Editing Chern class (section)

===Grothendieck axiomatic approach===

Alternatively, {{harvs|txt|authorlink=Alexander Grothendieck|first=Alexander|last=Grothendieck|year=1958}} replaced these with a slightly smaller set of axioms:

* Naturality:  (Same as above)
* Additivity: If <math> 0\to E'\to E\to E''\to 0</math> is an [[exact sequence]] of vector bundles, then <math>c(E)=c(E')\smile c(E'')</math>.
* Normalization:  If ''E'' is a [[line bundle]], then <math>c(E)=1+e(E_{\R})</math> where <math>e(E_{\R})</math> is the [[Euler class]] of the underlying real vector bundle.

He shows using the [[Leray–Hirsch theorem]] that the total Chern class of an arbitrary finite rank complex vector bundle can be defined in terms of the first Chern class of a tautologically-defined line bundle.

Namely, introducing the projectivization <math>\mathbb{P}(E)</math> of the rank ''n'' complex vector bundle ''E'' → ''B'' as the fiber bundle on ''B'' whose fiber at any point <math>b\in B</math> is the projective space of the fiber ''E<sub>b</sub>''. The total space of this bundle <math>\mathbb{P}(E)</math> is equipped with its tautological complex line bundle, that we denote <math>\tau</math>, and the first Chern class
<math display="block">c_1(\tau)=: -a</math>
restricts on each fiber <math>\mathbb{P}(E_b)</math> to minus the (Poincaré-dual) class of the hyperplane, that spans the cohomology of the fiber, in view of the cohomology of [[complex projective space]]s.

The classes
<math display="block">1, a, a^2, \ldots , a^{n-1}\in H^*(\mathbb{P}(E))</math>
therefore form a family of ambient cohomology classes restricting to a basis of the cohomology of the fiber. The [[Leray–Hirsch theorem]] then states that any class in <math>H^*(\mathbb{P}(E))</math> can be written uniquely as a linear combination of the 1, ''a'', ''a''<sup>2</sup>, ..., ''a''<sup>''n''−1</sup> with classes on the base as coefficients.

In particular, one may define the Chern classes of ''E'' in the sense of Grothendieck, denoted <math>c_1(E), \ldots c_n(E)</math> by expanding this way the class <math>-a^n</math>, with the relation:
<math display="block"> - a^n = c_1(E)\cdot a^{n-1}+ \cdots + c_{n-1}(E) \cdot a + c_n(E) .</math>

One then may check that this alternative definition coincides with whatever other definition one may favor, or use the previous axiomatic characterization.