Definition 76.52.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. An object $E$ of $D(\mathcal{O}_ X)$ is *perfect relative to $Y$* or *$Y$-perfect* if $E$ is pseudo-coherent (Cohomology on Sites, Definition 21.45.1) and $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_ Y)$ (Cohomology on Sites, Definition 21.46.1).

## 76.52 Relatively perfect objects

In this section we introduce a notion from [lieblich-complexes]. This notion has been discussed for morphisms of schemes in Derived Categories of Schemes, Section 36.35.

Please see Derived Categories of Schemes, Remark 36.35.14 for a discussion; here we just mention that $E$ being pseudo-coherent is the same thing as $E$ being pseudo-coherent relative to $Y$ by Lemma 76.45.4. Moreover, pseudo-coherence of $E$ implies $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$, see Derived Categories of Spaces, Lemma 75.13.6.

Example 76.52.2. Let $k$ be a field. Let $X$ be an algebraic space of finite presentation over $k$ (in particular $X$ is quasi-compact). Then an object $E$ of $D(\mathcal{O}_ X)$ is $k$-perfect if and only if it is bounded and pseudo-coherent (by definition), i.e., if and only if it is in $D^ b_{\textit{Coh}}(X)$ (by Derived Categories of Spaces, Lemma 75.13.7). Thus being relatively perfect does **not** mean “perfect on the fibres”.

The corresponding algebra concept is studied in More on Algebra, Section 15.83. We can link the notion for algebraic spaces with the algebraic notion as follows.

Lemma 76.52.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. The following are equivalent:

$E$ is $Y$-perfect,

for every commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r]_ g & V \ar[d] \\ X \ar[r]^ f & Y } \]where $U$, $V$ are schemes and the vertical arrows are étale, the complex $E|_ U$ is $V$-perfect in the sense of Derived Categories of Schemes, Definition 36.35.1,

for some commutative diagram as in (2) with $U \to X$ surjective, the complex $E|_ U$ is $V$-perfect in the sense of Derived Categories of Schemes, Definition 36.35.1,

for every commutative diagram as in (2) with $U$ and $V$ affine the complex $R\Gamma (U, E)$ is $\mathcal{O}_ Y(V)$-perfect.

**Proof.**
To make sense of parts (2), (3), (4) of the lemma, observe that the object $E|_ U$ of $D_\mathit{QCoh}(\mathcal{O}_ U)$ corresponds to an object $E_0$ of $D_\mathit{QCoh}(\mathcal{O}_{U_0})$ where $U_0$ denotes the scheme underlying $U$, see Derived Categories of Spaces, Lemma 75.4.2. Moreover, in this case $E_0$ is pseudo-coherent if and only if $E|_ U$ is pseudo-coherent, see Derived Categories of Spaces, Lemma 75.13.2. Also, $E|_ U$ locally has finite tor dimension over $f^{-1}\mathcal{O}_ Y|_ U = g^{-1}\mathcal{O}_ V$ if and only if $E_0$ locally has finite tor dimension over $g_0^{-1}\mathcal{O}_{V_0}$ by Derived Categories of Spaces, Lemma 75.13.4. Here $g_0 : U_0 \to V_0$ is the morphism of schemes representing $g : U \to V$ (notation as in Derived Categories of Spaces, Remark 75.6.3). Finally, observe that “being pseudo-coherent” is étale local and of course “having locally finite tor dimension” is étale local. Thus we see that it suffices to check $Y$-perfectness étale locally and by the above discussion we see that (1) implies (2) and (3) implies (1). Since part (4) is equivalent to (2) and (3) by Derived Categories of Schemes, Lemma 36.35.3 the proof is complete.
$\square$

Lemma 76.52.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. The full subcategory of $D(\mathcal{O}_ X)$ consisting of $Y$-perfect objects is a saturated^{1} triangulated subcategory.

