3.4 A tiny bit more intersection theory

Recall that, if \(X\) is a finite type separated scheme over a field \(k\), we have groups \(Z_i X\) of algebraic cycles, the groups \(\operatorname{\mathrm{Rat}}_i X\) of algebraic cycles which are rationally equivalent to \(0\), and the Chow groups \(A_i X = Z_i X/\operatorname{\mathrm{Rat}}_i X\).

I want to say some more about three topics in intersection theory that have to do with functoriality:

The Chow groups are covariantly functorial for proper morphisms. In other words, if \(f:X\to Y\) is a proper morphism, then we get an group homomorphism \(f_*:A_k X\to A_k Y\) called the proper pushforward.
The Chow groups are contravariantly functorial for flat morphisms. In other words, if \(f:X\to Y\) is flat and all of the nonempty fibers are \(n\)-dimensional, then we get an group homomorphism \(f^*:A_k Y\to A_{k+n} X\) called the flat pullback.
In general, there is no ring structure on the groups \(A_* X\). However, if \(X\) is smooth and equidimensional of dimension \(n\), then \(A^* X\) has the structure of a commutative graded ring. (Here \(A^k X = A_{n-k} X\).)
If \(f:X\to Y\) is an arbitrary morphism of smooth varieties, then we get a group homomorphism \(f^!:A^* Y\to A^* X\). In the case that \(X\) and \(Y\) are both equidimensional, then \(f^!:A^*Y \to A^*X\) is a ring homomorphism. More generally, if \(f:X\to Y\) is an local complete intersection (lci) morphism of arbitrary varieties of relative dimension \(d\), then we get a group homomorphism \(f^{!}:A_k Y\to A_{k+d} Y\).

Topics c and d take a lot of work to establish, and are really the main technical part of the theory, Fulton-MacPherson style intersection theory developed in [ 3 ] . But a and b are not as involved. It’s worth saying that the ring structure on the Chow groups of a smooth variety is really constructed by as a composition of the form

\begin{equation} A^k X\otimes A^j X \xrightarrow {\mathrm{prod}} A^{k+j} X\times X\xrightarrow {\Delta ^!} A^{k+j}, \end{equation}

3.16

where \(\mathrm{prod}\) takes a pair of cycles of the form \(([V],[W])\) with \(V\) and \(W\) irreducible to the cycle \([V\times W]\) and \(\Delta :X\to X\times X\) is the diagonal morphism. So, in the Fulton-MacPherson theory, d is more basic than c. So at least I can say what the maps are.