Flat pullback of cycles and of rational equivalence

Pure dimension. Let $V\subseteq Y$ be integral of dimension $d$ and put $S=f^{-1}(V)$. The base change $S\to V$ is flat and locally of finite type. Each generic point of an irreducible component of $S$ lies over the generic point of $V$: on affine charts, going down makes the contraction of a minimal prime minimal, and $V$ is integral. Choose finite-type affine charts $\mathrm{Spec}A\subseteq V$ and $\mathrm{Spec}B\subseteq S$ meeting such a generic point, with corresponding minimal prime $\mathfrak{p}\subset B$. Then $\mathfrak{p}\cap A=(0)$, and $(B\mathrm{/}\mathfrak{p}){\otimes }_A\mathrm{Frac}(A)$ is an integral component of the generic fibre. It has dimension $n$ by the pure-fibre hypothesis, so its function field has transcendence degree $n$ over $\mathrm{Frac}(A)$. The affine-domain dimension theorem and transcendence-degree additivity now give $\dim (B\mathrm{/}\mathfrak{p})={\mathrm{trdeg}}_k\mathrm{Frac}(B\mathrm{/}\mathfrak{p})={\mathrm{trdeg}}_k\mathrm{Frac}(A)+n=\dim A+n=d+n\mathrm{.}$ Every nonempty affine open of an integral locally finite-type $k$-scheme has the same function field and, by the affine-domain dimension theorem, the same dimension. The chain definition of dimension then shows the whole scheme has that dimension: any chain meets an affine neighbourhood of a point in its smallest member, where the intersections remain strict. Applying this to $V$ and to each component of $S$ identifies their global dimensions with the affine calculation above. Thus all components of $f^{-1}(V)$ have dimension $d+n$. If the preimage is empty its fundamental cycle is zero, which belongs to $Z_{d+n}(X)$; otherwise the dimension is $d+n$. In either case its fundamental cycle is a pure $(d+n)$-cycle.

Local rings over a divisor. Let $W\subseteq Y$ be integral of dimension $d+1$, let $r=x\mathrm{/}y\in k(W{)}^{\ast }$ with nonzero $x,y$ in the one-dimensional local domain $A={\mathcal{O}}_{W,\zeta }$ at a codimension-one point $\zeta$, and put $S=W{\times }_{Y}X$. For a codimension-one point $\xi$ of $S$ over $\zeta$, set $C={\mathcal{O}}_{S,\xi }$. The local map $A\to C$ is flat, $C$ is one-dimensional, and $x,y$ are nonzerodivisors in $C$. Every minimal prime $\mathfrak{p}$ of $C$ contracts to $(0)$ in $A$, so $C\mathrm{/}\mathfrak{p}$ is a one-dimensional domain and $r$ maps to its fraction field. The generic length of the corresponding component of $S$ is $n_{\mathfrak{p}}={\mathrm{\ell }}_{C_{\mathfrak{p}}}(C_{\mathfrak{p}})$. A prime filtration of $C$ has $n_{\mathfrak{p}}$ factors $C\mathrm{/}\mathfrak{p}$ for each minimal prime and only finite-length factors in addition.

Order calculation on the components. For a nonzerodivisor $x$, let ${\chi }_x(M)={\mathrm{\ell }}_C(\mathrm{coker}x)-{\mathrm{\ell }}_C(\ker x)$. This invariant is additive on short exact sequences, vanishes on finite-length factors, and on a one-dimensional domain factor $C\mathrm{/}\mathfrak{p}$ equals ${\mathrm{\ell }}_{C\mathrm{/}\mathfrak{p}}((C\mathrm{/}\mathfrak{p})\mathrm{/}x(C\mathrm{/}\mathfrak{p}))$. Since $x,y$ are injective on $C$, the prime filtration from step 2.1 gives ${\mathrm{\ell }}_C(C\mathrm{/}xC)-{\mathrm{\ell }}_C(C\mathrm{/}yC)={\sum }_{\mathfrak{p}}{n}_{\mathfrak{p}}{\mathrm{ord}}_{C\mathrm{/}\mathfrak{p}}(r)\mathrm{.}$.