**Proof.**
This follows from Cohomology on Sites, Lemmas 21.45.4, 21.45.6, 21.46.6, and 21.46.8.
$\square$

Lemma 76.52.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. A perfect object of $D(\mathcal{O}_ X)$ is $Y$-perfect. If $K, M \in D(\mathcal{O}_ X)$, then $K \otimes _{\mathcal{O}_ X}^\mathbf {L} M$ is $Y$-perfect if $K$ is perfect and $M$ is $Y$-perfect.

**Proof.**
Reduce to the case of schemes using Lemma 76.52.3 and then apply Derived Categories of Schemes, Lemma 36.35.5.
$\square$

Lemma 76.52.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let $g : Y' \to Y$ be a morphism of algebraic spaces over $S$. Set $X' = Y' \times _ Y X$ and denote $g' : X' \to X$ the projection. If $K \in D(\mathcal{O}_ X)$ is $Y$-perfect, then $L(g')^*K$ is $Y'$-perfect.

**Proof.**
Reduce to the case of schemes using Lemma 76.52.3 and then apply Derived Categories of Schemes, Lemma 36.35.6.
$\square$

Situation 76.52.7. Let $S$ be a scheme. Let $Y = \mathop{\mathrm{lim}}\nolimits _{i \in I} Y_ i$ be a limit of a directed system of algebraic spaces over $S$ with affine transition morphisms $g_{i'i} : Y_{i'} \to Y_ i$. We assume that $Y_ i$ is quasi-compact and quasi-separated for all $i \in I$. We denote $g_ i : Y \to Y_ i$ the projection. We fix an element $0 \in I$ and a flat morphism of finite presentation $X_0 \to Y_0$. We set $X_ i = Y_ i \times _{Y_0} X_0$ and $X = Y \times _{Y_0} X_0$ and we denote the transition morphisms $f_{i'i} : X_{i'} \to X_ i$ and $f_ i : X \to X_ i$ the projections.

Lemma 76.52.8. In Situation 76.52.7. Let $K_0$ and $L_0$ be objects of $D(\mathcal{O}_{X_0})$. Set $K_ i = Lf_{i0}^*K_0$ and $L_ i = Lf_{i0}^*L_0$ for $i \geq 0$ and set $K = Lf_0^*K_0$ and $L = Lf_0^*L_0$. Then the map

is an isomorphism if $K_0$ is pseudo-coherent and $L_0 \in D_\mathit{QCoh}(\mathcal{O}_{X_0})$ has (locally) finite tor dimension as an object of $D((X_0 \to Y_0)^{-1}\mathcal{O}_{Y_0})$

**Proof.**
For every quasi-compact and quasi-separated object $U_0$ of $(X_0)_{spaces, {\acute{e}tale}}$ consider the condition $P$ that

is an isomorphism where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. We will prove $P$ holds for each $U_0$.

Suppose that $(U_0 \subset W_0, V_0 \to W_0)$ is an elementary distinguished square in $(X_0)_{spaces, {\acute{e}tale}}$ and $P$ holds for $U_0, V_0, U_0 \times _{W_0} V_0$. Then $P$ holds for $W_0$ by Mayer-Vietoris for hom in the derived category, see Derived Categories of Spaces, Lemma 75.10.4.

We first consider $U_0 = W_0 \times _{Y_0} X_0$ with $W_0$ a quasi-compact and quasi-separated object of $(Y_0)_{spaces, {\acute{e}tale}}$. By the induction principle of Derived Categories of Spaces, Lemma 75.9.3 applied to these $W_0$ and the previous paragraph, we find that it is enough to prove $P$ for $U_0 = W_0 \times _{Y_0} X_0$ with $W_0$ affine. In other words, we have reduced to the case where $Y_0$ is affine. Next, we apply the induction principle again, this time to all quasi-compact and quasi-separated opens of $X_0$, to reduce to the case where $X_0$ is affine as well.

