The Stacks project

Proof. Say $S = \mathop{\mathrm{Spec}}(R)$ and $X = \mathop{\mathrm{Spec}}(A)$. Let $i$ correspond to the surjection $\alpha : R[x_1, \ldots , x_ n] \to A$ and let $X \to \mathbf{A}^ m_ S$ correspond to $\beta : R[y_1, \ldots , y_ m] \to A$. Choose $f_ j \in R[x_1, \ldots , x_ n]$ with $\alpha (f_ j) = \beta (y_ j)$ and $g_ i \in R[y_1, \ldots , y_ m]$ with $\beta (g_ i) = \alpha (x_ i)$. Then we get a commutative diagram

corresponding to the commutative diagram of closed immersions

Thus it suffices to show that under a closed immersion

an object $E$ of $D(\mathcal{O}_{\mathbf{A}^ m_ S})$ is $m$-pseudo-coherent if and only if $Rf_*E$ is $m$-pseudo-coherent. This follows from Derived Categories of Schemes, Lemma 36.12.5 and the fact that $f_*\mathcal{O}_{\mathbf{A}^ m_ S}$ is a pseudo-coherent $\mathcal{O}_{\mathbf{A}^{n + m}_ S}$-module. The pseudo-coherence of $f_*\mathcal{O}_{\mathbf{A}^ m_ S}$ is straightforward to prove directly, but it also follows from Derived Categories of Schemes, Lemma 36.10.2 and More on Algebra, Lemma 15.81.3. $\square$

Proof. Choose an affine open covering $S = \bigcup V_ i$ and for each $i$ an affine open covering $f^{-1}(V_ i) = \bigcup U_{ij}$ such that the equivalent conditions of Lemma 37.59.1 are satisfied for each of the pairs $(U_{ij} \to V_ i, E|_{U_{ij}})$. Since being quasi-coherent is local on $X$, we may assume that there exists an closed immersion $i : X \to \mathbf{A}^ n_ S$ such that $Ri_*E$ is $m$-pseudo-coherent on $\mathbf{A}^ n_ S$. By Derived Categories of Schemes, Lemma 36.10.1 this means that $H^ q(Ri_*E)$ is quasi-coherent for $q > m$. Since $i_*$ is an exact functor, we have $i_*H^ q(E) = H^ q(Ri_*E)$ is quasi-coherent on $\mathbf{A}^ n_ S$. By Morphisms, Lemma 29.4.1 this implies that $H^ q(E)$ is quasi-coherent as desired (strictly speaking it implies there exists some quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ such that $i_*\mathcal{F} = i_*H^ q(E)$ and then Modules, Lemma 17.13.4 tells us that $\mathcal{F} \cong H^ q(E)$ hence the result). $\square$

Proof. Write $S = \mathop{\mathrm{Spec}}(R)$, $V = D(f)$, $X = \mathop{\mathrm{Spec}}(A)$, and $U = D(g)$. Assume the equivalent conditions of Lemma 37.59.1 are satisfied for the pair $(X \to V, E)$.

Choose $R_ f[x_1, \ldots , x_ n] \to A$ surjective. Write $R_ f = R[x_0]/(fx_0 - 1)$. Then $R[x_0, x_1, \ldots , x_ n] \to A$ is surjective, and $R_ f[x_1, \ldots , x_ n]$ is pseudo-coherent as an $R[x_0, \ldots , x_ n]$-module. Thus we have

and we can apply Derived Categories of Schemes, Lemma 36.12.5 to conclude that the pushforward $E'$ of $E$ to $\mathbf{A}^{n + 1}_ S$ is $m$-pseudo-coherent.

