The Stacks project

Lemma 71.11.5. Let $S$ be a scheme. Let $g : X' \to X$ be a morphism of algebraic spaces over $S$ and let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_ X$-algebras. Then there is a canonical isomorphism

\[ r : \underline{\text{Proj}}_{X'}(g^*\mathcal{A}) \longrightarrow X' \times _ X \underline{\text{Proj}}_ X(\mathcal{A}) \]

as well as a corresponding isomorphism

\[ \theta : r^*\text{pr}_2^*\left(\bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_ X(\mathcal{A})}(d)\right) \longrightarrow \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_{X'}(g^*\mathcal{A})}(d) \]

of $\mathbf{Z}$-graded $\mathcal{O}_{\underline{\text{Proj}}_{X'}(g^*\mathcal{A})}$-algebras.

Proof. Let $F$ be the functor ( and let $F'$ be the corresponding functor defined using $g^*\mathcal{A}$ on $X'$. We claim there is a canonical isomorphism $r : F' \to X' \times _ X F$ of functors (and of course $r$ is the isomorphism of the lemma). It suffices to construct the bijection $r : F'(T) \to X'(T) \times _{X(T)} F(T)$ for quasi-compact schemes $T$ over $S$. First, if $\xi = (d', f', \mathcal{L}', \psi ')$ is a quadruple over $T$ for $F'$, then we can set $r(\xi ) = (f', (d', g \circ f', \mathcal{L}', \psi '))$. This makes sense as $(g \circ f')^*\mathcal{A}^{(d)} = (f')^*(g^*\mathcal{A})^{(d)}$. The inverse map sends the pair $(f', (d, f, \mathcal{L}, \psi ))$ to the quadruple $(d, f', \mathcal{L}, \psi )$. We omit the proof of the final assertion (hint: reduce to the case of schemes by ├ętale localization and apply Constructions, Lemma 27.16.10). $\square$

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