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49.5 Finite morphisms

In this section we collect some observations about the constructions in the previous sections for finite morphisms. Let $f : Y \to X$ be a finite morphism of locally Noetherian schemes. Let $\omega _{Y/X}$ be as in Remark 49.2.11.

The first remark is that

\[ f_*\omega _{Y/X} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f_*\mathcal{O}_ Y, \mathcal{O}_ X) \]

as sheaves of $f_*\mathcal{O}_ Y$-modules. Since $f$ is affine, this formula uniquely characterizes $\omega _{Y/X}$, see Morphisms, Lemma 29.11.6. The formula holds because for $\mathop{\mathrm{Spec}}(A) = U \subset X$ affine open, the inverse image $V = f^{-1}(U)$ is the spectrum of a finite $A$-algebra $B$ and hence

\[ H^0(U, f_*\omega _{Y/X}) = H^0(V, \omega _{Y/X}) = \omega _{B/A} = \mathop{\mathrm{Hom}}\nolimits _ A(B, A) = H^0(U, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f_*\mathcal{O}_ Y, \mathcal{O}_ X)) \]

by construction. In particular, we obtain a canonical evaluation map

\[ f_*\omega _{Y/X} \longrightarrow \mathcal{O}_ X \]

which is given by evaluation at $1$ if we think of $f_*\omega _{Y/X}$ as the sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f_*\mathcal{O}_ Y, \mathcal{O}_ X)$.

The second remark is that using the evaluation map we obtain canonical identifications

\[ \mathop{\mathrm{Hom}}\nolimits _ Y(\mathcal{F}, f^*\mathcal{G} \otimes _{\mathcal{O}_ Y} \omega _{Y/X}) = \mathop{\mathrm{Hom}}\nolimits _ X(f_*\mathcal{F}, \mathcal{G}) \]

functorially in the quasi-coherent module $\mathcal{F}$ on $Y$ and the finite locally free module $\mathcal{G}$ on $X$. If $\mathcal{G} = \mathcal{O}_ X$ this follows immediately from the above and Algebra, Lemma 10.14.4. For general $\mathcal{G}$ we can use the same lemma and the isomorphisms

\[ f_*(f^*\mathcal{G} \otimes _{\mathcal{O}_ Y} \omega _{Y/X}) = \mathcal{G} \otimes _{\mathcal{O}_ X} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f_*\mathcal{O}_ Y, \mathcal{O}_ X) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f_*\mathcal{O}_ Y, \mathcal{G}) \]

of $f_*\mathcal{O}_ Y$-modules where the first equality is the projection formula (Cohomology, Lemma 20.52.2). An alternative is to prove the formula affine locally by direct computation.

The third remark is that if $f$ is in addition flat, then the composition

\[ f_*\mathcal{O}_ Y \xrightarrow {f_*\tau _{Y/X}} f_*\omega _{Y/X} \longrightarrow \mathcal{O}_ X \]

is equal to the trace map $\text{Trace}_ f$ discussed in Section 49.3. This follows immediately by looking over affine opens.

The fourth remark is that if $f$ is flat and $X$ Noetherian, then we obtain

\[ \mathop{\mathrm{Hom}}\nolimits _ Y(K, Lf^*M \otimes _{\mathcal{O}_ Y} \omega _{Y/X}) = \mathop{\mathrm{Hom}}\nolimits _ X(Rf_*K, M) \]

for any $K$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$ and $M$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$. This follows from the material in Duality for Schemes, Section 48.12, but can be proven directly in this case as follows. First, if $X$ is affine, then it holds by Dualizing Complexes, Lemmas 47.13.1 and 47.13.91 and Derived Categories of Schemes, Lemma 36.3.5. Then we can use the induction principle (Cohomology of Schemes, Lemma 30.4.1) and Mayer-Vietoris (in the form of Cohomology, Lemma 20.33.3) to finish the proof.

[1] There is a simpler proof of this lemma in our case.

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