Choose an element $g' \in R[x_0, x_1, \ldots , x_ n]$ which maps to $g \in A$. Consider the surjection $R[x_0, \ldots , x_{n + 1}] \to R[x_0, \ldots , x_ n, 1/g']$. We obtain

where the lower left arrow is an open immersion and the lower right arrow is a closed immersion. We conclude as before that the pushforward of $E'|_{D(g')}$ to $\mathbf{A}^{n + 2}_ S$ is $m$-pseudo-coherent. Since this is also the pushforward of $E|_ U$ to $\mathbf{A}^{n + 2}_ S$ we conclude the lemma is true. $\square$

Proof. The implication (2) $\Rightarrow $ (1) is Lemma 37.59.4. 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

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

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. $\square$

Proof. The final statement is clear from the equivalence of (1) and (2). It is also clear that (2) implies (1). Assume (1) holds. Let $S = \bigcup V_ i$ and $f^{-1}(V_ i) = \bigcup U_{ij}$ be affine open coverings as in Definition 37.59.2. Let $U \subset X$ and $V \subset S$ be as in (2). By Lemma 37.59.5 it suffices to find a standard open covering $U = \bigcup U_ k$ of $U$ such that the equivalent conditions of Lemma 37.59.1 are satisfied for the pairs $(U_ k \to V, E|_{U_ k})$. In other words, for every $u \in U$ it suffices to find a standard affine open $u \in U' \subset U$ such that the equivalent conditions of Lemma 37.59.1 are satisfied for the pair $(U' \to V, E|_{U'})$. Pick $i$ such that $f(u) \in V_ i$ and then pick $j$ such that $u \in U_{ij}$. By Schemes, Lemma 26.11.5 we can find $v \in V' \subset V \cap V_ i$ which is standard affine open in $V'$ and $V_ i$. Then $f^{-1}V' \cap U$, resp. $f^{-1}V' \cap U_{ij}$ are standard affine opens of $U$, resp. $U_{ij}$. Applying the lemma again we can find $u \in U' \subset f^{-1}V' \cap U \cap U_{ij}$ which is standard affine open in both $f^{-1}V' \cap U$ and $f^{-1}V' \cap U_{ij}$. Thus $U'$ is also a standard affine open of $U$ and $U_{ij}$. By Lemma 37.59.4 the assumption that the equivalent conditions of Lemma 37.59.1 are satisfied for the pair $(U_{ij} \to V_ i, E|_{U_{ij}})$ implies that the equivalent conditions of Lemma 37.59.1 are satisfied for the pair $(U' \to V, E|_{U'})$. $\square$

Proof. Let $U$ and $V$ be as in (2) and choose a closed immersion $i : U \to \mathbf{A}^ n_ V$. A formal argument, using Lemma 37.59.6, shows it suffices to prove that $Ri_*(E|_ U)$ is $m$-pseudo-coherent if and only if $R\Gamma (U, E)$ is $m$-pseudo-coherent relative to $\mathcal{O}_ S(V)$. Say $U = \mathop{\mathrm{Spec}}(A)$, $V = \mathop{\mathrm{Spec}}(R)$, and $\mathbf{A}^ n_ V = \mathop{\mathrm{Spec}}(R[x_1, \ldots , x_ n]$. By the remarks preceding the lemma, $E|_ U$ is quasi-isomorphic to the complex of quasi-coherent sheaves on $U$ associated to the object $R\Gamma (U, E)$ of $D(A)$. Note that $R\Gamma (U, E) = R\Gamma (\mathbf{A}^ n_ V, Ri_*(E|_ U))$ as $i$ is a closed immersion (and hence $i_*$ is exact). Thus $Ri_*E$ is associated to $R\Gamma (U, E)$ viewed as an object of $D(R[x_1, \ldots , x_ n])$. We conclude as $m$-pseudo-coherence of $Ri_*(E|_ U)$ is equivalent to $m$-pseudo-coherence of $R\Gamma (U, E)$ in $D(R[x_1, \ldots , x_ n])$ by Derived Categories of Schemes, Lemma 36.10.2 which is equivalent to $R\Gamma (U, E)$ is $m$-pseudo-coherent relative to $R = \mathcal{O}_ S(V)$ by definition. $\square$

