@Jared : on the "other side of Scholze's formula", there is the trace of a certain "test function", say $f$. The latter function is obtained by "base change transfer" from another function $\phi$. While $\phi$ is undoubtedly purely local (involves cohomology of some deformation space for p-divisible groups), and while $f$ is characterized by purely local conditions (matching of orbital integrals), I wonder if the very existence of $f$ has been proved by local means ???

I'm 3 minutes late. The only thing I can add is that, in case you don't have Eisenbud at hand, but you have Lang's "Algebra", then the reference for infinite separable extensions is Prop VIII.4.1.

thanks, I have made corrections. By the way, would you think of a more elementary argument ?

Argh, I don't know how to edit the previous comment. I meant the product of these two subvectorspaces, not subsets. I.e. linear combinations of products.

Well, the product of these two subsets of $C(K)$, or if you prefer, the $(C(K)^H)^{opp}$ submodule of $C(K)$ for the right multiplcation that is spanned by S (which is a $C(K)$-submodule for the left multiplication).

I guess that if you understand the finite case, the compact case will be very similar. So let us suppose K finite. Here is a candidate : let S be any multiplicity one left-submodule of C(K) containing a copy of each simple left module. Then take $M=N:=S. C(K)^H$. I am much too lazy to check anything at the moment, but I'm pretty sure it works for H=K or H=\{e\}$ ! On the other hand this is maybe not as nice a construction as what you would hope for.