The Stacks project

Lemma 37.56.5. Let $X \to S$ be a finite type morphism of affine schemes. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$. Let $X = \bigcup U_ i$ be a standard affine open covering. The following are equivalent

  1. the equivalent conditions of Lemma 37.56.1 hold for the pairs $(U_ i \to S, E|_{U_ i})$,

  2. the equivalent conditions of Lemma 37.56.1 hold for the pair $(X \to S, E)$.

Proof. The implication (2) $\Rightarrow $ (1) is Lemma 37.56.4. Assume (1). Say $S = \mathop{\mathrm{Spec}}(R)$ and $X = \mathop{\mathrm{Spec}}(A)$ and $U_ i = D(f_ i)$. Write $1 = \sum f_ ig_ i$ in $A$. Consider the surjections

\[ R[x_ i, y_ i, z_ i] \to R[x_ i, y_ i, z_ i]/(\sum y_ iz_ i - 1) \to A. \]

which sends $y_ i$ to $f_ i$ and $z_ i$ to $g_ i$. Note that $R[x_ i, y_ i, z_ i]/(\sum y_ iz_ i - 1)$ is pseudo-coherent as an $R[x_ i, y_ i, z_ i]$-module. Thus it suffices to prove that the pushforward of $E$ to $T = \mathop{\mathrm{Spec}}(R[x_ i, y_ i, z_ i]/(\sum y_ iz_ i - 1))$ is $m$-pseudo-coherent, see Derived Categories of Schemes, Lemma 36.12.5. For each $i_0$ it suffices to prove the restriction of this pushforward to $W_{i_0} = \mathop{\mathrm{Spec}}(R[x_ i, y_ i, z_ i, 1/y_{i_0}]/(\sum y_ iz_ i - 1))$ is $m$-pseudo-coherent. Note that there is a commutative diagram

\[ \xymatrix{ X \ar[d] & U_{i_0} \ar[l] \ar[d] \\ T & W_{i_0} \ar[l] } \]

which implies that the pushforward of $E$ to $T$ restricted to $W_{i_0}$ is the pushforward of $E|_{U_{i_0}}$ to $W_{i_0}$. Since $R[x_ i, y_ i, z_ i, 1/y_{i_0}]/(\sum y_ iz_ i - 1)$ is isomorphic to a polynomial ring over $R$ this proves what we want. $\square$


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