Proof. By Morphisms, Lemma 29.45.4 the morphism $i$ is affine. Thus we may assume $S$, $Y$, and $X$ are affine. Say $S = \mathop{\mathrm{Spec}}(R)$, $Y = \mathop{\mathrm{Spec}}(A)$, and $X = \mathop{\mathrm{Spec}}(B)$. The condition means that $A/\text{rad}(A) \to B/\text{rad}(B)$ is surjective; here $\text{rad}(A)$ and $\text{rad}(B)$ denote the Jacobson radical of $A$ and $B$. As $B$ is of finite type over $A$, we can find $b_1, \ldots , b_ m \in \text{rad}(B)$ which generate $B$ as an $A$-algebra. Say $b_ j^ N = 0$ for all $j$. Consider the diagram of rings

which translates into a diagram

of affine schemes. By Lemma 37.59.6 we see that $E$ is $m$-pseudo-coherent relative to $S$ if and only if its pushforward to $\mathbf{A}^{n + m}_ S$ is $m$-pseudo-coherent. By Derived Categories of Schemes, Lemma 36.12.5 we see that this is true if and only if its pushforward to $T$ is $m$-pseudo-coherent. The same lemma shows that this holds if and only if the pushforward to $\mathbf{A}^ n_ S$ is $m$-pseudo-coherent. Again by Lemma 37.59.6 this holds if and only if $Ri_*E$ is $m$-pseudo-coherent relative to $S$. $\square$

Proof. Translation of the result of More on Algebra, Lemma 15.81.5 into the language of schemes. Observe that $R\pi _*$ indeed maps $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$ by Derived Categories of Schemes, Lemma 36.4.1. To do the translation use Lemma 37.59.6. $\square$

Proof. Immediate from Lemma 37.59.6 and Cohomology, Lemma 20.47.4. $\square$

Proof. Part (1) is immediate from the definition. To see part (3) we may work locally on $X$ (both properties are local). Thus we may assume $X$ and $S$ are affine. Choose a closed immersion $i : X \to \mathbf{A}^ n_ S$. Then we see that $\mathcal{F}$ is $(-1)$-pseudo-coherent relative to $S$ if and only if $i_*\mathcal{F}$ is $(-1)$-pseudo-coherent, which is true if and only if $i_*\mathcal{F}$ is an $\mathcal{O}_{\mathbf{A}^ n_ S}$-module of finite presentation, see Cohomology, Lemma 20.47.9. A module of finite presentation is quasi-coherent, see Modules, Lemma 17.11.2. By Morphisms, Lemma 29.4.1 we see that $\mathcal{F}$ is quasi-coherent if and only if $i_*\mathcal{F}$ is quasi-coherent. Having said this part (3) follows. The proof of (2) is similar but less involved. $\square$

Proof. Follows from Cohomology, Lemma 20.47.6 and the definitions. $\square$

Proof. Follows from Cohomology, Lemma 20.47.7 and the definitions. $\square$

Proof. Follows from Cohomology, Lemma 20.47.8 and the definitions. $\square$

Proof. The problem is local on $X$ and $X'$ hence we may assume $X$, $S$, $S'$, and $X'$ are affine. Choose a closed immersion $i : X \to \mathbf{A}^ n_ S$ and denote $i' : X' \to \mathbf{A}^ n_{S'}$ the base change to $S'$. Denote $g : X' \to X$ and $g' : \mathbf{A}^ n_{S'} \to \mathbf{A}^ n_ S$ the projections, so $E' = Lg^*E$. Since $X$ and $S'$ are tor-independent over $S$, the base change map (Cohomology, Remark 20.28.3) induces an isomorphism

