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Cited by 262Algebraic Geometry and Number TheoryHomotopy and Cohomology in Algebraic TopologyAdvanced Algebra and Geometry

The DOI matches its Crossref record; 1 of the 1 references it cites were found in a registry or an index; none of the 4 sources its claims cite is checked for support yet.

La théorie des classes de Chern

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Grothendieck, Alexander·1958·Bulletin de la Société mathématique de France
DOI 10.24033/bsmf.1501
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Abstract

L'accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath

Overview

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Written from the paper’s full text.

TL;DR

Grothendieck develops an axiomatic theory of Chern classes for algebraic vector bundles on nonsingular algebraic varieties, defining them in Chow rings via projective bundles and flag varieties. He proves that functoriality, normalization, and additivity characterize Chern classes, and connects the construction to K-theory, lambda-rings, and Riemann–Roch.

Key findings

  • Chern classes \(c_i(E)\in A^i(X)\) are defined for a rank \(p\) vector bundle \(E\) on a nonsingular quasi-projective variety \(X\) using the fundamental class in \(A(P(E))\) and the relation expressing its powers in terms of the \(c_i(E)\).
  • Theorem 1 shows the Chern classes satisfy functoriality \(c(f^{-1}E)=f^*c(E)\), normalization \(c(E)=1+p(\operatorname{cl}(E))\) for rank 1, and additivity \(c(E)=c(E')c(E'')\) for an exact sequence \(0\to E'\to E\to E''\to 0\); these three properties characterize the theory.
  • Theorem 2 states that for a regular section \(s\) of a rank \(p\) bundle \(E\), transversal to the zero section, the zero locus \(Y\) satisfies \(p_X(Y)=c_p(E)\); in Chow theory, if the zero cycle \(Z\) exists, its class is \(c_p(E)\).
  • The paper also introduces lambda-ring formalism and shows that the map from \(K(X)\) to \(A(X)\) is an isomorphism up to torsion, with applications to Riemann–Roch.

Methods

  • The paper is a theoretical, axiomatic construction: it fixes a category \(V\) of nonsingular algebraic spaces, a contravariant functor \(X\mapsto A(X)\) to graded anticommutative rings, a homomorphism \(P(X)\to A^1(X)\), and pushforward maps for closed subvarieties, subject to axioms A1–A4 and later A5 and condition V2.
  • Proofs use projective bundles, flag varieties, and the splitting principle: one reduces to completely split vector bundles, then uses Lemma 1, Lemma 2 and its corollary, Theorem 1, and Theorem 2.
  • No empirical data or sample are used; the work is purely algebraic-geometric.

Limitations

  • The paper states that for arbitrary nonsingular non-quasi-projective varieties, one cannot currently define a ring structure on \(A(X)\) or pullback morphisms satisfying the needed conditions, and it is not even certain this is possible; replacing \(A(X)\) by the graded ring associated to \(K(X)\) has compatibility only known in the quasi-projective case, with torsion difficulties disappearing after tensoring by \(\mathbb{Q}\).
  • The paper states that the first algebraic proof of Riemann–Roch is valid only in characteristic 0, that some results hold only up to torsion, and that the author does not know whether \(\varphi\) is an isomorphism, though it is so in degrees 1 and 2.
  • Evident from the text: the exposition is an appendix to the Borel–Serre memoir, some proofs are deferred (for example, formula (16) is not proved in form), and further questions are promised for later work; the OCR text is also fragmentary and garbled in places.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.24033/bsmf.1501

Cites

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Verification

Claims the paper makes

0 of 4 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § On peut aussi définir des applications (non additives !) ^: A->A (i^o)

    The kernel of the homomorphism φ from the Chow ring to GK(X) is a torsion group.

    “Nous allons montrer que le noyau de cet homomorphisme est un groupe de torsion.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § LA THÉORIE DES CLASSES DE CHERN

    Under the hypotheses of Lemma 2, the class p_X(Y_i) equals the product of the classes c_j for j=1,...,i.