If $X_0$ and $Y_0$ are affine, then we are back in the case of schemes which is proved in Derived Categories of Schemes, Lemma 36.35.8. The reader may use Derived Categories of Spaces, Lemmas 75.13.6, 75.4.2, 75.13.2, and 75.13.4 to accomplish the translation of the statement into a statement involving only schemes and derived categories of modules on schemes. $\square$

Lemma 76.52.9. In Situation 76.52.7 the category of $Y$-perfect objects of $D(\mathcal{O}_ X)$ is the colimit of the categories of $Y_ i$-perfect objects of $D(\mathcal{O}_{X_ i})$.

**Proof.**
For every quasi-compact and quasi-separated object $U_0$ of $(X_0)_{spaces, {\acute{e}tale}}$ consider the condition $P$ that the functor

is an equivalence where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. We observe that we already know this functor is fully faithful by Lemma 76.52.8. Thus it suffices to prove essential surjectivity.

Suppose that $(U_0 \subset W_0, V_0 \to W_0)$ is an elementary distinguished square in $(X_0)_{spaces, {\acute{e}tale}}$ and $P$ holds for $U_0, V_0, U_0 \times _{W_0} V_0$. We claim that $P$ holds for $W_0$. We will use the notation $U_ i = X_ i \times _{X_0} U_0$, $U = X \times _{X_0} U_0$, and similarly for $V_0$ and $W_0$. We will abusively use the symbol $f_ i$ for all the morphisms $U \to U_ i$, $V \to V_ i$, $U \times _ W V \to U_ i \times _{W_ i} V_ i$, and $W \to W_ i$. Suppose $E$ is an $Y$-perfect object of $D(\mathcal{O}_ W)$. Goal: show $E$ is in the essential image of the functor. By assumption, we can find $i \geq 0$, an $Y_ i$-perfect object $E_{U, i}$ on $U_ i$, an $Y_ i$-perfect object $E_{V, i}$ on $V_ i$, and isomorphisms $Lf_ i^*E_{U, i} \to E|_ U$ and $Lf_ i^*E_{V, i} \to E|_ V$. Let

the maps adjoint to the isomorphisms $Lf_ i^*E_{U, i} \to E|_ U$ and $Lf_ i^*E_{V, i} \to E|_ V$. By fully faithfulness, after increasing $i$, we can find an isomorphism $c : E_{U, i}|_{U_ i \times _{W_ i} V_ i} \to E_{V, i}|_{U_ i \times _{W_ i} V_ i}$ which pulls back to the identifications

Apply Derived Categories of Spaces, Lemma 75.10.8 to get an object $E_ i$ on $W_ i$ and a map $d : E_ i \to Rf_{i, *}E$ which restricts to the maps $a$ and $b$ over $U_ i$ and $V_ i$. Then it is clear that $E_ i$ is $Y_ i$-perfect (because being relatively perfect is an étale local property) and that $d$ is adjoint to an isomorphism $Lf_ i^*E_ i \to E$.

By exactly the same argument as used in the proof of Lemma 76.52.8 using the induction principle (Derived Categories of Spaces, Lemma 75.9.3) we reduce to the case where both $X_0$ and $Y_0$ are affine: first work with quasi-compact and quasi-separated objects in $(Y_0)_{spaces, {\acute{e}tale}}$ to reduce to $Y_0$ affine, then work with quasi-compact and quasi-separated object in $(X_0)_{spaces, {\acute{e}tale}}$ to reduce to $X_0$ affine. In the affine case the result follows from the case of schemes which is Derived Categories of Schemes, Lemma 36.35.9. The translation into the case for schemes is done by Lemma 76.52.3. $\square$

Lemma 76.52.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat, proper, and of finite presentation. Let $E \in D(\mathcal{O}_ X)$ be $Y$-perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ Y)$ and its formation commutes with arbitrary base change.

**Proof.**
The statement on base change is Derived Categories of Spaces, Lemma 75.21.4 (with $\mathcal{G}^\bullet $ equal to $\mathcal{O}_ X$ in degree $0$). Thus it suffices to show that $Rf_*E$ is a perfect object. We will reduce to the case where $Y$ is Noetherian affine by a limit argument.

