# The theory of Chern classes

*1958*

#### Translator’s note

*This page is a translation into English of the following:*

Grothendieck, A. “La théorie des classes de Chern.” *Bulletin de la Société Mathématique de France* **86** (1958), 137–154. DOI: `10.24033/bsmf.1501`

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*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `a5ddc8f`

# Introduction

In this appendix, we will develop an axiomatic theory of Chern classes that will allow us, in particular, to define the Chern classes of an algebraic vector bundle *formal properties* characterising a theory of Chern classes were brought to light), and on the other hand by an idea of Chern [2] that consists of using the multiplicative structure of the ring of cycle classes on the bundle of projective spaces *construction* of Chern classes.
We note that the exposition given here also applies to other settings than algebraic geometry, and recovers, for example, an entirely elementary theory of Chern classes for complex vector bundles on topological manifolds (and, from this, on any space for which the classification theorem of principal bundles with a structure group via a “classifying space” holds true).
Similarly, we will obtain, for a complex-analytic vector bundle

It appears that a satisfying theory of Chern classes in algebraic geometry was given, for the first time, by W.L. Chow (unpublished), using the Grassmannian.
The main aim of the current paper has been to eliminate the Grassmannian from the theory.
I have already shown [4] how the theory of Chern classes allows us to *recover* the structure of

# 1 Notation

In order to not expose ourselves to the complications arising from intersection theory, we will limit ourselves in what follows to considering only *non-singular* topological spaces.
We fix, once and for all, a base field

If

Let *flag of length i* in a vector space

*bundle on*X of flags of length i in

*flag manifold*

*completely split*. By this, we mean that this rank-

*factors*of the given splitting.

If

If *contravariant functor* in

With

With this, we can immediately verify that

Let *cycle* *zero cycle* of the section *transversal* to the subvariety *transversal to the zero section*.
We can express this property by a calculation:
since it is local on

# 2 The functor A(X)

In what follows, suppose that we have a category

If

Suppose further that we have the following data:

A contravariant functor

A homomorphism

For all

If

With this, we suppose that the following conditions are satisfied:

Let

Let

Let

Let

From these axioms we will prove two lemmas that will be useful in the next section.

Let

*Proof*. By §1, we can restrict, by induction on

Let

*Proof*. We will prove, by induction on

For

The following corollary will only be used in §3.

Under the conditions of Lemma 2, if, further,

The introduction of the operation

We note, first of all, that condition V1 is satisfied for all reasonable categories of algebraic spaces; it is satisfied, in particular, for the categories of arbitrary non-singular algebraic spaces, of non-singular quasi-projective algebraic spaces, and of non-singular projective algebraic spaces. The verification for the two latter cases presents no difficulty, and is left to the reader (the result being a particular case of a more general result concerning blown-up varieties).

We now given some particular cases where the conditions in this section are satisfied.

{\mathcal{V}} is the category of non-singular quasi-projective algebraic spaces, andA(X) is the ring of cycle classes onX under*rational equivalence*, with the usual definition off^* andf_* . Of course, we gradeA(X) by taking the class of a cycle onX that is everywhere of codimensionp to be of degree2p [so thatA(X) has only even degrees, as we would expect in a cohomological theory for a*commutative*graded ring]. The homomorphism\mathbf{P}(X)\to A^2(X) is an isomorphism, given by sending any rank-1 vector bundleL onX to the set of divisors of rational sections ofL that are not zero on any component ofX . For the theory of linear equivalence, including the verification of A1 to A4 (with only A1 not being immediate), see the exposés by Chevalley and Grothendieck in [4].The conditions that we demand are also satisfied if we take

A(X) to be the ring of cycle classes under*algebraic*equivalence, but, for a theory of Chern classes, we rather prefer to work with rational equivalence, which gives a finer theory.We cannot yet define a ring structure on the group

A(X) of cycle classes on an*arbitrary*(not necessarily quasi-projective) non-singular variety, nor, for a morphismf\colon X\to Y , a morphismf^*\colon A(Y)\to A(X) in such a way that the necessary conditions are satisfied. Further, it is not even certain that this might be possible. We can imagine replacing the ring of cycle classes (under rational equivalence) by the graded ring associated to the ringK(X) of classes of coherent algebraic sheaves onX , filtered in the natural way (by considering the dimension of the supports of the sheaves). Unfortunately, we would then have to prove that this filtration is compatible with the ring structure (and with the “inverse image” homomorphisms), which I only know how to do in the quasi-projective case, by using rational equivalence. However, it seems that these difficulties disappear when we tensor with the field of rationals\mathbb{Q} , i.e. if we ignore the phenomenons of torsion.

{\mathcal{V}} is the category of all non-singular algebraic spaces. IfX is such a variety, then we denote by\Omega_X^\bullet the sheaf of germs of regular differential forms onX , and byA(X) the cohomology group\operatorname{H}^\bullet(X,\Omega_X^\bullet) . We grade this group by taking\operatorname{H}^p(X,\Omega_X^q) to be of degreep+q , and we make this an algebra by means of the cup product. We thus obtain an anticommutative graded algebra, which is clearly a contravariant functor with respect toX . By following the formalism developed by Grothendieck in [3], we define in a natural way the homomorphismi_*\colon A(Y)\to A(X) associated to an injectioni\colon Y\to X (and it is probably possible to definei_* for every*proper*morphismi\colon Y\to X ).^{1}Finally, we define a morphism\mathbf{P}(X)\to\operatorname{H}^1(X,\Omega_X^1)\subset A^2(X) in a classical way, by writing, for example,\mathbf{P}(X)=\operatorname{H}^1(X,{\mathscr{O}}_X^\times) (where{\mathscr{O}}_X^\times denotes the sheaf of germs of invertible regular functions onX ), and by considering the homomorphismf\mapsto\mathrm{d}f/f from{\mathscr{O}}_X^\times to\Omega_X^1 . We can again easily verify that conditions A1 to A4 are satisfied, with A1 being a consequence of the Leray spectral sequence of the continuous map\mathbb{P}(E)\to X [the spectral sequence being trivial, as follows from considering from the class\xi_E on\mathbb{P}(E) .]The base field

