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107.19 The relative dualizing sheaf

This section serves mainly to introduce notation in the case of families of curves. Most of the work has already been done in the chapter on duality.

Let $f : X \to S$ be a family of curves. There exists an object $\omega _{X/S}^\bullet $ in $D_\mathit{QCoh}(\mathcal{O}_ X)$, called the relative dualizing complex, having the following property: for every base change diagram

\[ \xymatrix{ X_ U \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ U \ar[r]^ g & S } \]

with $U = \mathop{\mathrm{Spec}}(A)$ affine the complex $\omega _{X_ U/U}^\bullet = L(g')^*\omega _{X/S}^\bullet $ represents the functor

\[ D_\mathit{QCoh}(\mathcal{O}_{X_ U}) \longrightarrow \text{Mod}_ A,\quad K \longmapsto \mathop{\mathrm{Hom}}\nolimits _ U(Rf_*K, \mathcal{O}_ U) \]

More precisely, let $(\omega _{X/S}^\bullet , \tau )$ be the relative dualizing complex of the family as defined in Duality for Spaces, Definition 84.9.1. Existence is shown in Duality for Spaces, Lemma 84.9.5. Moreover, formation of $(\omega _{X/S}^\bullet , \tau )$ commutes with arbitrary base change (essentially by definition; a precise reference is Duality for Spaces, Lemma 84.9.6). From now on we will identify the base change of $\omega _{X/S}^\bullet $ with the relative dualizing complex of the base changed family without further mention.

Let $\{ S_ i \to S\} $ be an étale covering with $S_ i$ affine such that $X_ i = X \times _ S S_ i$ is a scheme, see Lemma 107.4.3. By Duality for Spaces, Lemma 84.10.1 we find that $\omega _{X_ i/S_ i}^\bullet $ agrees with the relative dualizing complex for the proper, flat, and finitely presented morphism $f_ i : X_ i \to S_ i$ of schemes discussed in Duality for Schemes, Remark 48.12.5. Thus to prove a property of $\omega _{X/S}^\bullet $ which is étale local, we may assume $X \to S$ is a morphism of schemes and use the theory developped in the chapter on duality for schemes. More generally, for any base change of $X$ which is a scheme, the relative dualizing complex agrees with the relative dualizing complex of Duality for Schemes, Remark 48.12.5. From now on we will use this identification without further mention.

In particular, let $\mathop{\mathrm{Spec}}(k) \to S$ be a morphism where $k$ is a field. Denote $X_ k$ the base change (this is a scheme by Spaces over Fields, Lemma 70.9.3). Then $\omega _{X_ k/k}^\bullet $ is isomorphic to the complex $\omega _{X_ k}^\bullet $ of Algebraic Curves, Lemma 53.4.1 (both represent the same functor and so we can use the Yoneda lemma, but really this holds because of the remarks above). We conclude that the cohomology sheaves $H^ i(\omega _{X_ k/k}^\bullet )$ are nonzero only for $i = 0, -1$. If $X_ k$ is Cohen-Macaulay and equidimensional of dimension $1$, then we only have $H^{-1}$ and if $X_ k$ is in addition Gorenstein, then $H^{-1}(\omega _{X_ k/k})$ is invertible, see Algebraic Curves, Lemmas 53.4.2 and 53.5.2.

Lemma 107.19.1. Let $X \to S$ be a family of curves with Cohen-Macaulay fibres equidimensional of dimension $1$ (Lemma 107.8.2). Then $\omega _{X/S}^\bullet = \omega _{X/S}[1]$ where $\omega _{X/S}$ is a pseudo-coherent $\mathcal{O}_ X$-module flat over $S$ whose formation commutes with arbitrary base change.

Proof. We urge the reader to deduce this directly from the discussion above of what happens after base change to a field. Our proof will use a somewhat cumbersome reduction to the Noetherian schemes case.

