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In Motivic cohomology of smooth geometrically cellular varieties (1999), Corollary 3.5, Bruno Kahn proves the following statement. Consider a cellular variety $X$ (i.e. it admits a filtration by closed subschemes $X_i$ such that the $X_i\setminus X_{i-1}$ are disjoint unions of affine spaces) over a field $k$ of characteristic 0 (I'm interested in $k=\mathbb C$). Then there is a natural isomorphism in $DM(k)$

$$M^c(X)\cong \bigoplus_i CH_i(X)\{i\} \hspace{1cm}(*)$$

where $M^c$ denotes the motive with compact support of $X$, $CH_i$ denotes the Chow group of cycles of dimension $i$ modulo linear equivalence, and $-\{i\}$ means $-\otimes \mathbb Z(i)[2i]$. This is a natural isomorphism of contravariant functors, by considering morphisms to be flat equidimensional maps.

Now, both $M^c$ and $CH_i$ are also covariant functors, if we instead take our maps to be proper.

Question: is there a natural isomorphism as in $(*)$, only now $M^c$ and $CH_i$ are understood to be covariant functors?

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  • $\begingroup$ Yes, this is fine. If you unravel what the isomorphism map is, you'll find it is the sum of maps $CH_i(X)\{i\}\to M^c(X)$ which are the adjoints to maps $CH_i(X)\to Hom(\mathbb Z\{i\},M^c(X))$ which take the class of a subvariety to the motivic Borel--Moore fundamental class of it in $X$. It suffices to check this is natural for proper maps, and it is, essentially because proper pushforward for Chow groups and for motives with compact support are defined analogously (using the degree of $f$). $\endgroup$
    – Bruno Stonek
    Commented Jun 14 at 10:06

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