The question is étale local on $Y$, hence we may assume $Y$ is affine. Say $Y = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits of Spaces, Lemma 70.7.1 there exists an $i$ and an algebraic space $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits of Spaces, Lemmas 70.6.13 and 70.6.12 we may assume $X_ i$ is proper and flat over $R_ i$. By Lemma 76.52.9 we may assume there exists a $R_ i$-perfect object $E_ i$ of $D(\mathcal{O}_{X_ i})$ whose pullback to $X$ is $E$. Applying Derived Categories of Spaces, Lemma 75.22.1 to $X_ i \to \mathop{\mathrm{Spec}}(R_ i)$ and $E_ i$ and using the base change property already shown we obtain the result. $\square$

Lemma 76.52.11. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $E, K \in D(\mathcal{O}_ X)$. Assume

$Y$ is quasi-compact and quasi-separated,

$f$ is proper, flat, and of finite presentation,

$E$ is $Y$-perfect,

$K$ is pseudo-coherent.

Then there exists a pseudo-coherent $L \in D(\mathcal{O}_ Y)$ such that

and the same is true after arbitrary base change: given

**Proof.**
Since $Y$ is quasi-compact and quasi-separated, the same is true for $X$. By Derived Categories of Spaces, Lemma 75.18.1 we can write $K = \text{hocolim} K_ n$ with $K_ n$ perfect and $K_ n \to K$ inducing an isomorphism on truncations $\tau _{\geq -n}$. Let $K_ n^\vee $ be the dual perfect complex (Cohomology on Sites, Lemma 21.48.4). We obtain an inverse system $\ldots \to K_3^\vee \to K_2^\vee \to K_1^\vee $ of perfect objects. By Lemma 76.52.5 we see that $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ is $Y$-perfect. Thus we may apply Lemma 76.52.10 to $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ and we obtain an inverse system

of perfect complexes on $Y$ with

Moreover, the formation of these complexes commutes with any base change, namely $Lg^*M_ n = Rf'_*((L(g')^*K_ n)^\vee \otimes _{\mathcal{O}_{X'}}^\mathbf {L} L(g')^*E) = Rf'_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L(g')^*K_ n, L(g')^*E)$.

As $K_ n \to K$ induces an isomorphism on $\tau _{\geq -n}$, we see that $K_ n \to K_{n + 1}$ induces an isomorphism on $\tau _{\geq -n}$. It follows that $K_{n + 1}^\vee \to K_ n^\vee $ induces an isomorphism on $\tau _{\leq n}$ as $K_ n^\vee = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, \mathcal{O}_ X)$. Suppose that $E$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\mathcal{O}_ Y$-modules. Then the same is true after any base change, see Derived Categories of Spaces, Lemma 75.20.7. We find that $K_{n + 1}^\vee \otimes _{\mathcal{O}_ X} E \to K_ n^\vee \otimes _{\mathcal{O}_ X} E$ induces an isomorphism on $\tau _{\leq n + a}$ and the same is true after any base change. Applying the right derived functor $Rf_*$ we conclude the maps $M_{n + 1} \to M_ n$ induce isomorphisms on $\tau _{\leq n + a}$ and the same is true after any base change. Choose a distinguished triangle

Pick $y \in |Y|$. Choose an elementary étale neighbourhood $(U, u) \to (Y, y)$; this is possible by Decent Spaces, Lemma 68.11.4. Take $Y'$ equal to the spectrum of the residue field at $u$. Pull back to see that $C_ n|_ U \otimes _{\mathcal{O}_ U}^\mathbf {L} \kappa (u)$ has nonzero cohomology only in degrees $\geq n + a$. By More on Algebra, Lemma 15.75.7 we see that the perfect complex $C_ n|_ U$ has tor amplitude in $[n + a, m_ n]$ for some integer $m_ n$ and after possibly shrinking $U$. Thus $C_ n$ has tor amplitude in $[n + a, m_ n]$ for some integer $m_ n$ (because $Y$ is quasi-compact). In particular, the dual perfect complex $C_ n^\vee $ has tor amplitude in $[-m_ n, -n - a]$.

