The Stacks project

Lemma 49.15.1. Let $f : Y \to X$ be a quasi-finite separated morphism of Noetherian schemes. For every pair of affine opens $\mathop{\mathrm{Spec}}(B) = V \subset Y$, $\mathop{\mathrm{Spec}}(A) = U \subset X$ with $f(V) \subset U$ there is an isomorphism

\[ H^0(V, f^!\mathcal{O}_ Y) = \omega _{B/A} \]

where $f^!$ is as in Duality for Schemes, Section 48.16. These isomorphisms are compatible with restriction maps and define a canonical isomorphism $H^0(f^!\mathcal{O}_ X) = \omega _{Y/X}$ with $\omega _{Y/X}$ as in Remark 49.2.11. Similarly, if $f : Y \to X$ is a quasi-finite morphism of schemes of finite type over a Noetherian base $S$ endowed with a dualizing complex $\omega _ S^\bullet $, then $H^0(f_{new}^!\mathcal{O}_ X) = \omega _{Y/X}$.

Proof. By Zariski's main theorem we can choose a factorization $f = f' \circ j$ where $j : Y \to Y'$ is an open immersion and $f' : Y' \to X$ is a finite morphism, see More on Morphisms, Lemma 37.42.3. By our construction in Duality for Schemes, Lemma 48.16.2 we have $f^! = j^* \circ a'$ where $a' : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_{Y'})$ is the right adjoint to $Rf'_*$ of Duality for Schemes, Lemma 48.3.1. By Duality for Schemes, Lemma 48.11.4 we see that $\Phi (a'(\mathcal{O}_ X)) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f'_*\mathcal{O}_{Y'}, \mathcal{O}_ X)$ in $D_\mathit{QCoh}^+(f'_*\mathcal{O}_{Y'})$. In particular $a'(\mathcal{O}_ X)$ has vanishing cohomology sheaves in degrees $< 0$. The zeroth cohomology sheaf is determined by the isomorphism

\[ f'_*H^0(a'(\mathcal{O}_ X)) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f'_*\mathcal{O}_{Y'}, \mathcal{O}_ X) \]

as $f'_*\mathcal{O}_{Y'}$-modules via the equivalence of Morphisms, Lemma 29.11.6. Writing $(f')^{-1}U = V' = \mathop{\mathrm{Spec}}(B')$, we obtain

\[ H^0(V', a'(\mathcal{O}_ X)) = \mathop{\mathrm{Hom}}\nolimits _ A(B', A). \]

As the zeroth cohomology sheaf of $a'(\mathcal{O}_ X)$ is a quasi-coherent module we find that the restriction to $V$ is given by $\omega _{B/A} = \mathop{\mathrm{Hom}}\nolimits _ A(B', A) \otimes _{B'} B$ as desired.

The statement about restriction maps signifies that the restriction mappings of the quasi-coherent $\mathcal{O}_{Y'}$-module $H^0(a'(\mathcal{O}_ X))$ for opens in $Y'$ agrees with the maps defined in Lemma 49.2.3 for the modules $\omega _{B/A}$ via the isomorphisms given above. This is clear.

Let $f : Y \to X$ be a quasi-finite morphism of schemes of finite type over a Noetherian base $S$ endowed with a dualizing complex $\omega _ S^\bullet $. Consider opens $V \subset Y$ and $U \subset X$ with $f(V) \subset U$ and $V$ and $U$ separated over $S$. Denote $f|_ V : V \to U$ the restriction of $f$. By the discussion above and Duality for Schemes, Lemma 48.20.9 there are canonical isomorphisms

\[ H^0(f_{new}^!\mathcal{O}_ X)|_ V = H^0((f|_ V)^!\mathcal{O}_ U) = \omega _{V/U} = \omega _{Y/X}|_ V \]

We omit the verification that these isomorphisms glue to a global isomorphism $H^0(f_{new}^!\mathcal{O}_ X) \to \omega _{Y/X}$. $\square$


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