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.

  • $\begingroup$ The only point of Deligne's appendix is to introduce a slick way to create the abstract formalism of coherent duality theory without spending a lot of time under the hood (and correspondingly without having a concrete interpretation of anything, though Verdier later tried to extract the concreteness from it anyway). So referring to Deligne's appendix seems to be nothing more or less than appealing to the existence of the duality theory as a black box. Have you looked at the general statements of Grothendieck's local duality theorems? Why doesn't the proof of (**) you see work in general? $\endgroup$ – BCnrd Aug 18 '10 at 4:54
  • $\begingroup$ @BCnrd: thanks for your comment. I did look, but didn't feel confident enough to be absolutely sure that all the details would go through. However, it might be possible that the general case has already appeared somewhere, hence the question. $\endgroup$ – Hailong Dao Aug 18 '10 at 5:46
  • $\begingroup$ Hailong, since you say you're happy to take $X$ finite over a complete local noetherian ring, I then wonder what you are using to replace Chow groups (which are really for things of finite type over a field). So I'm not sure what is meant to play the role of $\tau$ in your setting. Good night, $\endgroup$ – BCnrd Aug 18 '10 at 6:40
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    $\begingroup$ @BCnrd: You can certainly define Chow groups in general (see Fulton Section 20) or this MO question mathoverflow.net/questions/17634/…. $\tau$ can also be defined in general, see Fulton 20.1 or Section 1 of this paper: springerlink.com/content/4h3pqre2t80q378g. Technically, $\tau$ has to be defined using the regular scheme $X$ embeds to, but I am willing to fix that. $\endgroup$ – Hailong Dao Aug 18 '10 at 7:00

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