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Let $k$ be an algebraically closed field and let $X$ be a finite type $k$-scheme that is Cohen-Macaulay and equidimensional. Under these assumptions there is a relative dualizing sheaf $\omega_{X/k}$ that is an $\mathcal{O}_X$-module.

  • Is $\omega_{X/k}$ Cohen-Macaulay?
  • Is $\omega_{X/k}$ at least (S$_1$)? (Equivalently, does $\omega_{X/k}$ have no embedded associated primes?)
  • What if we assume that $X$ is of dimension $1$ (but is possibly nonreduced)?

If $X$ is of dimension $1$ and is reduced (and hence generically $k$-smooth), then a positive answer to all these questions may be inferred from Lemma 5.2.1 of Conrad's book "Grothendieck Duality and Base Change".

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    $\begingroup$ Yes, it is: see Bourbaki, Algèbre Commutative 10, §9, Proposition 1 (not yet translated in english, I am afraid). $\endgroup$ – abx Dec 25 '15 at 5:45
  • $\begingroup$ The reference suggested by abx is very similar to the argument in my answer upon giving the proof that the notion of "dualizing module" in Bourbaki agrees with Grothendieck's notion of dualizing sheaf in the affine CM setting (but with a bit more efficiency by invoking how depth is encoded by Ext's against the residue field, rather than by going through a slicing argument as in my answer, though the latter underlies the proof of the former). $\endgroup$ – nfdc23 Dec 25 '15 at 15:50
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    $\begingroup$ BTW, the equidimensional assumption is superfluous. If $X$ is CM, then each connected component is equidimensional. $\endgroup$ – Sándor Kovács Dec 26 '15 at 0:51
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This is an immediate application of the behavior of dualizing complexes relative to finite morphisms (such as closed immersions).

To explain this, first recall that if $f:Y \rightarrow Z$ is a finite morphism between noetherian schemes and $\omega$ is a dualizing complex on $Z$ then $f^{!}(\omega)$ is a dualizing complex on $Y$, where the functor $f^{!} = \mathbf{R}\mathscr{H}om_Z(f_{\ast}(O_Y),\cdot)$ on $D^+_{\rm{qc}}(Z) = D({\rm{QCoh}}(Z))$ is viewed with values in $D^+_{\rm{qc}}(Y) = D^+({\rm{QCoh}}(Y))$ in the evident manner. Recall also that dualizing complexes are of Zariski-local nature, to be used tacitly below.

Now consider a Cohen-Macaulay scheme $Z$ with pure dimension $n\ge 1$, so a "normalized" dualizing complex on $Z$ (i.e., a dualizing complex whose associated "codimension" function coincides with the usual one) is given by $\omega_Z[n]$ for dualizing sheaves $\omega_Z$ on $Z$ (which in turn are unique up to tensoring against a line bundle). Suppose $a \in O_Z(Z)$ is $O_Z$-regular (i.e., nowhere a zero-divisor on $O_Z$) and vanishes somewhere, so $Y := V(a) \subset Z$ is non-empty and CM of pure dimension $n-1$. Then for the inclusion $j:Y \hookrightarrow Z$, a "normalized" dualizing complex $\omega_Y[n-1]$ on $Y$ is given by $j^{!}(\omega_Z[n])$. This says that $\mathscr{E}xt^i_Z(j_{\ast}O_Y, \omega_Z)$ vanishes for $i \ne 1$ and that the quasi-coherent $j_{\ast}O_Y$-module $\mathscr{E}xt^1_Z(j_{\ast}O_Y, \omega_Z)$ is a dualizing sheaf on $Y$ when "viewed" as a quasi-coherent $O_Y$-module.

The long exact sequence for $\mathscr{E}xt^{\bullet}_Z(\cdot, \omega_Z)$'s arising from the short exact sequence $$0 \rightarrow O_Z \stackrel{a}{\rightarrow} O_Z \rightarrow j_{\ast}O_Y \rightarrow 0$$ now yields that $\omega_Z$ has vanishing $a$-torsion (i.e., $a$ is $\omega_Z$-regular) and that $\omega_Z/(a \cdot\omega_Z) \simeq \mathscr{E}xt^1_Z(j_{\ast}O_Y, \omega_Z) =: \omega_Y$ is a dualizing sheaf on $Y$. This sets up an inductive formalism that we shall now use.

