# Dualizing sheaf in mixed characteristic for regular schemes.

I've been looking many places, but everything I find seems to either talk about (a) varieties or (b) extremely general situations with dualizing complexes. As I am not in the situation of (a) (i.e. over a field), and I don't yet understand (b), I was wondering if there was some place which talks about the middle ground. Specifically:

Let $X$ be a Noetherian, regular scheme of finite type over $\mathbb{Z}_p$, but not smooth. Then I've read in many places (even with just Cohen-Macaulay) that the dualizing complex is just a sheaf. Is there an "elementary" description of this sheaf?

Any references would be greatly appreciated.

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Assume $X$ is faithfully flat over the base dvr $R$ (or else you're over a field), and connected so the fibers of $X$ over Spec($R$) have a common pure dimension $d$. Any affine open in $X$ is Zariski-closed in an affine space over $R$, so more generally consider an open cover of $X$ by opens $U_i$ which admit closed immersions $j_i:U_i \hookrightarrow Y_i$ over $R$ where $Y_i$ is $R$-smooth of pure relative dimension $n_i$. Then ${\mathcal{Ext}}^{n_i-d}_{Y_i}(O_{U_i},\Omega^{d_i}_{Y_i/R})$ is a coherent $O_{U_i}$-module and these glue (very indirectly!) to give $\omega_{X/R}$. – user30180 Mar 5 '13 at 2:30
Typo: $\Omega^{d_i}_{Y_i/R}$ should be $\Omega^{n_i}_{Y_i/R}$ above. – user30180 Mar 5 '13 at 2:31
Thanks, this is exactly what I was looking for. Do you have a reference for this? – rghthndsd Mar 5 '13 at 14:40
Do you want a reference just that such Ext's glue to make a sheaf, or that it also has specific useful properties (such as for global duality or local duality)? And is the reference permitted to dualizing complexes? Is your $X$ of interest projective or quasi-projective? – user30180 Mar 6 '13 at 14:14
Typo: "permitted to use dualizing" – user30180 Mar 6 '13 at 14:15