    “Sous ces conditions, on a A-ro=n£,-l^'^/?Nous démontrerons par récurrence surJÇi^j^p) que (*) px(r,)=f[^. 1 ^ ;' ^ j”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: There is a category V of nonsingular algebraic spaces satisfying condition V1: if X is in V and E is a vector bundle on X, then the projective bundle P(E) is in V.
    • The authors state this as an assumption: There are data a, b, c: a contravariant functor A from V to graded anticommutative rings with unit; a functorial homomorphism p_X: P(X)->A^1(X); and, for each closed subvariety Y of X in V of constant codimension p, a homomorphism i_*: A(Y)->A(X) increasing degrees by p.
    • The authors state this as an assumption: Axiom A1 holds: for X in V and a rank-p vector bundle E on X, the elements 1, E_E, ..., E_E^{p-1} form a basis of A(P(E)) over A(X).
    • The authors state this as an assumption: Axiom A2 holds: for a rank-1 vector bundle L on X in V and a regular section s transversal to the zero section whose zero locus Y is in V, p_X(Y)=p_X(cl_X(L)).
  • Findingreasoning
    § Additional information

    Let X∈V, E a rank p vector bundle on X, and s a regular section of E transverse to the zero section; let Y be the zero set of s, a closed nonsingular subvariety of codimension p. Then p_*(Y) = c_p(E).

    “THÉORÈME 2. == Soient ^TçV, E un fibre vectoriel de rang p sur X^ s une section régulière de E transversale à la section nulle^ Y V ensemble des zéros de s (qui est donc un sous-espace fermé non singulier de codimension p de X).Sous ces conditions^ on a (18) p^(r)==c,(E).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The category V satisfies condition (V2): for every X in V and every closed nonsingular subspace Y of X, Y is in V.
  • Findingreasoning
    § LA THÉORIE DES CLASSES DE CHERN

    Under the same assumptions, the Chern classes satisfy normalization: if E is a rank-1 vector bundle on X in V, then c(E)=1+p_X(cl_X(E)).

    “(ii) Normalisation.-Si E est un fibre vectoriel de rang i surJTeV, alors on a (4) c(E)=:i+p^(clx(E)).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: There is a category V of nonsingular algebraic spaces satisfying condition V1: if X is in V and E is a vector bundle on X, then the projective bundle P(E) is in V.
    • The authors state this as an assumption: There are data a, b, c: a contravariant functor A from V to graded anticommutative rings with unit; a functorial homomorphism p_X: P(X)->A^1(X); and, for each closed subvariety Y of X in V of constant codimension p, a homomorphism i_*: A(Y)->A(X) increasing degrees by p.
    • The authors state this as an assumption: Axiom A1 holds: for X in V and a rank-p vector bundle E on X, the elements 1, E_E, ..., E_E^{p-1} form a basis of A(P(E)) over A(X).
    • The authors state this as an assumption: Axiom A2 holds: for a rank-1 vector bundle L on X in V and a regular section s transversal to the zero section whose zero locus Y is in V, p_X(Y)=p_X(cl_X(L)).
  • Findingnone given
    § Additional information

    The set s(X) is the zero set of a section transverse to the zero section of a suitable vector bundle on P(E), namely L^p ⊗ f^{-1}(E/S), where S is the trivial rank-one subbundle of E defined by s.

    “en notant que s ( ^) est l'ensemble des zéros d'une section (transversale à la section nulle) d'un fibre vectoriel convenable sur P(É}, savoir le fibre L^^f-^Ê/S), où S est le sous-fibre (trivial) de ran^ i de h défini par la section ,ç.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The category V satisfies condition (V2): for every X in V and every closed nonsingular subspace Y of X, Y is in V.
  • Probably not a substantial claim. 6
  • The quote only partly states this claim. 1
  • It may state two separate claims as one. 7

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Findingreasoning
    § 5. Le cycle des zéros (

    In all the preceding, calculations involved only even-degree elements in A(X), which implies they were essentially commutative.

