analytique r en chacun de ses points”
§ ANNALES DE L'INSTITUT FOURIER > COROLLAIRE 2. -Les anneaux Oy; et Mp ont même dimension.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“COROLLAIRE 2. -Les anneaux Oy; et Mp ont même dimension.”
§ ANNALES DE L'INSTITUT FOURIER > COROLLAIRE 2. -Les anneaux Oy; et Mp ont même dimension.
What the sources say
- Not matchedcites [16] [16] P. Samuel. Algèbre locale. Mém. Sci. Math., 123, Paris, 1953.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“la correspondance entre ces deux catégories de faisceaux laisse invariants les groupes de cohomologie”
§ ANNALES DE L'INSTITUT FOURIER > INTRODUCTION
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In the paper
“II existe sur X une structure d'espace analytique et une seule telle que, pour toute carte ç : V ->-U, Fensemble Z-owert V soit ouvert, et y soit un isomorphisme analytique de V”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Rests on an assumption: A polynomial is continuous for the usual topology and holomorphic.
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In the paper
“PROPOSITION 5. -Soient X une variété algébrique^ et U une partie de X. Si U est ï-owerte et ï-dense dans X, alors U est dense dans X.”
§ ANNALES DE L'INSTITUT FOURIER > COROLLAIRE 2. -Les anneaux Oy; et Mp ont même dimension.
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In the paper
“PROPOSITION 6. ^-Pour qu'une variété algébrique X soit complète, il faut et il suffit quelle soit compacte.”
§ ANNALES DE L'INSTITUT FOURIER > PROPOSITION 6. ^-Pour qu'une variété algébrique X soit complète, il faut et il suffit quelle soit compacte.
What the sources say
- Not matchedcites [7] [7] W-L. Chow. On the projective embedding of homogeneous varieties. Lefschetz's volume, Princeton, 1956.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [19] [19] J.-P. Serre. Sur la cohomologie des variétés algébriques. J. de Maths. Pures et Appl., 35, 1956.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Rests on an assumption: Every projective algebraic variety is compact in the usual topology.
- Not matched
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In the paper
“PROPOSITION 9. -Soient T et X deux variétés algébriques^ et soit p : T --X une application régulière bijective.Si p est un isomorphisme analytique de T sur X, cest aussi un isomorphisme birégulier.”
§ ANNALES DE L'INSTITUT FOURIER > PROPOSITION 6. ^-Pour qu'une variété algébrique X soit complète, il faut et il suffit quelle soit compacte.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Rests on an assumption: The fraction field of the local ring of an irreducible component of an algebraic variety at a point is the field of rational functions on that component.
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In the paper
“Si X et Y sont deux variétés algébriques, on a (X x Y)^ X^1 x Y 71 .”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIn the paper
“Le théorème 2 revient donc à affirmer que io£ est bijectif.Or £ est bijectif d'après le théorème 1 (qui est applicable parce que A) est cohérent, d'après [18], n° 14), et i est bijectif d'après le lemme 6, c.q.f.d.”
§ ANNALES DE L'INSTITUT FOURIER > On a alors :
What the sources say
- Not matchedcites [18] [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, pp. 197-278.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“la dimension (analytiquec'est-à-dire la moitié de la dimension topologique) de X en x est la borne supérieure des dimensions des X;.On observera que cette dimension coïncide avec la dimension (au sens de Krull) de l'anneau local W)^”
§ ANNALES DE L'INSTITUT FOURIER > 4. Voisinage d'un point dans un espace analytique.
