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I've seen and used the following map from the algebraic $K$-theory to the differential forms on a scheme $X$: $$ K_n(X) \to H^0(X,\Omega^n_X)$$ sending $K_1(X)\ni f\mapsto d\log f$, and extending to a map of algebras under cup product/exterior product. It is easier to define this map on the subgroup of Milnor $K$-theory, where I see it most often defined, but Bloch defines the map on the full algebraic $K$-groups, e.g. using Chern classes in his paper On the tangent space of Quillen $K$-theory (where it's attributed to "secret papers of Gersten").

Whatever the definition, the functoriality under pullbacks is easy to see. However, I'm strongly convinced that it should also be true that it is functorial for finite pushforward, i.e. via the Bass-Tate transfer map on $K$-theory, and the usual trace for differential forms. Does anyone have a reference or proof of this?

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    $\begingroup$ I don’t understand your map. Usually the $d\log$ map is from the $K$-theory sheaf $\mathcal{K}_{n,X}$ to $\Omega_{X}^{n}$, and of course $K_{n}(X)$ is typically larger than $H^{0}(X,\mathcal{K}_{n,X})$. $\endgroup$
    – Oli Gregory
    Commented Jul 31, 2023 at 21:20
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    $\begingroup$ is there not a map from $K_n(X)$ to $H^0(X,\mathcal{K}_{n,X})$? anyway, you can also take $X$ to be affine if you want, and ask the same question. $\endgroup$
    – xir
    Commented Jul 31, 2023 at 22:01
  • $\begingroup$ Yes you’re right, thanks. $\endgroup$
    – Oli Gregory
    Commented Aug 1, 2023 at 10:23
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    $\begingroup$ I'm guessing this Bloch map coincides with $\pi_n$ of the Dennis trace map from K-theory to Hochschild homology. For the Dennis trace the desired functoriality is clear, since it is even functorial in the dg-category of perfect complexes. $\endgroup$
    – Marc Hoyois
    Commented Aug 1, 2023 at 13:14

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This answer just amounts to adding a reference to Marc Hoyois' comment: there is a discussion of precisely this on pages 393-394 in Scholl's An introduction to Kato's Euler systems, London Math. Soc. Lecture Note Ser., 254 Cambridge University Press, Cambridge, 1998, 379–460.

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