Once we show $\omega _{X/S}^\bullet = \omega _{X/S}[1]$ with $\omega _{X/S}$ flat over $S$, the statement on base change will follow as we already know that formation of $\omega _{X/S}^\bullet $ commutes with arbitrary base change. Moreover, the pseudo-coherence will be automatic as $\omega _{X/S}^\bullet $ is pseudo-coherent by definition. Vanishing of the other cohomology sheaves and flatness may be checked étale locally. Thus we may assume $f : X \to S$ is a morphism of schemes with $S$ affine (see discussion above). Write $S = \mathop{\mathrm{lim}}\nolimits S_ i$ as a cofiltered limit of affine schemes $S_ i$ of finite type over $\mathbf{Z}$. Since $\mathcal{C}\! \mathit{urves}^{CM, 1}$ is locally of finite presentation over $\mathbf{Z}$ (as an open substack of $\mathcal{C}\! \mathit{urves}$, see Lemmas 107.8.2 and 107.5.3), we can find an $i$ and a family of curves $X_ i \to S_ i$ whose pullback is $X \to S$ (Limits of Stacks, Lemma 100.3.5). After increasing $i$ if necessary we may assume $X_ i$ is a scheme, see Limits of Spaces, Lemma 68.5.11. Since formation of $\omega _{X/S}^\bullet $ commutes with arbitrary base change, we may replace $S$ by $S_ i$. Doing so we may and do assume $S_ i$ is Noetherian. Then $f$ is clearly a Cohen-Macaulay morphism (More on Morphisms, Definition 37.20.1) by our assumption on the fibres. Also then $\omega _{X/S}^\bullet = f^!\mathcal{O}_ S$ by the very construction of $f^!$ in Duality for Schemes, Section 48.16. Thus the lemma by Duality for Schemes, Lemma 48.23.3. $\square$

Definition 107.19.2. Let $f : X \to S$ be a family of curves with Cohen-Macaulay fibres equidimensional of dimension $1$ (Lemma 107.8.2). Then the $\mathcal{O}_ X$-module

\[ \omega _{X/S} = H^{-1}(\omega _{X/S}^\bullet ) \]

studied in Lemma 107.19.1 is called the relative dualizing sheaf of $f$.

In the situation of Definition 107.19.2 the relative dualizing sheaf $\omega _{X/S}$ has the following property (which moreover characterizes it locally on $S$): for every base change diagram

\[ \xymatrix{ X_ U \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ U \ar[r]^ g & S } \]

with $U = \mathop{\mathrm{Spec}}(A)$ affine the module $\omega _{X_ U/U} = (g')^*\omega _{X/S}$ represents the functor

\[ \mathit{QCoh}(\mathcal{O}_{X_ U}) \longrightarrow \text{Mod}_ A,\quad \mathcal{F} \longmapsto \mathop{\mathrm{Hom}}\nolimits _ A(H^1(X, \mathcal{F}), A) \]

This follows immediately from the corresponding property of the relative dualizing complex given above. In particular, if $A = k$ is a field, then we recover the dualizing module of $X_ k$ as introduced and studied in Algebraic Curves, Lemmas 53.4.1, 53.4.2, and 53.5.2.

Lemma 107.19.3. Let $X \to S$ be a family of curves with Gorenstein fibres equidimensional of dimension $1$ (Lemma 107.12.2). Then the relative dualizing sheaf $\omega _{X/S}$ is an invertible $\mathcal{O}_ X$-module whose formation commutes with arbitrary base change.

Proof. This is true because the pullback of the relative dualizing module to a fibre is invertible by the discussion above. Alternatively, you can argue exactly as in the proof of Lemma 107.19.1 and deduce the result from Duality for Schemes, Lemma 48.25.10. $\square$


Comments (3)

Comment #4948 by Min on

What is in the functor the dualizing sheaf represents? Should it be ?

Comment #4949 by Min on

What is T in the functor the dualizing sheaf represents? Should it be U?


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