What the sources say
- Not matchedcites [14] [14] W. Rückert. Zum Eliminationsproblem der Potenzreihenideale. Math. Ann., 107, 1933, pp. 259-281.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [3] [3] H. Cartan. Séminaire E. N. S., 1953-1954.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [16] [16] P. Samuel. Algèbre locale. Mém. Sci. Math., 123, Paris, 1953.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“THÉORÈME 2 . -Si S et Ç sont deux faisceaux algébriques cohérents sur X, tout homomorphisme analytique de y dans (V 1 provient d'un homomorphisme algébrique de 9 dans fî, et d'un seul.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
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In the paper
“PROPOSITION 26.-Pour tout A-module de type fini E, Vhomomorphisme £ défini ci-dessus est bijectif.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
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In the paper
“PROPOSITION 16. -Si X est compacte^ Inapplication £ est injective.”
§ ANNALES DE L'INSTITUT FOURIER > 20. Espaces fibres algébriques et espaces fibres analytiques.
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“PROPOSITION 12. -Si X est non singulière, les nombres de Betti de X et de X e7 sont les mêmes.”
§ ANNALES DE L'INSTITUT FOURIER > 18. Caractère algébrique des nombres de Betti.
What the sources say
- Not matchedcites [8] [8] P. Dolbeault. Sur la cohomologie des variétés analytiques complexes. C. R., 236, 1953, pp. 175-177.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“La proposition précédente montre en particulier que, si <1 est un faisceau d'idéaux de 0, le faisceau ^h n'est autre que le faisceau d'idéaux de 3€ engendré par les éléments de 3.”
§ ANNALES DE L'INSTITUT FOURIER > As c As.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“PROPOSITION 1. -a) Le faisceau 3€^ est un faisceau cohérent d'anneaux ([18], n° 15).”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
What the sources say
- Not matchedcites [18] [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, pp. 197-278.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“b) Pour tout faisceau algébrique ^, Vhomomorphisme a : 9 -^^h est injectif.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
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In the paper
“PROPOSITION 7. -Si f: X ->-Y est une application régulière d'une variété algébrique X dans une variété algébrique Y, l'adhérence et la Z-adhérence de /*(X) dans Y coïncident.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
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“LEMME 6. -Uhomomorphisme i : JV 1 -^ 3^ est bijectif.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
What the sources say
- Not matchedcites [18] [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, pp. 197-278.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“LEMME 5. -Le théorème 1 est vrai pour le faisceau 0(n).”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
What the sources say
- Not matchedcites [18] [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, pp. 197-278.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“la prop.3 du n° 6 nous permet d'identifier le faisceau 0' ainsi obtenu à un sous-faisceau du faisceau X des germes de fonctions holomorphes sur X”
§ ANNALES DE L'INSTITUT FOURIER > As c As.
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In the paper
“Mais l'idéal 3,c(Y) est intersection d'idéaux premiers, qui correspondent aux composantes irréductibles de Y passant par rr.”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“On voit facilement que la connaissance de 3f^ détermine X au voisinage de x”
§ ANNALES DE L'INSTITUT FOURIER > 4. Voisinage d'un point dans un espace analytique.
What the sources say
- Not matchedcites [3] [3] H. Cartan. Séminaire E. N. S., 1953-1954.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“c) Si S est un faisceau algébrique cohérent, ^ est un faisceau analytique cohérent.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
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In the paper
“PROPOSITION 18. -5i G est le groupe linéaire général GL/,(C), l'application £ est bijective.”
§ ANNALES DE L'INSTITUT FOURIER > 20. Espaces fibres algébriques et espaces fibres analytiques.
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In the paper
“PROPOSITION 4. -L'idéal A^(Y) est engendré par 0(<lc(Y)).”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
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In the paper
“PROPOSITION 10. -a) Le foncteur ^h est un foncteur exact.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“si alors 9 est algébrique cohérent, et si x est un point de X, on peut trouver une suite exacte : 0<?-^0P-^g?-^0,valable dans un Z-voisinage U de x.D'après a), on en déduit une suite exacte : 3^_^^p_^_^o, valable sur U. Comme U est un voisinage de x, et que le faisceau d'anneaux 3€ est cohérent (prop.1, n° 3), ceci montre bien que ^ est cohérent”
§ ANNALES DE L'INSTITUT FOURIER > As c As.
