Proposition
Let \(X\) be a scheme.
- For all \(\alpha \in A_k(X)\),
- \(s_i(E) \cap \alpha = 0\) when \(i < 0\)
- \(s_0(E)\cap \alpha = \alpha\)
- (Commutativity) Let \(E\) and \(F\) be vector bundles on \(X\). Then for all \(\alpha \in A_k(X)\) and \(i,j\) we have \[s_i(E) \cap (s_j(F) \cap \alpha) = s_j(F)\cap s_i(E)\cap \alpha\]
- (Proper pushforward compatability) If \(f:X'\to X\) is proper, \(E\) is a vector bundle on \(X\) and \(\alpha \in A_*(X')\), then \[f_*\big(s_i(f^*E)\cap \alpha) = s_i(E)\cap f_*(\alpha)\big)\]
- (Flat pullback compatability) If \(f:X' \to X\) is flat, \(E\) is a vector bundle on \(X\) and \(\alpha \in A_*(X)\), then \[f^*(s_i(E)\cap \alpha) = s_i(f^*E)\cap f^*(\alpha)\]
- If \(E\) is a line bundle on \(X\), then \[s_1(E)\cap \alpha = -c_1(E)\cap \alpha\].
Proof
- (5): Let \(\mathcal L\) be the invertible sheaf corresponding to \(E\). Recall that tensoring with an invertible sheaf doesn’t affect \(\Proj\); that is, for any locally free sheaf \(\mathcal E\) of \(\mathcal O_X\)-modules, \(\Proj_X(\Sym \mathcal E) = \Proj_X(\Sym(E\otimes \mathcal L)) \). Thus \(X = \Proj(\Sym \mathcal O_X) = \Proj\Sym \mathcal L = P\) and \(p:P\to X\) is the identity. Note that \(E = \mathcal O_E(-1)\), so \[s_1(E)\cap \alpha = p_*(c_1(\mathcal O_E(1)) \cap p^*\alpha) = c_1(E^\vee)\cap \alpha = -c_1(E)\cap \alpha.\]