Proper pushforward commutes with flat pullback

Proper pushforward of the cycle of a sheaf. Let $\mathcal{F}$ be coherent on $X$ with support of dimension at most $d$. Then $f_{\ast }[\mathcal{F}{]}_d=[f_{\ast }\mathcal{F}{]}_d$ in $Z_d(Y)$. Indeed, let $\eta$ be the generic point of a $d$-dimensional component of $\mathrm{Supp}(f_{\ast }\mathcal{F})$; every point of $\mathrm{Supp}\mathcal{F}$ mapping to $\eta$ is generic in a $d$-dimensional support component, because otherwise its closure would have dimension less than $d$ and map onto a neighbourhood of $\eta$ of dimension $d$, which is impossible; hence the fibre of the support over $\eta$ is finite. Give that support the closed scheme structure defined by $\mathrm{Ann}\mathcal{F}$; $\mathcal{F}$ is a coherent sheaf on this closed subscheme. Its restricted morphism to $Y$ is proper and is quasi-finite over $\eta$, so after removing the closed image of its non-quasi-finite locus it is finite near $\eta$ by [L3]. Then $(f_{\ast }\mathcal{F}{)}_{\eta }\cong {⨁}_{\xi \mapsto \eta }{\mathcal{F}}_{\xi }$ is a finite module of finite length over the local ring ${\mathcal{O}}_{Y,\eta }$, and a composition series over each local ring gives ${\mathrm{\ell }}_{{\mathcal{O}}_{Y,\eta }}((f_{\ast }\mathcal{F}{)}_{\eta })={\sum }_{\xi \mapsto \eta }[\kappa (\xi ):\kappa (\eta )]{\mathrm{\ell }}_{{\mathcal{O}}_{X,\xi }}({\mathcal{F}}_{\xi })$, with the residue degree weights because the simple factors of ${\mathcal{F}}_{\xi }$ acquire composition factors of $\kappa (\xi )$ over $\kappa (\eta )$; comparing with the norm-degree definition of $f_{\ast }[\mathcal{F}{]}_d$ proves the identity. If the image of a component has dimension less than $d$, both $d$-cycles vanish at that component.

Flat pullback of the cycle of a sheaf. For $\mathcal{F}$ coherent on $Y$ supported in dimension at most $d$ one has $g^{\ast }[\mathcal{F}{]}_d=[g^{\ast }\mathcal{F}{]}_{d+n}$ in $Z_{d+n}(Y^{\mathrm{\prime }})$: at a generic point $\xi$ of a top-dimensional component of the preimage of its support, with $\eta =g(\xi )$, the stalk of $g^{\ast }\mathcal{F}$ is ${\mathcal{F}}_{\eta }{\otimes }_{{\mathcal{O}}_{Y,\eta }}{\mathcal{O}}_{Y^{\mathrm{\prime }},\xi }$. Tensoring a composition series of the finite-length module ${\mathcal{F}}_{\eta }$ with this flat local algebra shows that its length is ${\mathrm{\ell }}_{{\mathcal{O}}_{Y,\eta }}({\mathcal{F}}_{\eta }){\mathrm{\ell }}_{{\mathcal{O}}_{Y^{\mathrm{\prime }},\xi }}({\mathcal{O}}_{Y^{\mathrm{\prime }},\xi }\mathrm{/}{\mathfrak{m}}_{\eta }{\mathcal{O}}_{Y^{\mathrm{\prime }},\xi })$. The second factor is the generic fibre multiplicity and can exceed one for a nonreduced fibre.

Flat base change for the direct image. For a quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{G}$ there is a natural isomorphism $g^{\ast }{f}_{\ast }\mathcal{G}\cong f_{\ast }^{\mathrm{\prime }}{g}^{\mathrm{\prime }\ast }\mathcal{G}$: on an affine open $\mathrm{Spec}A\subseteq Y^{\mathrm{\prime }}$ whose image in $Y$ is contained in an affine open over which $f$ is proper and $\mathcal{G}$ is quasi-coherent, the preimage under $f$ has a finite affine cover with affine finite intersections (properness gives separatedness and quasi-compactness), the Čech complex computes the degree-zero sections by [L3], tensoring with $A$ over the base ring is exact, and the comparison maps agree on overlaps; hence they glue.

Compatibility on cycles. Apply steps 1.1, 1.2 and 1.3 to $\mathcal{F}={\mathcal{O}}_V$ for an integral closed subscheme $V\subseteq X$ of dimension $d$: $g^{\ast }{f}_{\ast }[V]=g^{\ast }[f_{\ast }{\mathcal{O}}_V{]}_d=[g^{\ast }{f}_{\ast }{\mathcal{O}}_V{]}_{d+n}=[f_{\ast }^{\mathrm{\prime }}{g}^{\mathrm{\prime }\ast }{\mathcal{O}}_V{]}_{d+n}=f_{\ast }^{\mathrm{\prime }}[g^{\mathrm{\prime }\ast }{\mathcal{O}}_V{]}_{d+n}=f_{\ast }^{\mathrm{\prime }}{g}^{\mathrm{\prime }\ast }[V]$, where the middle equality is step 1.3 applied to the quasi-coherent sheaf ${\mathcal{O}}_V$ and the outer equalities are steps 1.1 and 1.2; the finite-support and locally finite cases are handled by the same identities term by term. Since both sides are additive, $g^{\ast }{f}_{\ast }=f_{\ast }^{\mathrm{\prime }}{g}^{\mathrm{\prime }\ast }$ on cycles and therefore on Chow groups by [L2].