Naturality of the Chow ring and the projection formula

Pullback is a unital ring homomorphism. By [L1] the graph pullback $f^{\ast }={\mathrm{\Gamma }}_f^{!}{\mathrm{pr}}_Y^{\ast }$ is a composite of refined Gysin operations, which preserve codimension by [L2]; it is unital because ${\mathrm{\Gamma }}_f^{!}{\mathrm{pr}}_Y^{\ast }[Y]=[{\mathrm{\Gamma }}_f]=[X]$ up to the canonical identification of the graph with $X$, and the smooth-section identity gives ${\mathrm{\Gamma }}_f^{!}{\mathrm{pr}}_X^{\ast }=1$. To see that $f^{\ast }$ is multiplicative, write $\alpha =c[Y]$ and $\beta =d[Y]$ in the operational description of [L1]: restriction of operational classes is a ring map by [L2], so $f^{\ast }(\alpha \beta )=f^{\ast }(cd[Y])=cd[X]=f^{\ast }\alpha f^{\ast }\beta$. Composition $(gf{)}^{\ast }=f^{\ast }{g}^{\ast }$ follows from the composition theorem for refined Gysin applied to the composable graphs, with the projection identity of [L2].

Agreement with flat pullback. If $f$ is flat, apply the regular-immersion/smooth-projection identity of [L2] to ${\mathrm{\Gamma }}_f:X\hookrightarrow X{\times }_{k}Y$ followed by ${\mathrm{pr}}_Y:X{\times }_{k}Y\to Y$. Its composite is $f$, flat of relative dimension $\dim X-\dim Y$, so the identity gives ${\mathrm{\Gamma }}_f^{!}{\mathrm{pr}}_Y^{\ast }=f^{\ast }$ for the flat pullback; hence $f^{\ast }$ agrees with the flat pullback of [L3] on classes of the stated dimensions.

Proper pushforward. If $f$ is proper, [L3] gives functoriality $(gf{)}_{\ast }=g_{\ast }{f}_{\ast }$ and the codimension shift $\dim Y-\dim X$ on nonzero cycle classes, since a proper pushforward of a $p$-dimensional class is supported on images of dimension at most $p$ and the norm-degree formula is multiplicative in towers. The projection formula is the corresponding statement of [L1], with the identification $A^p=A_{\dim -p}$: writing $\alpha =c[Y]$ in operational form, $f_{\ast }(c\beta )=c{f}_{\ast }\beta$ is the proper axiom of bivariant classes, which is exactly $f_{\ast }(f^{\ast }\alpha \beta )=\alpha f_{\ast }\beta$.