# The theory of Chern classes

#### 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.

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

Version: `a5ce874`

# 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It appears that a satisfying theory of Chern classes in algebraic geometry was given, for the first time, by W.L. Chow (unpublished), using the Grassmannian.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* *cycle of zeros* 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

# Bibliography

*Trans. Amer. Math. Soc*.

**85**(1957), 181–207.

*Amer. J. Math.*

**75**(1953), 565–597.

*Séminaire Bourbaki*.

**9**(1956–1957).

**1**(1956–1958).