Let \(L\) be a line bundle on a scheme \(X\). We do not assume \(X\) to be integral, so \(L\) need not correspond to a Cartier divisor. However, for any \(k\)-dimensional subvariety \(V\subset X\), the restriction \(L|_V\) of \(L\) to \(V\) is isomorphic to \(\mathcal O_V(C)\) for some Cartier divisor \(C\) on \(V\). Every Cartier divisor corresponds to a Weil divisor, call it \([C]\) in this case, which represents an element in \(A_{k-1}(V) \subset A_{k-1}(X)\). This proceedure gives us a way of realizing the first Chern class of a line bundle on \(X\) as a map \(A_k(X)\to A_{k-1}(X):\)
Definition
Let \(L\) be a line bundle on \(X\). The first Chern class of \(L\) is a homomorphism \(c_1(L)\cap - :A_k(X)\to A_{k-1}(X)\) for each \(1\leq k \leq \dim X\) defined as follows:
- Take a \(k\)-dimensional subvariety \(V\) of \(X\)
- Let \(C\) be the Cartier divisor (defined up to linear equivalence) such that \(L|_V \cong \mathcal O_V(C)\)
- Let \([C]\) be the Weil divisor determined by \(C\)
- Notice that \([C]\in A_{k-1}(X)\)
- Define \(c_1(L)\cap [V] = [C]\).
- Since \(A_k(X)\) is generated by cycles \([V]\), extend by linearity to get the desired map.