The Stacks project

37.53 Relative pseudo-coherence

This section is the analogue of More on Algebra, Section 15.80 for schemes. We strongly urge the reader to take a look at that section first. Although we have developed the material in this section and the material on pseudo-coherent complexes in Cohomology, Sections 20.43, 20.44, 20.45, and 20.46 for arbitrary complexes of $\mathcal{O}_ X$-modules, if $X$ is a scheme then working exclusively with objects in $D_\mathit{QCoh}(\mathcal{O}_ X)$ greatly simplifies many of the lemmmas and arguments, often reducing the problem at hand immediately to the algebraic counterpart. Moreover, one of the first thing we do is to show that being relatively pseudo-coherent implies the cohomology sheaves are quasi-coherent, see Lemma 37.53.3. Hence, on a first reading we suggest the reader work exclusively with objects in $D_\mathit{QCoh}(\mathcal{O}_ X)$.

Lemma 37.53.1. Let $X \to S$ be a finite type morphism of affine schemes. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$. The following are equivalent

  1. for some closed immersion $i : X \to \mathbf{A}^ n_ S$ the object $Ri_*E$ of $D(\mathcal{O}_{\mathbf{A}^ n_ S})$ is $m$-pseudo-coherent, and

  2. for all closed immersions $i : X \to \mathbf{A}^ n_ S$ the object $Ri_*E$ of $D(\mathcal{O}_{\mathbf{A}^ n_ S})$ is $m$-pseudo-coherent.

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

\[ \xymatrix{ R[x_1, \ldots , x_ n, y_1, \ldots , y_ m] \ar[d]^{x_ i \mapsto g_ i} \ar[rr]_-{y_ j \mapsto f_ j} & & R[x_1, \ldots , x_ n] \ar[d] \\ R[y_1, \ldots , y_ m] \ar[rr] & & A } \]

corresponding to the commutative diagram of closed immersions

\[ \xymatrix{ \mathbf{A}^{n + m}_ S & \mathbf{A}^ n_ S \ar[l] \\ \mathbf{A}^ m_ S \ar[u] & X \ar[u] \ar[l] } \]

Thus it suffices to show that under a closed immersion

\[ f : \mathbf{A}^ m_ S \to \mathbf{A}^{n + m}_ S \]

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 15.80.3. $\square$

Recall that if $f : X \to S$ is a morphism of scheme which is locally of finite type, then for every pair of affine opens $U \subset X$ and $V \subset S$ such that $f(U) \subset V$, the ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is of finite type (Morphisms, Lemma 29.15.2). Hence there always exist closed immersions $U \to \mathbf{A}^ n_ V$ and the following definition makes sense.

Definition 37.53.2. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Fix $m \in \mathbf{Z}$.

  1. We say $E$ is $m$-pseudo-coherent relative to $S$ if there exists 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 37.53.1 are satisfied for each of the pairs $(U_{ij} \to V_ i, E|_{U_{ij}})$.

  2. We say $E$ is pseudo-coherent relative to $S$ if $E$ is $m$-pseudo-coherent relative to $S$ for all $m \in \mathbf{Z}$.

  3. We say $\mathcal{F}$ is $m$-pseudo-coherent relative to $S$ if $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_ X)$ is $m$-pseudo-coherent relative to $S$.

  4. We say $\mathcal{F}$ is pseudo-coherent relative to $S$ if $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_ X)$ is pseudo-coherent relative to $S$.

If $X$ is quasi-compact and $E$ is $m$-pseudo-coherent relative to $S$ for some $m$, then $E$ is bounded above. If $E$ is pseudo-coherent relative to $S$, then $E$ has quasi-coherent cohomology sheaves.

Lemma 37.53.3. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. If $E$ in $D(\mathcal{O}_ X)$ is $m$-pseudo-coherent relative to $S$, then $H^ i(E)$ is a quasi-coherent $\mathcal{O}_ X$-module for $i > m$. If $E$ is pseudo-coherent relative to $S$, then $E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$.