    “1° Dans tout ce qui précède, on n'a eu à calculer qu'avec des éléments de degré pair dans les A(A)^ ce qui implique en particulier que tous les calculs étaient essentiellement commutatifs.”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § LA THÉORIE DES CLASSES DE CHERN

    Under the same assumptions, the Chern classes satisfy additivity: for an exact sequence 0->E'->E->E''->0 of vector bundles on X in V, c(E)=c(E')c(E'').

    “(iii) Additivité.-Soient ^TçV, o -> E 1 -> E -^ E”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: There is a category V of nonsingular algebraic spaces satisfying condition V1: if X is in V and E is a vector bundle on X, then the projective bundle P(E) is in V.
    • The authors state this as an assumption: There are data a, b, c: a contravariant functor A from V to graded anticommutative rings with unit; a functorial homomorphism p_X: P(X)->A^1(X); and, for each closed subvariety Y of X in V of constant codimension p, a homomorphism i_*: A(Y)->A(X) increasing degrees by p.
    • The authors state this as an assumption: Axiom A1 holds: for X in V and a rank-p vector bundle E on X, the elements 1, E_E, ..., E_E^{p-1} form a basis of A(P(E)) over A(X).
    • The authors state this as an assumption: Axiom A2 holds: for a rank-1 vector bundle L on X in V and a regular section s transversal to the zero section whose zero locus Y is in V, p_X(Y)=p_X(cl_X(L)).

  • It may state two separate claims as one.

    Findingreasoning
    § 5. Le cycle des zéros (

    In the complex case, conditions A1 to A4 are satisfied, with A1 following from the Leray spectral sequence of P(E)->X.

    “On vérifie encore sans difficulté les conditions Al à A ^, la condition Al étant une conséquence de la suite spectrale de Leray de l'application continue P{E) -^X [suite spectrale qui est triviale, comme il résulte de la considération de la classer sur P(jE).]”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Interpretationnone given
    § 5. Le cycle des zéros (

    The axioms A1 to A4 remain satisfied if each A(X) is replaced by the direct sum of its A^i(X).

    “On notera d'ailleurs que les axiomes A là A h-restent satisfaits si l'on remplace chaque A(^f) par la somme directe des A^Ç^Y).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Interpretationnone given
    § On peut aussi définir des applications (non additives !) ^: A->A (i^o)

    The λ-ring A is special: the operations λ^i on products and on λ^j(x) are given by universal integer polynomials in the appropriate λ-operations.

    “En fait, le À-anneau A est un À-anneau spéciale par quoi on entend que les //(.yj) peuvent s'exprimer par des polynômes universels à coefficients entiers en fonction des À 7 (.a”),À^j), et que les V (^'(.-r))peuvent s'exprimer par les polynômes universels à coefficients entiers en fonction des ^{x).”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Interpretationreasoning
    § phisme de K(A) sur A (A) à torsion près.

    For proving intersection formulas in A(X), if torsion is neglected, one can compute in GK(X), equivalently in A^*(X), reducing to alternating sums of Tor of sheaves; this is the method 'without moving cycles'.

    “Ce qui précède montre que pour prouver des formules d'intersection dans y4(^t), on peut, si l'on néglige la torsion, faire les calculs dans GK(^T), c'est-à-dire en définitive dans A^(JT), i. e. on est ramené à calculer des sommes alternées de Tor de faisceaux : c'est la méthode « sans faire bouger les cycles ».”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

    Premisereasoning
    § 5. Le cycle des zéros (

    In the complex case, the definitions of b and c are well known, and properties A1 to A4 are well known, with A2 following from the Leray spectral sequence.

