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115.25 Dualizing modules on regular proper models

In Semistable Reduction, Situation 55.9.3 we let $\omega _{X/R}^\bullet = f^!\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$ be the relative dualizing complex of $f : X \to \mathop{\mathrm{Spec}}(R)$ as introduced in Duality for Schemes, Remark 48.12.5. Since $f$ is Gorenstein of relative dimension $1$ by Semistable Reduction, Lemma 55.9.2 we can use Duality for Schemes, Lemmas 48.25.10, 48.21.7, and 48.25.4 to see that

\[ \omega _{X/R}^\bullet = \omega _ X[1] \]

for some invertible $\mathcal{O}_ X$-module $\omega _ X$. This invertible module is often called the relative dualizing module of $X$ over $R$. Since $R$ is regular (hence Gorenstein) of dimension $1$ we see that $\omega _ R^\bullet = R[1]$ is a normalized dualizing complex for $R$. Hence $\omega _ X = H^{-2}(f^!\omega _ R^\bullet )$ and we see that $\omega _ X$ is not just a relative dualizing module but also a dualizing module, see Duality for Schemes, Example 48.22.1. Thus $\omega _ X$ represents the functor

\[ \textit{Coh}(\mathcal{O}_ X) \to \textit{Sets},\quad \mathcal{F} \mapsto \mathop{\mathrm{Hom}}\nolimits _ R(H^1(X, \mathcal{F}), R) \]

by Duality for Schemes, Lemma 48.22.5. This gives an alternative definition of the relative dualizing module in Semistable Reduction, Situation 55.9.3. The formation of $\omega _ X$ commutes with arbitrary base change (for any proper Gorenstein morphism of given relative dimension); this follows from the corresponding fact for the relative dualizing complex discussed in Duality for Schemes, Remark 48.12.5 which goes back to Duality for Schemes, Lemma 48.12.4. Thus $\omega _ X$ pulls back to the dualizing module $\omega _ C$ of $C$ over $K$ discussed in Algebraic Curves, Lemma 53.4.2. Note that $\omega _ C$ is isomorphic to $\Omega _{C/K}$ by Algebraic Curves, Lemma 53.4.1. Similarly $\omega _ X|_{X_ k}$ is the dualizing module $\omega _{X_ k}$ of $X_ k$ over $k$.

Lemma 115.25.1. In Semistable Reduction, Situation 55.9.3 the dualizing module of $C_ i$ over $k$ is

\[ \omega _{C_ i} = \omega _ X(C_ i)|_{C_ i} \]

where $\omega _ X$ is as above.

Proof. Let $t : C_ i \to X$ be the closed immersion. Since $t$ is the inclusion of an effective Cartier divisor we conclude from Duality for Schemes, Lemmas 48.9.7 and 48.14.2 that we have $t^!(\mathcal{L}) = \mathcal{L}(C_ i)|_{C_ i}$ for every invertible $\mathcal{O}_ X$-module $\mathcal{L}$. Consider the commutative diagram

\[ \xymatrix{ C_ i \ar[r]_ t \ar[d]_ g & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(k) \ar[r]^ s & \mathop{\mathrm{Spec}}(R) } \]

Observe that $C_ i$ is a Gorenstein curve (Semistable Reduction, Lemma 55.9.2) with invertible dualizing module $\omega _{C_ i}$ characterized by the property $\omega _{C_ i}[0] = g^!\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}$. See Algebraic Curves, Lemma 53.4.1, its proof, and Algebraic Curves, Lemmas 53.4.2 and 53.5.2. On the other hand, $s^!(R[1]) = k$ and hence

\[ \omega _{C_ i}[0] = g^! s^!(R[1]) = t^!f^!(R[1]) = t^!\omega _ X \]

Combining the above we obtain the statement of the lemma. $\square$

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