Editing Chern class (section)

=== Axiomatic description ===
There is another construction of Chern classes which take values in the algebrogeometric analogue of the cohomology ring, the [[Chow ring]].

Let <math>X</math> be a nonsingular quasi-projective variety of dimension <math>n</math>. It can be shown that there is a unique theory of Chern classes which assigns an algebraic vector bundle <math>E \to X</math> to elements <math>c_i(E) \in A^i(X)</math> called Chern classes, with Chern polynomial <math>c_t(E)=c_0(E) + c_1(E)t + \cdots + c_n(E)t^n</math>, satisfying the following (similar to [[Chern_class#Grothendieck_axiomatic_approach |Grothendieck's axiomatic approach]]). <ref>{{harvnb|Hartshorne|loc=Appendix A. 3 Chern Classes.}}</ref>
# If for a Cartier divisor <math>D</math>, we have <math>E \cong \mathcal{O}_X(D)</math>, then <math>c_t(E) = 1+Dt</math>.
# If <math>f: X' \to X</math> is a morphism, then <math>c_i(f^*E) = f^* c_i(E)</math>.
# If <math>0 \to E' \to E \to E'' \to 0</math> is an exact sequence of vector bundles on <math>X</math>, the Whitney sum formula holds: <math>c_t(E) = c_t(E')c_t(E'')</math>.

<!-- the previously listed axioms were not entirely correct---the Chern classes do NOT give a ring homomorphism from the K-group to the Chow ring (e.g., see the Whitney sum formula turning addition in the K-group into multiplication in the Chow ring, rather than preserving addition). To get a ring homomorphism, you need to use the Chern character instead. -->

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These are nice but are actually already done earlier; merge them with the early section
=== Abstract computations using formal [roperties ===
==== Direct sums of line bundles ====
Using these relations we can make numerous computations for vector bundles. First, notice that if we have line bundles <math>\mathcal{L},\mathcal{L}'</math> we can form a short exact sequence of vector bundles
<math display="block">0 \to \mathcal{L} \to \mathcal{L}\oplus\mathcal{L}' \to \mathcal{L}' \to 0</math>
Using properties <math>1</math> and <math>2</math> we have that
<math display="block">\begin{align}
c(\mathcal{L}\oplus\mathcal{L}') &= c(\mathcal{L})c(\mathcal{L}') \\
&= (1+c_1(\mathcal{L}))(1+c_1(\mathcal{L}')) \\
&= 1 + c_1(\mathcal{L}) + c_1(\mathcal{L}') + c_1(\mathcal{L})c_1(\mathcal{L}') 
\end{align}</math>
By induction, we have
<math display="block">c(\bigoplus^n_{i=1} \mathcal{L}_i) = c(\mathcal{L}_1)\cdots c(\mathcal{L}_n)</math>

====Duals of line bundles====
Since line bundles on a smooth projective variety <math>X</math> are determined by a divisor class <math>[D]</math> and the dual line bundle is determined by the negative divisor class <math>-[D]</math>, we have that
<math display="block">c_1(\mathcal{L}) = -c_1(\mathcal{L}^*)</math>

===Tangent bundle of projective space===
This can be applied to the Euler sequence for projective space
<math display="block">0 \to \mathcal{O}_{\mathbb{P}^n} \to \mathcal{O}_{\mathbb{P}^n}(1)^{\oplus (n+1)} \to \mathcal{T}_{\mathbb{P}^n} \to 0</math>
to compute
<math display="block">\begin{align}
c(\mathcal{O}_{\mathbb{P}^n})c(\mathcal{T}_{\mathbb{P}^n}) &= c(\mathcal{O}_{\mathbb{P}^n}(1)^{\oplus (n+1)}) \\
c(\mathcal{T}_{\mathbb{P}^n})&= (1 + H)^{n+1} \\
&= {n+1 \choose 0}1 + {n+1 \choose 1}H + \cdots + {n+1 \choose n}H^n
\end{align}</math>
where <math>H</math> is the class of a degree one hyperplane. Also, notice that <math>H^{n+1}=0</math> in the chow ring of <math>\mathbb{P}^n</math>.
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