Let $L_ n = M_ n^\vee $ be the dual perfect complex. The conclusion from the discussion in the previous paragraph is that $L_ n \to L_{n + 1}$ induces isomorphisms on $\tau _{\geq -n - a}$. Thus $L = \text{hocolim} L_ n$ is pseudo-coherent, see Derived Categories of Spaces, Lemma 75.18.1. Since we have

(Cohomology on Sites, Lemma 21.48.8) and since $R\mathop{\mathrm{lim}}\nolimits $ commutes with $Rf_*$ we find that

This proves the formula over $Y$. Since the construction of $M_ n$ is compatible with base chance, the formula continues to hold after any base change. $\square$

Remark 76.52.12. The reader may have noticed the similarity between Lemma 76.52.11 and Derived Categories of Spaces, Lemma 75.23.3. Indeed, the pseudo-coherent complex $L$ of Lemma 76.52.11 may be characterized as the unique pseudo-coherent complex on $Y$ such that there are functorial isomorphisms

compatible with boundary maps for $\mathcal{F}$ ranging over $\mathit{QCoh}(\mathcal{O}_ Y)$. If we ever need this we will formulate a precise result here and give a detailed proof.

Lemma 76.52.13. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ such that the structure morphism $f : X \to S$ is flat and locally of finite presentation. Let $E$ be a pseudo-coherent object of $D(\mathcal{O}_ X)$. The following are equivalent

$E$ is $S$-perfect, and

$E$ is locally bounded below and for every point $s \in S$ the object $L(X_ s \to X)^*E$ of $D(\mathcal{O}_{X_ s})$ is locally bounded below.

**Proof.**
Since everything is local we immediately reduce to the case that $X$ and $S$ are affine, see Lemma 76.52.3. This case is handled by Derived Categories of Schemes, Lemma 36.35.13.
$\square$

Lemma 76.52.14. Let $A$ be a ring. Let $X$ be an algebraic space separated, of finite presentation, and flat over $A$. Let $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$. If $R \Gamma (X, E \otimes ^\mathbf {L} K)$ is perfect in $D(A)$ for every perfect $E \in D(\mathcal{O}_ X)$, then $K$ is $\mathop{\mathrm{Spec}}(A)$-perfect.

**Proof.**
By Lemma 76.51.5, $K$ is pseudo-coherent relative to $A$. By Lemma 76.45.4, $K$ is pseudo-coherent in $D(\mathcal{O}_ X)$. By Derived Categories of Spaces, Proposition 75.29.4 we see that $K$ is in $D^-(\mathcal{O}_ X)$. Let $\mathfrak {p}$ be a prime ideal of $A$ and denote $i : Y \to X$ the inclusion of the scheme theoretic fibre over $\mathfrak {p}$, i.e., $Y$ is a scheme over $\kappa (\mathfrak p)$. By Lemma 76.52.13, we will be done if we can show $Li^*(K)$ is bounded below. Let $G \in D_{perf} (\mathcal{O}_ X)$ be a perfect complex which generates $D_\mathit{QCoh}(\mathcal{O}_ X)$, see Derived Categories of Spaces, Theorem 75.15.4. We have

The first equality uses that $Li^*$ preserves perfect objects and duals and Cohomology on Sites, Lemma 21.48.4; we omit some details. The second equality follows from Derived Categories of Spaces, Lemma 75.20.4 as $X$ is flat over $A$. It follows from our hypothesis that this is a perfect object of $D(\kappa (\mathfrak {p}))$. The object $Li^*(G) \in D_{perf}(\mathcal{O}_ Y)$ generates $D_\mathit{QCoh}(\mathcal{O}_ Y)$ by Derived Categories of Spaces, Remark 75.15.5. Hence Derived Categories of Spaces, Proposition 75.29.4 now implies that $Li^*(K)$ is bounded below and we win. $\square$

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