    “siquement par dualité de Poincaré, et les propriétés Al à A k sont bien connues (A 2 résultant encore de la suite spectrale de Leray).”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Interpretationnone given
    § LA THÉORIE DES CLASSES DE CHERN

    At present one cannot define on the group A(X) of cycle classes of an arbitrary nonsingular variety (not necessarily quasi-projective) a ring structure and, for a morphism f, a pullback f*: A(Y)->A(X) satisfying the required conditions; it is not even certain that this is possible.

    “On ne peut pas pour l'instant définir dans le groupe A(/f) des classes de cycles sur une variété non singulière quelconque (non nécessairement quasi projective) une structure d'anneau, et pour un morphisme f ' .X-> Y un morphisme /* : A ( Y) -:>A(A')^ de façon à satisfaire aux conditions qu'il faut.Et il n'est pas même certain que cela soit possible.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationcitation
    § On peut aussi définir des applications (non additives !) ^: A->A (i^o)

    If X is quasi-projective, the filtration of K(X) is compatible with its ring structure, and the map Z↦y(Z) from the Chow group to K(X) is compatible with filtrations and induces a ring homomorphism φ from the Chow ring A(X) to the associated graded GK(X).

    “Utilisant ceci, et la proposition 8 du présent rapport de BOREL-SERRE, on trouve que si A est quasi projectile, la fdtration de K{A) est compatible avec sa structure d^anneau; l'homomorphisme Z-^y(Z) du groupe A(A) des classes de cycles dans K(A) est compatible avec les filtrations, et définit, par passage aux gradués associés, un homomorphisme c? d ) anneaux gradués de Vanneau de Chow A (A) sur V”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites BOREL-SERRE

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

    Premisenone given
    § LA THÉORIE DES CLASSES DE CHERN

    Condition V1 holds for all reasonable categories of algebraic spaces, in particular for all nonsingular algebraic spaces, for nonsingular quasi-projective algebraic spaces, and for nonsingular projective algebraic spaces.

    “Notons d'abord que la condition V 1 est satisfaite pour toutes les catégories raisonnables d'espaces algébriques, et en tous cas pour la catégorie de tous les espaces algébriques non singuliers, pour la catégorie des espaces algébriques non singuliers quasi projectifs, et pour la catégorie des espaces algébriques non singuliers projectifs.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim.

    Interpretationnone given
    § LA THÉORIE DES CLASSES DE CHERN

    For a theory of Chern classes, rational equivalence is preferable because it gives a finer theory.

    “Mais pour une théorie des classes de Chern, on a intérêt à travailler avec l'équivalence rationnelle, qui donne une théorie plus fine.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationnone given
    § Abstract

    The same axiomatic treatment works outside algebraic geometry and yields an elementary theory of Chern classes for complex vector bundles on topological varieties, and on all spaces where the classification theorem for principal bundles via a classifying space holds.

    “On notera que l'exposé donné ici vaut dans d'autres cadres que celui de la Géométrie algébrique et redonne par exemple une théorie entièrement élémentaire des classes de Chern pour les fibres vectoriels complexes sur des variétés topologiques (et partant, sur tous les espaces pour lesquels le théorème de classification des fibres principaux à groupe structural à l'aide d'un « espace classiflant )>”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: There is a category V of nonsingular algebraic spaces satisfying condition V1: if X is in V and E is a vector bundle on X, then the projective bundle P(E) is in V.
    • The authors state this as an assumption: There are data a, b, c: a contravariant functor A from V to graded anticommutative rings with unit; a functorial homomorphism p_X: P(X)->A^1(X); and, for each closed subvariety Y of X in V of constant codimension p, a homomorphism i_*: A(Y)->A(X) increasing degrees by p.
    • The authors state this as an assumption: Axiom A1 holds: for X in V and a rank-p vector bundle E on X, the elements 1, E_E, ..., E_E^{p-1} form a basis of A(P(E)) over A(X).
    • The authors state this as an assumption: Axiom A2 holds: for a rank-1 vector bundle L on X in V and a regular section s transversal to the zero section whose zero locus Y is in V, p_X(Y)=p_X(cl_X(L)).
12 of 12

  • The authors state this as an assumption.

