The Stacks project

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

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

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

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

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

of $f_*\mathcal{O}_ Y$-modules where the first equality is the projection formula (Cohomology, Lemma 20.54.2). 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

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

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.