What the sources say
- Not matchedcites [18] [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, pp. 197-278.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“En fait, nous allons démontrer un résultat un peu plus précis, à savoir que tout isomorphisme analytique © : E -'-E / est un isomorphisme algébrique (c'est-à-dire régulier).”
§ ANNALES DE L'INSTITUT FOURIER > 20. Espaces fibres algébriques et espaces fibres analytiques.
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In the paper
“PROPOSITION 8 . -Soient X et Y deux variétés algébriques, et soit f: X-^ Y une application holomorphe de X dans Y. Si le graphe T de f est un sous-ensemble ï-localement fermé [i.e.une sous-variété algébrique) de X X Y, Inapplication f est régulière.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
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In the paper
“LEMME 7 . -Soit E un hyperplan de Pr(C), et soit A) un faisceau analytique cohérent sur E. On a H^E^, Jfc(M)) = 0 pour q > 0 et n assez grand.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
What the sources say
- Not matchedcites [3] [3] H. Cartan. Séminaire E. N. S., 1953-1954.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [8] [8] P. Dolbeault. Sur la cohomologie des variétés analytiques complexes. C. R., 236, 1953, pp. 175-177.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Not matched
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In the paper
“Comme toute fonction régulière est holomorphe”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Reste à démontrer l'existence de <?.Je dis que l'on peut se borner au cas où X est un espace projectifP^C).”
- Also stated in: § On a alors : · plain
“Si l'on suppose le théorème 3 démontré pour l'espace X, il existe donc un faisceau algébrique cohérent Ê sur X tel que fi^ soit isomorphe à M)^.”
§ ANNALES DE L'INSTITUT FOURIER > On a alors :
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In the paper
“La proposition 3 montre en particulier que 0: 0^ -^ 3-6^ est injectif”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“nous savons que 0: Ô^u -^ 3€^u est bijectif”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“LEMME 2 . -Soit f: X ->-Y une application régulière d'une variété algébrique X dans une variété algébrique Y, et supposons que jf(X) soit Z-dense dans Y. Il existe alors une partie U c /'(X) qui est Z-ouverte et Z-dense dans Y.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
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In the paper
“LEMME 4. -Le théorème 1 est vrai pour le faisceau 0.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
What the sources say
- Not matchedcites [18] [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, pp. 197-278.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [8] [8] P. Dolbeault. Sur la cohomologie des variétés analytiques complexes. C. R., 236, 1953, pp. 175-177.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“PROPOSITION 29. -Soient A et B deux anneaux locaux, soit a un idéal de A, et soit 0 un homomorphisme de A dans B. Si 0 vérifie Vhypothèse de la proposition 28, il en est de même de r homomorphisme de A/û dans B/0(a)B défini par 0 {ce qui montre que le couple (A/a, B/0(a)B) est un couple plat).”
§ ANNALES DE L'INSTITUT FOURIER > References
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesNone of the sources cited for this claim have been checked yet.
In the paper
“PROPOSITION 27. -Soit A un anneau locale et soit A son complété. Le couple (A, A) est plat.”
§ ANNALES DE L'INSTITUT FOURIER > References
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“COROLLAIRE.-Soit E un A-module de type fini.Si un sousmodule F de E vérifie la relation E = F + mE, on a E = F.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“PROPOSITION 17 . -5i G est le groupe additif C, Inapplication £ est bijective.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear? Based: unclearThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“PROPOSITION 21. -U homomorphisme i défini ci-dessus est bijectif lorsque A est un anneau noethérien, E est un Amodule de type fini, et B est A-plat.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“PROPOSITION 24. -Si un A.-module de type fini E vérifie la relation E = niE, on a E = 0.”