Namely, for a point $x' \in X'$ lying over $x \in X$ the base change map on stalks at $x'$ is the map

coming from the closed immersions $i$ and $i'$. Note that the source is quasi-isomorphic to a localization of $E_ x \otimes _{\mathcal{O}_{S, s}}^\mathbf {L} \mathcal{O}_{S', s'}$ which is isomorphic to the target as $\mathcal{O}_{X', x'}$ is isomorphic to (the same) localization of $\mathcal{O}_{X, x} \otimes _{\mathcal{O}_{S, s}}^\mathbf {L} \mathcal{O}_{S', s'}$ by assumption. We conclude the lemma holds by an application of Cohomology, Lemma 20.47.3. $\square$

Proof. The problem is local on $X$. Thus we may assume $X$, $Y$, and $S$ are affine. Arguing as in the proof of More on Algebra, Lemma 15.81.13 we can find a commutative diagram

Observe that

by Cohomology, Lemma 20.54.4. By assumption and the fact that $Y$ is affine, we can represent $Ri_*\mathcal{O}_ X = i_*\mathcal{O}_ X$ by a complexes of finite free $\mathcal{O}_{\mathbf{A}_ Y^ n}$-modules $\mathcal{F}^\bullet $, with $\mathcal{F}^ q = 0$ for $q > 0$ (details omitted; use Derived Categories of Schemes, Lemma 36.10.2 and More on Algebra, Lemma 15.81.7). By assumption $E$ is bounded above, say $H^ q(E) = 0$ for $q > a$. Represent $E$ by a complex $\mathcal{E}^\bullet $ of $\mathcal{O}_ Y$-modules with $\mathcal{E}^ q = 0$ for $q > a$. Then the derived tensor product above is represented by $\text{Tot}(p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^\bullet )$.

Since $j$ is a closed immersion, the functor $j_*$ is exact and $Rj_*$ is computed by applying $j_*$ to any representating complex of sheaves. Thus we have to show that $j_*\text{Tot}(p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^\bullet )$ is $m$-pseudo-coherent as a complex of $\mathcal{O}_{\mathbf{A}^{n + m}_ S}$-modules. Note that $\text{Tot}(p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^\bullet )$ has a filtration by subcomplexes with successive quotients the complexes $p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^ q[-q]$. Note that for $q \ll 0$ the complexes $p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^ q[-q]$ have zero cohomology in degrees $\leq m$ and hence are $m$-pseudo-coherent. Hence, applying Lemma 37.59.10 and induction, it suffices to show that $p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^ q[-q]$ is pseudo-coherent relative to $S$ for all $q$. Note that $\mathcal{F}^ q = 0$ for $q > 0$. Since also $\mathcal{F}^ q$ is finite free this reduces to proving that $p^*\mathcal{E}^\bullet $ is $m$-pseudo-coherent relative to $S$ which follows from Lemma 37.59.15 for instance. $\square$

Proof. The question is local on $X$, hence we may assume $X$, $Y$, and $S$ are affine. Arguing as in the proof of More on Algebra, Lemma 15.81.13 we can find a commutative diagram

The assumption that $\mathcal{O}_ Y$ is pseudo-coherent relative to $S$ implies that $\mathcal{O}_{\mathbf{A}^ m_ Y}$ is pseudo-coherent relative to $\mathbf{A}^ m_ S$ (by flat base change; this can be seen by using for example Lemma 37.59.15). This in turn implies that $j_*\mathcal{O}_{\mathbf{A}^ n_ Y}$ is pseudo-coherent as an $\mathcal{O}_{\mathbf{A}^{n + m}_ S}$-module. Then the equivalence of the lemma follows from Derived Categories of Schemes, Lemma 36.12.5. $\square$

Proof. The equivalence of (1) and (2) is Lemma 37.59.9. The equivalence of (2) and (3) follows from Lemma 37.59.17 applied to $\text{id} : P \to P$ provided we can show that $\mathcal{O}_ P$ is pseudo-coherent relative to $S$. This follows from More on Algebra, Lemma 15.82.4 and the definitions. $\square$