When is the pullback in Chow groups defined?

Question

This is the first time I ask a question on Mathoverflow, so I apologize in advance if it is not suitable/a duplicate/otherwise inappropriated.

I am thinking about Voevodsky's category of motives and I realized that in his presheaves with transfers formalism pullbacks for Chow groups are defined for arbitrary maps of smooth schemes. Precisely if $f:X\to Y$ is a map of smooth schemes and $\alpha\in CH_*(Y)$ he defines

$$f^*\alpha = (pr_1)_*(\Gamma_f \cdot (pr_2)^*\alpha)$$

where $pr_1,pr_2$ are the projection maps from $X\times Y$ and $\Gamma_f$ is the closed subscheme of $X\times Y$ determined by the graph of $f$ (note that $(pr_1)_*$ is well defined more or less because the restriction of $pr_1$ to $\Gamma_f$ is an isomorphism).

After researching a bit I found a paper by Bloch ("Algebraic cycles and Higher K-theory") he seems to define the pullback for all maps with smooth target. However in classical treatments of intersection theory I've only seen $f^*$ defined for flat maps.

Q: In what generality is the pullback of cycle classes defined?

Answer 1

You have to distinguish between pullbacks of cycles, pullbacks on Chow groups, and pullbacks of relative cycles.

• You cannot always pull back cycles. If $f: Y\to X$ is a morphism of (arbitrary) schemes and $Z\subset X$ is an elementary cycle, $f^*(Z)$ is defined provided that $Z$ is ``in good position with respect to $f$'', which simply means that $f^{-1}(Z)$ has the same codimension in $Y$ as $Z$ had in $X$; this is always true if $f$ is flat. The formula for $f^*(Z)$ is the one you wrote, where the intersection uses Serre's multiplicities.

• Pullbacks on Chow groups are defined at least for arbitrary maps between smooth schemes over fields (see below for various generalizations). This is because of Chow's moving lemma, which says that you can always replace a cycle by a rationally equivalent one which is in good position with respect to a given map.

• If $X$ is of finite type over $S$ regular, an equidimensional relative cycle on $X/S$ is a cycle on $X$ which is equidimensional and dominant over (a connected component of) $S$. For instance a finite correspondence from $X$ to $Y$ over $S$ is such a cycle on $X\times_SY/X$, which is moreover finite over $X$. Equidimensional relative cycles can be pulled back along any map $T\to S$. There is a notion of relative cycle which need not be equidimensional and these notions can be extended to singular schemes, but it gets technical, see Suslin-Voevodsky http://www.math.uiuc.edu/K-theory/0035/susvoe2.pdf or Cisinski-Déglise http://arxiv.org/pdf/0912.2110v3.

With hard work, the functoriality on Chow groups can be extended to Bloch's cycle complexes $z^r(X,*)$, and hence to higher Chow groups. This was done by Bloch for smooth schemes over a field, by Levine for smooth schemes over a Dedekind domain (https://www.uni-due.de/~bm0032/publ/ChowMovLemFinal.pdf), and recently by Spitzweck for smooth schemes over variable Dedekind domains (http://arxiv.org/pdf/1207.4078.pdf). A huge advantage of Voevodsky's definition of the motivic complexes is that the functoriality is apparent, since these complexes are defined in terms of equidimensional cycles. At the end of the day Voevodsky's complex is equivalent to Bloch's, but that requires a moving lemma (which is known for fields but not for more general Dedekind domains). The basic idea for the comparison is that, up to controlled rational equivalence, a codimension $r$ cycle on $X$ is as good as a codimension $r$ cycles on $X\times\mathbb{A}^r$ (by homotopy invariance), and any such can be moved to be of relative dimension zero over $X$.