Proper pushforward of cycles and the norm formula

The cycle pushforward. The closure $W$ of $f(V)$ is closed by properness and irreducible, and we give it the reduced structure, so $W$ is integral with function field $K$; $\dim W\le \dim V$ and $[L:K]$ is finite when $\dim W=\dim V$ by [F1]. The displayed formula therefore defines a graded homomorphism $f_{\ast }:Z_d(X)\to Z_d(Y)$, since integral closed subschemes of a fixed dimension form a basis of $Z_d$ and the coefficient is an integer. For a dense open $U\subseteq V$ one has $\overline{f(U)}=W$ and $k(U)=k(V)$, so the definition is insensitive to replacing $V$ by a dense open.

The finite norm-order formula. Let $(A,\mathfrak{m})$ be a one-dimensional Noetherian local domain with fraction field $K$, and let $B$ be a finite domain over $A$ with fraction field $L$, finite over $K$. The ring $B$ is semilocal; for $r\in L^{\ast }$, ${\mathrm{ord}}_A(N_{L\mathrm{/}K}(r))={\sum }_{\mathfrak{q}\in \mathrm{Max}(B)}[\kappa (\mathfrak{q}):\kappa (A)]{\mathrm{ord}}_{B_{\mathfrak{q}}}(r)\mathrm{.}$ To prove this, call a finite torsion-free $A$-submodule of $L$ spanning $L$ a full lattice. Any two full lattices are commensurable, so for such lattices set $\delta (M,N)={\mathrm{\ell }}_A(M\mathrm{/}(M\cap N))-{\mathrm{\ell }}_A(N\mathrm{/}(M\cap N))\mathrm{.}$ Length additivity makes $\delta$ additive in chains of lattices; a $K$-linear isomorphism preserves it. Consequently $g\mapsto \delta (M,gM)$ is a homomorphism ${\mathrm{GL}}_n(K)\to \mathbb{Z}$, independent of the chosen full lattice $M$, where $n=[L:K]$. For $M=A^n$, a diagonal matrix has index the sum of the orders of its diagonal entries. For an elementary transvection $E_{ij}(u)$ with $u\in K$, put $I=\{a\in A:ua\in A\}$. The intersection of $A^n$ and $E_{ij}(u)A^n$ has the same coordinates as $A^n$ except for the $j$th coordinate, which is $I$; both quotients by this intersection are isomorphic to $A\mathrm{/}I$, so the index is zero. Gaussian elimination therefore gives $\delta (M,gM)={\mathrm{ord}}_A(\det g)$ for every $g\in {\mathrm{GL}}_n(K)$. Apply this to multiplication by $r$ on the lattice $B$: its determinant is $N_{L\mathrm{/}K}(r)$. Write $r=a\mathrm{/}b$ with nonzero $a,b\in B$. Translation by $b$ and additivity of $\delta$ give ${\mathrm{ord}}_A(N_{L\mathrm{/}K}(r))={\mathrm{\ell }}_A(B\mathrm{/}aB)-{\mathrm{\ell }}_A(B\mathrm{/}bB)\mathrm{.}$ For any nonzero $a\in B$, the quotient $B\mathrm{/}aB$ is a zero-dimensional Noetherian ring, hence has finite length; as a $B$-module it has a composition series whose simple factors are the residue fields at maximal ideals of $B$. Thus ${\mathrm{\ell }}_A(B\mathrm{/}aB)={\sum }_{\mathfrak{q}}[\kappa (\mathfrak{q}):\kappa (A)]{\mathrm{\ell }}_{B_{\mathfrak{q}}}(B_{\mathfrak{q}}\mathrm{/}a{B}_{\mathfrak{q}})\mathrm{.}$ Subtracting the analogous equality for $b$ proves the formula by the definition of the order function.

Equal-dimensional images. Suppose $\dim W=\dim V=d+1$. Let $T\subseteq W$ be an integral closed subscheme of dimension $d$, with generic point $\zeta$. The fibre of $V\to W$ over $\zeta$ is zero-dimensional: a positive-dimensional component would have closure of dimension at least $d+1$ in the integral scheme $V$, hence would be all of $V$, contradicting dominance. Thus $V\to W$ is quasi-finite at every point over $\zeta$. Its quasi-finite locus is open, and properness lets us shrink around $\zeta$ so that the restriction is proper quasi-finite, hence finite by [F4]. The resulting finite algebra over $A={\mathcal{O}}_{W,\zeta }$ is a domain finite over $A$, and its maximal ideals correspond to the codimension-one subschemes of $V$ mapping onto $T$. The formula in step 1.2 says that the coefficient of $[T]$ in ${\mathrm{div}}_W(N_{k(V)\mathrm{/}k(W)}(r))$ is ${\sum }_{Z\mapsto T}[k(Z):k(T)]{\mathrm{ord}}_{{\mathcal{O}}_{V,Z}}(r),$ which is exactly the coefficient of $[T]$ in $f_{\ast }{\mathrm{div}}_V(r)$. Codimension-one subschemes of $V$ whose image has dimension less than $d$ push forward to zero and do not contribute to the divisor on $W$. Equality at every such $T$ proves $f_{\ast }{\mathrm{div}}_V(r)={\mathrm{div}}_W(N_{k(V)\mathrm{/}k(W)}(r))\mathrm{.}$.