§ ANNALES DE L'INSTITUT FOURIER > References
What the sources say
- Not matchedcites [15] [15] P. Samuel. Commutative Algebra (Notes by D. Herzig). Cornell Univ., 1953.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passes- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“PROPOSITION 30. -Soient A et A 7 deux anneaux locaux, soit 0 un homomorphisme de A dans A 7 vérifiant l'hypothèse de la proposition 28, et soit E un A-module de type fini.”
§ ANNALES DE L'INSTITUT FOURIER > References
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“PROPOSITION 28. -Supposons que 0 : A -^ B soit bifectif, et identifions A à un sous-anneau de B au moyen de 0. Le couple (A, B) est alors un couple plat.”
§ ANNALES DE L'INSTITUT FOURIER > References
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“COROLLAIRE. -Tout espace analytique compact possède au plus une structure de variété algébrique.”
§ ANNALES DE L'INSTITUT FOURIER > 18. Caractère algébrique des nombres de Betti.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“COROLLAIRE.-Soit V une variété projectile, non singulière, définie sur un corps de nombres algébriques K. Les variétés complexes X obtenues à partir de V en plongeant K dans G ont des nombres de Betti indépendants du plongement choisi”
§ ANNALES DE L'INSTITUT FOURIER > 18. Caractère algébrique des nombres de Betti.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“PROPOSITION 13. -Tout sous-ense.nble analytique fermé de V espace projectif est algébrique.”
§ ANNALES DE L'INSTITUT FOURIER > References
What the sources say
- Not matchedcites [6] [6] W-L. Chow. On compact complex analytic varieties. Amer. J. of Maths., 71, 1949, pp. 893-914.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“PROPOSITION 15. -Toute application holomorphe f Sune variété algébrique compacte X dans une variété algébrique Y est régulière.”
§ ANNALES DE L'INSTITUT FOURIER > References
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“PROPOSITION 20.-Soit G un sous-groupe algébrique du groupe GL^(C) vérifiant la condition suivante:(R) -II existe une section rationnelle GL^(C)/G-^GL,,(C).Alors, pour toute variété projectile X, l'application : ^(X.^H^X^) est bijective.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“a) lorsque G est résoluble, en vertu d'un théorème de Rosenlicht, [13];”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
What the sources say
- Not matchedcites [13] [13] M. Rosenlicht. Some basic theorems on algebraic groups. Amer J. of Maths., 78, 1956, pp. 401-443.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“b) lorsque G == SL^(C), la section rationnelle étant alors évidente ;”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“PROPOSITION 19 . -Soit X une variété algébrique compacte, et soit P un espace fibre principal analytique^ de groupe structural H, et de base X.Pour que P soit algébrique, il faut et il suffit quil en soit ainsi de l'espace fibre P XH G déduit de P en étendant le groupe structural de H à G.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“PROPOSITION 14. -Si X est une variété algébrique^ tout sous- ensemble analytique compact X 7 de X est algébrique.”
§ ANNALES DE L'INSTITUT FOURIER > References
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes
- It may state two separate claims as one. 7
- The quote only partly states this claim. 4
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“Pour n = 1, GL^(C) coïncide avec le groupe multiplicatif C*; si l'on suppose que X est une variété normale, le groupe H 1 (X, fî) coïncide avec le groupe des classes de diviseurs localement linéairement équivalents à zéro (cf.[20], § 3) et la proposition 18 signifie que tout espace fibre analytique de base X et de groupe structural C* provient d'un tel diviseur.”
§ ANNALES DE L'INSTITUT FOURIER > 20. Espaces fibres algébriques et espaces fibres analytiques.