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 37.53.1 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). $\square$

Next, we prove the condition of relative pseudo-coherence localizes well.

Lemma 37.53.4. Let $S$ be an affine scheme. Let $V \subset S$ be a standard open. Let $X \to V$ be a finite type morphism of affine schemes. Let $U \subset X$ be an affine open. Let $E$ be an object of $D(\mathcal{O}_ X)$. If the equivalent conditions of Lemma 37.53.1 are satisfied for the pair $(X \to V, E)$, then the equivalent conditions of Lemma 37.53.1 are satisfied for the pair $(U \to S, E|_ U)$.

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 37.53.1 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

\[ X \to \mathbf{A}^ n_ V \to \mathbf{A}^{n + 1}_ S \]

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

\[ \xymatrix{ X \ar[d] & U \ar[d] \ar[l] \ar[dr] \\ \mathbf{A}^{n + 1}_ S & D(g')\ar[l] \ar[r] & \mathbf{A}^{n + 2}_ S } \]

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. $\square$

Lemma 37.53.5. 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 37.53.1 hold for the pairs $(U_ i \to S, E|_{U_ i})$,

  2. the equivalent conditions of Lemma 37.53.1 hold for the pair $(X \to S, E)$.

Proof. The implication (2) $\Rightarrow $ (1) is Lemma 37.53.4. 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. $\square$

Lemma 37.53.6. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D(\mathcal{O}_ X)$. Fix $m \in \mathbf{Z}$. The following are equivalent

  1. $E$ is $m$-pseudo-coherent relative to $S$,

  2. for every affine opens $U \subset X$ and $V \subset S$ with $f(U) \subset V$ the equivalent conditions of Lemma 37.53.1 are satisfied for the pair $(U \to V, E|_ U)$.

Moreover, if this is true, then for every open subschemes $U \subset X$ and $V \subset S$ with $f(U) \subset V$ the restriction $E|_ U$ is $m$-pseudo-coherent relative to $V$.

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 37.53.2. Let $U \subset X$ and $V \subset S$ be as in (2). By Lemma 37.53.5 it suffices to find a standard open covering $U = \bigcup U_ k$ of $U$ such that the equivalent conditions of Lemma 37.53.1 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 37.53.1 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 37.53.4 the assumption that the equivalent conditions of Lemma 37.53.1 are satisfied for the pair $(U_{ij} \to V_ i, E|_{U_{ij}})$ implies that the equivalent conditions of Lemma 37.53.1 are satisfied for the pair $(U' \to V, E|_{U'})$. $\square$

For objects of the derived category whose cohomology sheaves are quasi-coherent, we can relate relative $m$-pseudo-coherence to the notion defined in More on Algebra, Definition 15.80.4. We will use the fact that for an affine scheme $U = \mathop{\mathrm{Spec}}(A)$ the functor $R\Gamma (U, -)$ induces an equivalence between $D_\mathit{QCoh}(\mathcal{O}_ U)$ and $D(A)$, see Derived Categories of Schemes, Lemma 36.3.5. This functor is compatible with pullbacks: if $E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ U)$ and $A \to B$ is a ring map corresponding to a morphism of affine schemes $g : V = \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A) = U$, then $R\Gamma (V, Lg^*E) = R\Gamma (U, E) \otimes _ A^\mathbf {L} B$. See Derived Categories of Schemes, Lemma 36.3.8.

Lemma 37.53.7. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Fix $m \in \mathbf{Z}$. The following are equivalent

  1. $E$ is $m$-pseudo-coherent relative to $S$,

  2. there exists 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 complex of $\mathcal{O}_ X(U_{ij})$-modules $R\Gamma (U_{ij}, E)$ is $m$-pseudo-coherent relative to $\mathcal{O}_ S(V_ i)$, and

  3. for every affine opens $U \subset X$ and $V \subset S$ with $f(U) \subset V$ the complex of $\mathcal{O}_ X(U)$-modules $R\Gamma (U, E)$ is $m$-pseudo-coherent relative to $\mathcal{O}_ S(V)$.

