Hom(f_! f^! \one, \Hhat) and Hom(\one, f_* f^* \Hhat)
Actually I was starting to write this email much earlier today, only to see that I made a foolish mistake. I thought: since we have a map f_! \r f_, we get a map f^! \r f^ for free (adjointness) reasons. However, this is wrong, as I now think. At least there is no formal reason for this to exist. Anyhow, I think I can prove the agreement for f : X \r S such that f is projective and X (and S) are regular. The point is, the two groups are isomorphic, respectively, to
Hom(f^! \one, f^! \Hhat) and Hom(f^* \one, f^* \Hhat)
We know f^* \one is isomorphic to f^! \one (d)[2d] by relative and absolute purity. Imagine for a second that f is smooth. Then we know for any object E in DM_Beilinson: f^* E = f^! E (d)[2d]. This includes E=H_Beilinson, E=H_hat. We are slightly less fortunate, since we want it to hold for f only between regular schemes. The point is to show the commutativity of a diagram featuring f^! H_B, f^* H_B (d)[2d], f^! Deligne, f^* Deligne (d)[2d]. Generically (on S), f is smooth, and the regulator factors over K-theory of open subschemes which are the fibers over open subschemes U \subset S. So generically, one part of the comparison is dealt with by the previous argument. This brings us to the following question (I’m now too lazy to write down the diagrams, they are currently on my blackboard): assume X \r S is smooth and projective. We can either take the relative purity isomorphism, or combine the absolute (X \subset P^n_S) and the relative one (P^n_S \r S). Do the two comparison isomorphisms agree? I think: yes, they do agree: the absolute purity iso is induced by product (in K-theory) with some expression involving the relative normal bundle to the regular embedding X \subset P^n_S, while the relative ones involve the tangent bundle of the smooth maps. In such a situation, we ought to have a short exact sequence of these bundles, much as for a composition of regular embeddings (see e.g. Fulton, Intersection Theory book, B.7.4). This will translate into the commutativity of the necessary diagram.
nLab page on On the two definitions of cohomology