Lemma 48.11.1 (0BUZ)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.11: Right adjoint of pushforward for finite morphisms
Lemma 48.11.1 (cite)

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Lemma 48.11.1. With notation as above. The functor $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ Y, -)$ is a right adjoint to the restriction functor $\textit{Mod}(f_*\mathcal{O}_ Y) \to \textit{Mod}(\mathcal{O}_ X)$. For an affine open $U \subset X$ we have

\[ \Gamma (U, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ Y, \mathcal{F})) = \mathop{\mathrm{Hom}}\nolimits _ A(B, \mathcal{F}(U)) \]

where $A = \mathcal{O}_ X(U)$ and $B = \mathcal{O}_ Y(f^{-1}(U))$.

Proof. Adjointness follows from Modules, Lemma 17.22.3. As $f$ is affine we see that $f_*\mathcal{O}_ Y$ is the quasi-coherent sheaf corresponding to $B$ viewed as an $A$-module. Hence the description of sections over $U$ follows from Schemes, Lemma 26.7.1. $\square$

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