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 37.53.6, 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. $\square$

Lemma 37.53.8. Let $i : X \to Y$ morphism of schemes locally of finite type over a base scheme $S$. Assume that $i$ induces a homeomorphism of $X$ with a closed subset of $Y$. Let $E$ be an object of $D(\mathcal{O}_ X)$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $Ri_*E$ is $m$-pseudo-coherent relative to $S$.

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

\[ \xymatrix{ B & R[x_ i, y_ j]/(y_ j^ N) \ar[l] & R[x_ i, y_ j] \ar[l] \\ A \ar[u] & R[x_ i] \ar[l] \ar[u] \ar[ru] } \]

which translates into a diagram

\[ \xymatrix{ X \ar[d] \ar[r] & T \ar[d] \ar[r] & \mathbf{A}^{n + m}_ S \ar[ld] \\ Y \ar[r] & \mathbf{A}^ n_ S } \]

of affine schemes. By Lemma 37.53.6 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 37.53.6 this holds if and only if $Ri_*E$ is $m$-pseudo-coherent relative to $S$. $\square$

Lemma 37.53.9. Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite type over a base scheme $S$. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $R\pi _*E$ is $m$-pseudo-coherent relative to $S$.

Proof. Translation of the result of More on Algebra, Lemma 15.80.5 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 37.53.6. $\square$

Lemma 37.53.10. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $(E, E', E'')$ be a distinguished triangle of $D(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$.

  1. If $E$ is $(m + 1)$-pseudo-coherent relative to $S$ and $E'$ is $m$-pseudo-coherent relative to $S$ then $E''$ is $m$-pseudo-coherent relative to $S$.

  2. If $E, E''$ are $m$-pseudo-coherent relative to $S$, then $E'$ is $m$-pseudo-coherent relative to $S$.

  3. If $E'$ is $(m + 1)$-pseudo-coherent relative to $S$ and $E''$ is $m$-pseudo-coherent relative to $S$, then $E$ is $(m + 1)$-pseudo-coherent relative to $S$.

Moreover, if two out of three of $E, E', E''$ are pseudo-coherent relative to $S$, the so is the third.

Proof. Immediate from Lemma 37.53.6 and Cohomology, Lemma 20.44.4. $\square$

Lemma 37.53.11. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Then

  1. $\mathcal{F}$ is $m$-pseudo-coherent relative to $S$ for all $m > 0$,

  2. $\mathcal{F}$ is $0$-pseudo-coherent relative to $S$ if and only if $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module,

  3. $\mathcal{F}$ is $(-1)$-pseudo-coherent relative to $S$ if and only if $\mathcal{F}$ is quasi-coherent and finitely presented relative to $S$.

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 20.44.9. 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. $\square$

Lemma 37.53.12. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $m \in \mathbf{Z}$. Let $E, K$ be objects of $D(\mathcal{O}_ X)$. If $E \oplus K$ is $m$-pseudo-coherent relative to $S$ so are $E$ and $K$.

Proof. Follows from Cohomology, Lemma 20.44.6 and the definitions. $\square$

Lemma 37.53.13. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $m \in \mathbf{Z}$. Let $\mathcal{F}^\bullet $ be a (locally) bounded above complex of $\mathcal{O}_ X$-modules such that $\mathcal{F}^ i$ is $(m - i)$-pseudo-coherent relative to $S$ for all $i$. Then $\mathcal{F}^\bullet $ is $m$-pseudo-coherent relative to $S$.

