3.4.6 Ring structure for smooth \(X\)
Another important thing to say about the ring structure on \(A^* X\) when \(X\) is smooth is that disjoint cycles have \(0\) product. In fact, suppose \(A\) and \(B\) are closed subvarieties, with \([A]\in Z^a X\), \([B]\in Z^b X\). Then we can write \([A][B]\in A^{n+m} X\) in the form \(\sum _{i} n_i[V_i]\) with \(n_i\in \mathbb {Z}\) and \(V_i\) subvarieties contained in \(A\cap B\). In particular, if \(A\cap B =\emptyset \), then \([A][B] = 0\). Moreover, if \(A\cap B\) is equidimensional of the “expected dimension" \(\dim X - \dim A - \dim B\), then the \(V_i\) are the components of the scheme theoretic intersection \(V_i\) and the \(n_i\) are positive integers. See [ 3 , Proposition 8.2 ] .