Fulton's "Intersection theory" book contains the following fact (example 18.3.19):

Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a dualizing sheaf $\omega_Z$) and is of dimension $n$. If $E$ is locally free coherent sheaf on $X$, then: $$\tau_k(E) = (-1)^{n-k}\tau_k(E^{\vee}\otimes \omega_X) \ \ (*)$$ in $A_k(X)_{\mathbb Q}$, the $k$-th Chow group of $X$ with rational coefficients. Here $\tau: K_0(X) \to A_*(X)_{\mathbb Q}$ is the generalized Riemann-Roch homomorphism.

The formula follows from a more general one for complexes with coherent cohomology (and without Cohen-Macaulayness): $$ \sum (-1)^i\tau_k(\mathcal H^i(C^{\cdot})) = \(-1)^k\sum(-1)^i\tau_k(\mathcal H^i(RHom(C^{\cdot},\omega^{\cdot}_X))) \ \ (**)$$

In a proof I would like to use (*) in a more general setting:

Does anyone know a reference for

`(**)`

or`(*)`

when $X$ is imbeddable in a regular scheme, not necessarily over a field (I am willing to assume $X$ is finite over some complete regular local ring)?

The original source of (**) (Fulton-MacPherson "Categorical framework for study of singular spaces") hints that a generalization is possible, then refers to Delign's appendix of Hartshorne "Residues and Duality"! We all know that fleshing out the details there is non-trivial, however.