Proof. Follows from Cohomology, Lemma 20.44.7 and the definitions. $\square$

Lemma 37.53.14. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_ X)$. If $E$ is (locally) bounded above and $H^ i(E)$ is $(m - i)$-pseudo-coherent relative to $S$ for all $i$, then $E$ is $m$-pseudo-coherent relative to $S$.

Proof. Follows from Cohomology, Lemma 20.44.8 and the definitions. $\square$

Lemma 37.53.15. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_ X)$ which is $m$-pseudo-coherent relative to $S$. Let $S' \to S$ be a morphism of schemes. Set $X' = X \times _ S S'$ and denote $E'$ the derived pullback of $E$ to $X'$. If $S'$ and $X$ are Tor independent over $S$, then $E'$ is $m$-pseudo-coherent relative to $S'$.

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

\[ Ri'_*(Lg^*E) = L(g')^*Ri_*E \]

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

\[ E_ x \otimes _{\mathcal{O}_{\mathbf{A}^ n_ S, x}}^\mathbf {L} \mathcal{O}_{\mathbf{A}^ n_{S'}, x'} \longrightarrow E_ x \otimes _{\mathcal{O}_{X, x}}^\mathbf {L} \mathcal{O}_{X', x'} \]

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 20.44.3. $\square$

Lemma 37.53.16. Let $f : X \to Y$ be a morphism of schemes locally of finite type over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_ Y)$. Assume

  1. $\mathcal{O}_ X$ is pseudo-coherent relative to $Y$1, and

  2. $E$ is $m$-pseudo-coherent relative to $S$.

Then $Lf^*E$ is $m$-pseudo-coherent relative to $S$.

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 15.80.13 we can find a commutative diagram

\[ \xymatrix{ X \ar[r]_ i \ar[d]_ f & \mathbf{A}^ d_ Y \ar[r]_ j \ar[ld]^ p & \mathbf{A}^{n + d}_ S \ar[ld] \\ Y \ar[r] & \mathbf{A}^ n_ S } \]

Observe that

\[ Ri_* Lf^*E = Ri_* Li^* Lp^*E = Lp^*E \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}}^\mathbf {L} Ri_*\mathcal{O}_ X \]

by Cohomology, Lemma 20.51.4. 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 15.80.7). 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 37.53.10 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 37.53.15 for instance. $\square$

Lemma 37.53.17. Let $f : X \to Y$ be a morphism of schemes locally of finite type over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_ X)$. Assume $\mathcal{O}_ Y$ is pseudo-coherent relative to $S$2. Then the following are equivalent

  1. $E$ is $m$-pseudo-coherent relative to $Y$, and

  2. $E$ is $m$-pseudo-coherent relative to $S$.

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 15.80.13 we can find a commutative diagram

\[ \xymatrix{ X \ar[r]_ i \ar[d]_ f & \mathbf{A}^ m_ Y \ar[r]_ j \ar[ld]^ p & \mathbf{A}^{n + m}_ S \ar[ld] \\ Y \ar[r] & \mathbf{A}^ n_ S } \]

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 37.53.15). 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. $\square$

Lemma 37.53.18. Let

\[ \xymatrix{ X \ar[rd] \ar[rr]_ i & & P \ar[ld] \\ & S } \]

be a commutative diagram of schemes. Assume $i$ is a closed immersion and $P \to S$ flat and locally of finite presentation. Let $E$ be an object of $D(\mathcal{O}_ X)$. Then the following are equivalent

  1. $E$ is $m$-pseudo-coherent relative to $S$,

  2. $Ri_*E$ is $m$-pseudo-coherent relative to $S$, and

  3. $Ri_*E$ is $m$-pseudo-coherent on $P$.

Proof. The equivalence of (1) and (2) is Lemma 37.53.9. The equivalence of (2) and (3) follows from Lemma 37.53.17 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 15.81.4 and the definitions. $\square$

[1] This means $f$ is pseudo-coherent, see Definition 37.54.2.
[2] This means $Y \to S$ is pseudo-coherent, see Definition 37.54.2.

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