k is the field of complex numbers,{\mathcal{V}} is the category of non-singular algebraic spaces, andA(X)=\operatorname{H}^\bullet(X,\mathbb{Z}) (withX being endowed with its “usual” topology). The definition of b. (either by Poincaré duality on divisor classes, or as an obstruction class in the classical exact sequence0\to\mathbb{Z}\to{\mathscr{O}}_X\to{\mathscr{O}}_X^\times\to0 of sheaves onX , endowed with its usual topology) is well known. The definition of c. classically comes from Poincaré duality, and properties A1 to A4 are well known (with A1 again following from the Leray spectral sequence).

# 3 Definition and fundamental properties of Chern classes

Let

The *Chern classes* of

and call *(total) Chern class* of

Suppose that we have the data of a., b., and c. from the previous section, satisfying axioms A1 to A4. Then the Chern classes (defined by (1)) satisfy the following conditions:

*Functoriality.*— Letf\colon X\to Y be a morphism in{\mathcal{V}} , and letE be a vector bundle onY . Thenc(f^{-1}(E)) = f^*(c(E)) \tag{3} [wheref^{-1}(E) denotes the vector bundle onX given by the inverse image ofE underf ].*Normalisation.*— IfE is a rank-1 vector bundle onX\in{\mathcal{V}} , thenc(E) = 1+p_X(\operatorname{cl}_X(E)). \tag{4} *Additivity.*— LetX\in{\mathcal{V}} , and let0\to E'\to E\to E''\to 0 be an exact sequence of vector bundles onX . Thenc(E) = c(E')c(E''). \tag{5}

Furthermore, properties i., ii., and iii. entirely *characterise* Chern classes.

*Proof*. We first prove the *uniqueness* of a theory of Chern classes satisfying properties i., ii., and iii.
Let

We now show that the Chern classes defined by (1) do indeed satisfy i., ii., and iii.

*Proof of i.* —
Let

*Proof of ii.* —
Suppose that

*Proof of iii.* —
With the set-up of i., let

So let

Let

*Proof of Theorem 1.* —
So we are under the conditions of the corollary to Lemma 2, which implies that

This finishes the proof of Theorem 1.

Let

*Proof*. Passing to a flag variety, as per usual, we can reduce to the case where

# 4 Remarks and various addenda

In all the above, we have only needed to work with elements of even degree in

Let

for

We can also define (non-additive!) maps

which also implies that the map

if

In fact, the *special*

It is the detailed study of ^{2}
This proof is, for now, only valid in characteristic

That said, we now return to the commutative graded rings *completed Chern class* of

We can thus also say that *the completed Chern class \widetilde{c}(E) defines a homomorphism of \lambda-rings from K(X) to \widetilde{A(X)}*.

*Application to the Chow ring.* —
Let *filtration* on

Using this, and Proposition 8 of the previous article by Borel–Serre^{3}, we find that *if X is quasi-projective, then the filtration of K(X) is compatible with its ring structure*;
the homomorphism

*a homomorphism*\varphi of graded rings from the Chow ring A(X) to the graded ring GK(X) associated to K(X) . We will show that

*the kernel of this homomorphism is a torsion group*. For this, consider the homomorphism

*a homomorphism*\psi from GK(X) to the graded ring A'(X) associated to the filtered ring \widetilde{A(X)} , which can itself be identified (as a graded group) with A(X) . [As for its multiplicative structure, we can formally verify that it is given by the product

in degree

If

[The first case in (16) simply says that *a priori* when we note that, since the restriction of

The equations in (15) indeed show that * \varphi and \psi are isomorphisms, up to a torsion group*.
(I do not know if

*.*\widetilde{c} is an isomorphism, up to torsion, from K(X) to A(X)

The above shows that, to prove intersection formulas in ^{4}, which gives (by passing to the associated graded objects) that

# 5 The zero cycles of a regular section of a vector bundle

In this section, we assume that the category

For all

Furthermore, we assume that the following axiom is verified:

Let

Let

With the above notation, if

*Proof*. Let

Since

So let

From this, we will deduce

Let

*Proof*. Consider

Equation (18) then follows from (17), since

In fact, the above proof proves the following formula, which holds for *every* regular section

Suppose that we are in the setting of Chow theory (

Equation (17) holds true even if we replace the zero section

# Bibliography

*Trans. Amer. Math. Soc*.

**85**(1957), 181–207.

*Amer. J. Math.*

**75**(1953), 565–597.

*Séminaire Bourbaki*.

**9**(1956–1957).

**1**(1956–1958).

(Note added during editing). This homomorphism

i_* is now defined in full generality.↩︎[Trans.] This is referring to Borel, A; Serre, J.-P. “Le théorème de Riemann–Roch.”

*Bull. Soc. Math. France***86**(1958), 97–136.↩︎[Trans.] This is once again referring to paper by Borel and Serre mentioned in the previous translator footnote.↩︎

[Trans.] This is once again referring to paper by Borel and Serre mentioned in the previous translator footnotes.↩︎