    There is a category V of nonsingular algebraic spaces satisfying condition V1: if X is in V and E is a vector bundle on X, then the projective bundle P(E) is in V.

    “On suppose donnée par la suite une catégorie V d'espaces algébriques non singuliers (les morphismes dans cette catégorie étant les morphismes d'espace algébrique).La seule 'condition imposée à V étant (Vl).Si Xç.V, et si E est un fibre vectoriel sur X^ alors P(E)ç.V.”

  • The authors state this as an assumption.

    There are data a, b, c: a contravariant functor A from V to graded anticommutative rings with unit; a functorial homomorphism p_X: P(X)->A^1(X); and, for each closed subvariety Y of X in V of constant codimension p, a homomorphism i_*: A(Y)->A(X) increasing degrees by p.

    “On suppose qu'on a de plus les données suivantes : a. Un foncteur contravariant X-> A (X) de V dans la catégorie des anneaux gradués avec unité anticommutatifs ; b.Un homomorphisme fonctoriel px ' -'P {X)-^ A^-ÇX) {X'ç^r)”

  • The authors state this as an assumption.

    Axiom A1 holds: for X in V and a rank-p vector bundle E on X, the elements 1, E_E, ..., E_E^{p-1} form a basis of A(P(E)) over A(X).

    “Al.Soient ^TeV, E un fibre vectoriel de rang p sur X^ P{E) le fibre projectif associé, Ï.E l'élément de A(P(E)) défini par ^==p(cl(Ls)).Considérons A(P(E)) comme un module à gauche sur A(A') grâce a l'homomorphisme /* : A {X)->A (P(E)) associé à la projection /: P(E)-^A”

  • The authors state this as an assumption.

    Axiom A2 holds: for a rank-1 vector bundle L on X in V and a regular section s transversal to the zero section whose zero locus Y is in V, p_X(Y)=p_X(cl_X(L)).

    “A 2. Soient JTçV, L un fibre vectoriel de rang i sur V, s une section régulière de L transversale à la section nulle et telle que l'espace Y des zéros de s appartienne à V. Alors on a px(r)=px(cixW).”

  • The authors state this as an assumption.

    The category V satisfies condition (V2): for every X in V and every closed nonsingular subspace Y of X, Y is in V.

    “(V2) Pour tout AçV et tout sous-espace fermé non singulier J de JT, on a VçV.”
5 of 5

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Metadata

Authors
Grothendieck, A.
Corresponding author
Grothendieck, Alexander
Venue
Bulletin de la Société mathématique de France
Publisher
Societe Mathematique de France
Published
1958
Volume
79
Pages
137-154
ISSN
  • 0037-9484 (ISSN-L)
  • 2102-622X
DOI
10.24033/bsmf.1501
OpenAlex
W251777577
Cited by
259
References
0
Citations by year
2026: 2 · 2025: 4 · 2024: 1 · 2023: 5 · 2022: 8 · 2021: 3 · 2020: 3 · 2019: 5 · 2018: 4 · 2017: 3 · 2016: 6 · 2015: 10 · 2014: 4 · 2013: 4 · 2012: 14
FWCI
1.14
Type
journal article
Language
fr
Open access
Locations
  • Bulletin de la Société mathématique de France (publishedVersion)
  • French digital mathematics library (Numdam) (submittedVersion)
Topics
  • Physical Sciences › Mathematics › Geometry and Topology › Algebraic Geometry and Number Theory
  • Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
  • Physical Sciences › Mathematics › Algebra and Number Theory › Advanced Algebra and Geometry
Keywords
Chern classes, characteristic classes, vector bundles, differential geometry, algebraic topology

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