In the original setup with a CM scheme $X$ of finite type over a field $k$ (which may be arbitrary) such that $X$ has pure dimension $n$, we can run such a slicing argument $n$ times on a sufficiently small Zariski-open neighborhood of any given closed point $x$ of $X$ (upon "spreading out" a system of parameters in the $n$-dimensional CM local ring $O_{X,x}$) to deduce that any system of parameters in the maximal ideal of $O_{X,x}$ is a regular sequence for the stalk $\omega_{X,x}$ of any dualizing sheaf $\omega_X$ for $X$ (such as a relative dualizing sheaf $\omega_{X/k}$, since $k$ is regular). Thus, $\omega_X$ has CM stalks at all closed points, so it is CM at all points (by localization) and hence is a CM coherent sheaf.

[An alternative approach would be to use the link to local duality on stalks to carry out arguments directly on Spec($O_{X,x}$) without needing to perform spreading-out, but the above seems more direct given the context of the question as posed.]

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  • $\begingroup$ Thank you. You probably mean the long exact sequence for $\mathscr{E}xt^{\bullet}_Z(\cdot, \omega_Z)$ though before the display. $\endgroup$ – O-Ren Ishii Dec 25 '15 at 17:56
  • $\begingroup$ @O-RenIshii: Good catch, now fixed. $\endgroup$ – nfdc23 Dec 25 '15 at 19:25
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Actually, it is even better than that. Here are the facts:

  1. $\omega_X$ is always $S_2$, even if $X$ is not CM
  2. If $X$ is $S_2$, then $X$ is CM if and only if $\omega_X$ is CM.

You can find a proof of these for instance on page 181 in Kollár-Mori. (They only state these for projective schemes, so you might need to work a little on the second statement if you need it in more general settings, but in your situation it should be enough using CM-ification on a projective closure of a neighbourhood of any point).

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  • $\begingroup$ Thank you. I am a little bit puzzled by the reference you give though. I thought that $\omega_{X/k}$ exists as a sheaf only if $X$ is Cohen-Macaulay (otherwise $\omega_{X/k}$ is a complex in some derived category), whereas the $\omega_X$ discussed in Kollar-Mori seems to be a sheaf without the CM assumption on $X$. Is the catch that Kollar-Mori only use the non-derived duality as the defining property of $\omega_X$, so $\omega_X$ has a chance to exist even when $\omega_{X/k}$ does not (but both agree when $\omega_{X/k}$ exists)? Could you clarify? $\endgroup$ – O-Ren Ishii Dec 25 '15 at 18:06
  • $\begingroup$ Kollár–Mori use [p.180] the same definition that Hartshorne uses: a dualising sheaf is an object representing the functor $H^n(X, -)^\vee$ (on $\underline{\operatorname{Coh}}_X$). Thus, it need not be a dualising sheaf in the sense that you're thinking of. In fact, Hartshorne proves [Thm III.7.6] that this is equivalent to $X$ being CM and equidimensional. $\endgroup$ – R. van Dobben de Bruyn Dec 25 '15 at 19:09
  • $\begingroup$ Sorry, when I say Hartshorne, I mean Algebraic Geometry, not Residues and Duality. $\endgroup$ – R. van Dobben de Bruyn Dec 25 '15 at 19:15
  • $\begingroup$ I seemed to remember that CM-ification is problematic in general: which existence result(s) do you have in mind (if any) ? $\endgroup$ – Matthieu Romagny Dec 25 '15 at 20:28
  • $\begingroup$ @MatthieuRomagny: I guess you are right, there is something to be worked out here. I was thinking of using ams.org/journals/tran/2000-352-06/S0002-9947-00-02603-9, but it does not provide a strong CM-ification. I still think this approach should work. $\endgroup$ – Sándor Kovács Dec 26 '15 at 1:00

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