What the sources say
- Not matchedcites [20] [20] A. Weil. Fibre-spaces in algebraic geometry (Notes by A. Wallace). Chicago Univ., 1952.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Partly supportedcites [12] [12] K. Kodaira and D. C. Spencer. Divisor class groups on algebraic varieties. Proc. Nat. Acad. Sci. U. S. A., 39, 1953, pp. 872-877. sourceExperimental — accuracy not yet measured
In the opening discussion and Theorem 1 (pages not supplied), the paper identifies divisor classes with complex line bundles and proves that every such bundle comes from a divisor. Its stated setting is a non-singular projective algebraic variety, however; it does not establish the cited identification of H¹(X,O*) with those divisor classes for an arbitrary normal X. The statement about Proposition 18 is the citing author's own interpretation, not a result attributed to this source. (LLM, conf=0.84)
Tests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“X^ est un espace localement compact dénombrable à Vinfini.”
§ ANNALES DE L'INSTITUT FOURIER > 5. Définition de l'espace analytique associé à une variété algébrique.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“faisceaux algébriques cohérents et faisceaux analytiques cohérents se correspondent biunivoquement”
§ ANNALES DE L'INSTITUT FOURIER > INTRODUCTION
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Le faisceau 3^ est un faisceau de 3-6-modules, c'est-à-dire un faisceau analytique', l'injection 0'-^ 3^ définit un homomorphisme canonique a : 9' ->-9^.”
§ ANNALES DE L'INSTITUT FOURIER > As c As.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“THÉORÈME 3 . -Pour tout faisceau analytique cohérent jll» sur X 71 , il existe un faisceau algébrique cohérent ^ sur X tel que ^ soit isomorphe à JIL.De plus, cette propriété détermine 9 de façon unique, à un isomorphisme près.”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Pour tout a;çX, on a donc 3^ == 3^; les faisceaux 9 et y ne diffèrent que par leur topologie (9' n'est pas autre chose que V image réciproque de 9 par l'application continue X^X).”
§ ANNALES DE L'INSTITUT FOURIER > As c As.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“En vertu du lemme 8, il existe un entier n tel que M)(n) soit isomorphe à un faisceau quotient d'un faisceau 3^, et M) est donc isomorphe à un quotient de Wo (-n^.”
§ ANNALES DE L'INSTITUT FOURIER > 16. Démonstration du théorème 3. Les faisceaux ll(n).
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“Les deux faisceaux en question sont nuls en dehors de Y^; il nous suffira donc de montrer que leurs restrictions à Y 71 sont isomorphes.”
§ ANNALES DE L'INSTITUT FOURIER > 10. Prolongement d'un faisceau.
What the sources say
- Not matchedcites [18] [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, pp. 197-278.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe quote only partly states this claim.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Ces trois théorèmes signifient que la théorie des faisceaux analytiques cohérents sur X^ coïncide essentiellement avec celle des faisceaux algébriques cohérents sur X. Bien entendu, ils tiennent à ce que X est une variété projectile, et sont inexacts même pour une variété affine.”
§ ANNALES DE L'INSTITUT FOURIER > On a alors :
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“c) lorsque G == Sj^(C), n = 2m; dans ce cas, l'espace homogène GL^(C)/G est l'espace des formes alternées non dégénérées”
§ ANNALES DE L'INSTITUT FOURIER > £: E^JL^Ê.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe quote only partly states this claim.
The paper relies on this without support.
Every projective algebraic variety is compact in the usual topology.
“comme toute variété projective est compacte pour la topologie usuelle”
The paper relies on this without support.
The fraction field of the local ring of an irreducible component of an algebraic variety at a point is the field of rational functions on that component.
“le corps des quotients de A;, soit K(, n'est donc pas autre chose que le corps des fonctions rationnelles sur la variété irréductible X^.”
The paper relies on this without support.
A polynomial is continuous for the usual topology and holomorphic.