Functoriality. For the identity morphism the formula is $[k(V):k(V)][V]=[V]$. For proper composable $f:X\to Y$, $g:Y\to Z$ and an integral $V\subseteq X$ with closure $W=\overline{f(V)}$ and closure $T=\overline{g(W)}$, the function field degrees multiply in the tower $k(T)\subseteq k(W)\subseteq k(V)$ when all three dimensions agree, both sides give $[k(V):k(T)][T]$, and if either dimension drops then both composites give zero; a proper morphism carries a locally finite family of integral closed subschemes to a locally finite family, so the identity extends to locally finite cycles.

Dimension drop. If $\dim W\le \dim V-2$, every codimension-one subscheme $Z\subset V$ has dimension $\dim V-1$, while $\dim f(Z)\le \dim W\le \dim V-2<\dim Z$; hence its pushforward is zero. Suppose instead that $\dim V=d+1$ and $\dim W=d$. The generic fibre $C=V{\times }_W\mathrm{Spec}k(W)$ is a proper integral curve over $K=k(W)$. Codimension-one subschemes of $V$ that dominate $W$ correspond to closed points $x$ of $C$, and their contribution to the coefficient of $[W]$ in $f_{\ast }{\mathrm{div}}_V(r)$ is ${\sum }_{x\in C\text{ closed}}[k(x):K]{\mathrm{ord}}_{{\mathcal{O}}_{C,x}}(r)\mathrm{.}$ This degree is zero. Indeed, if $r$ is constant its divisor is zero. Otherwise let $\mathrm{\Gamma }$ be the closure of the graph of the rational map $r:C⇢{\mathbb{P}}_K^1$. The projections $p:\mathrm{\Gamma }\to C$ and $q:\mathrm{\Gamma }\to {\mathbb{P}}_K^1$ are proper; $p$ is birational, while $q$ is nonconstant, hence quasi-finite and finite. Since the local rings of ${\mathbb{P}}_K^1$ are fields or discrete valuation rings and $\mathrm{\Gamma }$ is integral, the finite morphism $q$ is flat of some degree $e$. The equal-dimensional formula of step 2.1 gives $p_{\ast }{\mathrm{div}}_{\mathrm{\Gamma }}(r)={\mathrm{div}}_C(r)$, and ${\mathrm{div}}_{\mathrm{\Gamma }}(r)=q^{\ast }[0]-q^{\ast }[\mathrm{\infty }]$. Both fibres have degree $e$ over $K$, so the displayed sum is $e-e=0$. This proves $f_{\ast }{\mathrm{div}}_V(r)=0$ in the remaining case as well.

Base change and finite flat normalization. Consider a cartesian square with $f:X\to Y$ proper and $g:Y^{\mathrm{\prime }}\to Y$ flat of pure relative dimension $n$, and write $f^{\mathrm{\prime }}:X^{\mathrm{\prime }}=X{\times }_Y{Y}^{\mathrm{\prime }}\to Y^{\mathrm{\prime }}$ and $g^{\mathrm{\prime }}:X^{\mathrm{\prime }}\to X$. For an integral $d$-dimensional $V\subseteq X$, put $W=\overline{f(V)}$ and $e=\dim V-\dim W$. If $e>0$, both $g^{\ast }{f}_{\ast }[V]$ and $f_{\ast }^{\mathrm{\prime }}{g}^{\mathrm{\prime }\ast }[V]$ are zero: every component of the flat pullback of $V$ has dimension $d+n$, while its image has dimension at most $\dim W+n<d+n$. If $e=0$, set $m=[k(V):k(W)]$. For each component $W_i$ of $g^{-1}(W)$, let $C_i$ be the Artinian local ring of $W{\times }_Y{Y}^{\mathrm{\prime }}$ at its generic point. Flat pullback gives coefficient $\mathrm{\ell }(C_i)$ on $[W_i]$. The generic algebra of $V{\times }_Y{Y}^{\mathrm{\prime }}$ over $C_i$ is $C_i{\otimes }_{k(W)}k(V)$, a free $C_i$-module of rank $m$. Decomposing it into its Artinian local factors shows that the sum of the generic lengths of the components above $W_i$, weighted by their function-field degrees over $k(W_i)$, is $m\mathrm{\ell }(C_i)$. These are precisely the coefficients of $[W_i]$ in $f_{\ast }^{\mathrm{\prime }}{g}^{\mathrm{\prime }\ast }[V]$ and $g^{\ast }{f}_{\ast }[V]$, respectively. Hence the base-change identity holds on cycles and therefore on Chow groups. Finally, if $f$ is finite flat of constant degree $n$ between integral schemes of the same dimension, then $f$ is dominant and $[k(X):k(Y)]=n$, so $f_{\ast }[X]=n[Y]$ by step 1.1.