Flat length and divisor identity. The flat local length formula [L4], applied to $A\mathrm{/}xA$ and $A\mathrm{/}yA$, gives ${\mathrm{\ell }}_C(C\mathrm{/}xC)-{\mathrm{\ell }}_C(C\mathrm{/}yC)={\mathrm{ord}}_A(r){\mathrm{\ell }}_C(C\mathrm{/}{\mathfrak{m}}_{A}C)\mathrm{.}$ The last factor is the generic length in the flat pullback of the codimension-one cycle at $\xi$. Thus the coefficient of $f^{\ast }({\mathrm{div}}_W(r))$ at $\xi$ equals the coefficient of ${\sum }_j{n}_j{\mathrm{div}}_{W_j}(r{\mathrm{\mid }}_{W_j})$. At the generic point of $W$, $r$ is a unit and both coefficients vanish. Summing over codimension-one points and components proves $f^{\ast }({\mathrm{div}}_W(r))={\sum }_j{n}_j{\mathrm{div}}_{W_j}(r{\mathrm{\mid }}_{W_j}),$ with each component pushed to $X$.

Descent to Chow groups. By definition ${\mathrm{Rat}}_d(Y)$ is generated by the cycles $(i_W{)}_{\ast }{\mathrm{div}}_W(r)$ for integral $W\subseteq Y$ of dimension $d+1$ and $r\in k(W{)}^{\ast }$. The cycle-level pullback is additive, and step 4.1, applied to the closed immersion of $W$ into $Y$ and its flat base change, sends each generator to a sum of principal divisor cycles on the components of $f^{-1}(W)$. That sum lies in ${\mathrm{Rat}}_{d+n}(X)$, so $f^{\ast }$ descends to $A_d(Y)\to A_{d+n}(X)$.

Functoriality. For the identity morphism the preimage of an integral subscheme is itself. Let $f:X\to Y$ and $g:Y\to T$ be composable flat morphisms of the stated kind. The two scheme-theoretic preimages of an integral $V\subseteq T$ agree, and the fibre dimensions add: $n_{g\circ f}=n_g+n_f$. At a generic point of each top-dimensional component, the pullback coefficient is the length of the corresponding local tensor product; associativity of tensor products and length additivity give the same coefficient for the composite and the two successive pullbacks. Thus $(g\circ f{)}^{\ast }[V]=f^{\ast }{g}^{\ast }[V]$ on cycles and on Chow groups. For an open immersion $U\hookrightarrow T$, the intersection of an integral $V\subseteq T$ with $U$ is either empty or a dense open integral subscheme of the same dimension, so pullback is restriction of cycles.

Localization. Let $j:U\hookrightarrow X$ be an open immersion and $i:Z=X\setminus U\hookrightarrow X$ the complementary reduced closed subscheme. A cycle supported on $Z$ restricts to zero. Every integral closed $V\subseteq U$ is a dense open subscheme of its closure $\bar{V}\subseteq X$, with the same function field, so $j^{\ast }[\bar{V}]=[V]$ and restriction is surjective. If a cycle $\alpha$ represents a class restricting to zero, then as cycles on $U$ it is a finite (or locally finite) sum ${\sum }_a(i_{V_a}{)}_{\ast }{\mathrm{div}}_{V_a}(r_a)$. The functions extend to the same function fields on the closures ${\bar{V}}_a$, and their divisors restrict to the stated divisors on $U$. Hence $\gamma =\alpha -{\sum }_a(i_{{\bar{V}}_a}{)}_{\ast }{\mathrm{div}}_{{\bar{V}}_a}(r_a)$ restricts to the zero cycle on $U$ and is supported on $Z$. The family of closures is locally finite: for every affine open $N\subseteq X$, the open $N\cap U$ is quasi-compact because $N$ is noetherian. A locally finite family meets a quasi-compact open in only finitely many members, and if $N$ meets ${\bar{V}}_a$, then the open set $N$ meets $V_a$. Thus only finitely many closures meet each such $N$. Hence $\gamma =i_{\ast }{\gamma }_Z$ for a (locally finite) cycle on $Z$, proving exactness in the middle.