“comme un polynôme est continu pour la topologie usuelle (resp.holomorphe)”
Citations
Cites
Some Basic Theorems on Algebraic Groups
Rosenlicht, M. (1956) · American Journal of Mathematics · doi:10.2307/2372523 · Cited by 495
On Compact Complex Analytic Varieties
Chow, W.-L. (1949) · American Journal of Mathematics · doi:10.2307/2372375 · Cited by 132
Divisor class groups on algebraic varieties
Kodaira, K., & Spencer, D. C. (1953) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.39.8.872 · Cited by 39
Zum Eliminationsproblem der Potenzreihenideale
Rückert, W. (1932) · Springer eBooks · doi:10.1007/978-3-662-41142-1_1 · Cited by 23
Fibre-spaces in algebraic geometry
WEIL, A. A. (1979) · Springer eBooks · doi:10.1007/978-1-4757-1705-1_51 · Cited by 9
Flagged first, then by citations
Research works printed but not linked: 7
- [1] H. Cartan. Idéaux et modules de fonctions analytiques de variables complexes. Bull. Soc. Math. France, 78, 1950, pp. 29-64.
- [5] H. Cartan et J.-P. Serre. Un théorème de finitude concernant les variétés analytiques compactes. C. R., 237, 1953, pp. 128-130.
- [8] P. Dolbeault. Sur la cohomologie des variétés analytiques complexes. C. R., 236, 1953, pp. 175-177.
- [9] J. Frenkel. Cohomologie à valeurs dans un faisceau non abélien. C. R., 240, 1955, pp. 2368-2370.
- [11] K. Kodaira. On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties). Ann. of Maths., 60, 1954, pp. 28-48.
- [17] P. Samuel. Méthodes d'algèbre abstraite en géométrie algébrique. Ergebn. der Math., Springer, 1955.
- [18] J.-P. Serre. Faisceaux algébriques cohérents. Ann. of Maths., 61, 1955, pp. 197-278.
Cited by
Sorted by citations
The Arithmetic of Elliptic Curves
Silverman, J. H. (1986) · Graduate Texts in Mathematics · doi:10.1007/978-1-4757-1920-8 · Cited by 4,044
Grothendieck, A. (1964) · Publications mathématiques de l IHÉS · doi:10.1007/bf02684747 · Cited by 2,994
Deligne, P. (1974) · Publications mathématiques de l IHÉS · doi:10.1007/bf02685881 · Cited by 1,201
Space of Unitary Vector Bundles on a Compact Riemann Surface
Seshadri, C. S. (1967) · Annals of Mathematics · doi:10.2307/1970444 · Cited by 872
Variation of hodge structure: The singularities of the period mapping
Schmid, W. (1973) · Inventiones Mathematicae · doi:10.1007/bf01389674 · Cited by 817
Metadata
- Authors
- Serre, J.-P.
- Corresponding author
- Serre, Jean-Pierre
- ORCID
- Serre, Jean-Pierre
- Venue
- Annales de l'Institut Fourier
- Publisher
- MathDoc/Centre Mersenne
- Published
- 1956
- Volume
- 6
- Pages
- 1-42
- ISSN
- 0373-0956 (ISSN-L)
- 1777-5310
- OpenAlex
- W2092940386
- Cited by
- 645
- References
- 4
- Citations by year
- 2026: 6 · 2025: 12 · 2024: 15 · 2023: 22 · 2022: 19 · 2021: 62 · 2020: 28 · 2019: 26 · 2018: 27 · 2017: 20 · 2016: 16 · 2015: 9 · 2014: 15 · 2013: 10 · 2012: 12
- FWCI
- 6.67
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Annales de l’institut Fourier (publishedVersion)
- Annals of the Fourier Institute (Institut Fourier) (submittedVersion)
- French digital mathematics library (Numdam) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Algebraic Geometry and Number Theory
- Physical Sciences › Mathematics › Geometry and Topology › Geometry and complex manifolds
- Physical Sciences › Mathematics › Algebra and Number Theory › Advanced Algebra and Geometry
- Keywords
- analytic spaces, coherent sheaves, projective varieties, cohomology groups, Lefschetz theorem
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-09-28, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-09-28 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (gpt-6-sol).
Statistics and arithmetic are recomputed, not judged.
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