The Stacks project

36.35 Relatively perfect objects

In this section we introduce a notion from [lieblich-complexes].

Definition 36.35.1. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. An object $E$ of $D(\mathcal{O}_ X)$ is perfect relative to $S$ or $S$-perfect if $E$ is pseudo-coherent (Cohomology, Definition 20.44.1) and $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_ S)$ (Cohomology, Definition 20.45.1).

Please see Remark 36.35.14 for a discussion.

Example 36.35.2. Let $k$ be a field. Let $X$ be a scheme 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 Lemma 36.10.3). Thus being relatively perfect does not mean “perfect on the fibres”.

The corresponding algebra concept is studied in More on Algebra, Section 15.82. We can link the notion for schemes with the algebraic notion as follows.

Lemma 36.35.3. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. The following are equivalent

  1. $E$ is $S$-perfect,

  2. for any affine open $U \subset X$ mapping into an affine open $V \subset S$ the complex $R\Gamma (U, E)$ is $\mathcal{O}_ S(V)$-perfect.

  3. 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 $R\Gamma (U_{ij}, E)$ is $\mathcal{O}_ S(V_ i)$-perfect.

Proof. Being pseudo-coherent is a local property and “locally having finite tor dimension” is a local property. Hence this lemma immediately reduces to the statement: if $X$ and $S$ are affine, then $E$ is $S$-perfect if and only if $K = R\Gamma (X, E)$ is $\mathcal{O}_ S(S)$-perfect. Say $X = \mathop{\mathrm{Spec}}(A)$, $S = \mathop{\mathrm{Spec}}(R)$ and $E$ corresponds to $K \in D(A)$, i.e., $K = R\Gamma (X, E)$, see Lemma 36.3.5.

Observe that $K$ is $R$-perfect if and only if $K$ is pseudo-coherent and has finite tor dimension as a complex of $R$-modules (More on Algebra, Definition 15.82.1). By Lemma 36.10.2 we see that $E$ is pseudo-coherent if and only if $K$ is pseudo-coherent. By Lemma 36.10.5 we see that $E$ has finite tor dimension over $f^{-1}\mathcal{O}_ S$ if and only if $K$ has finite tor dimension as a complex of $R$-modules. $\square$

Lemma 36.35.4. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. The full subcategory of $D(\mathcal{O}_ X)$ consisting of $S$-perfect objects is a saturated1 triangulated subcategory.

Lemma 36.35.5. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. A perfect object of $D(\mathcal{O}_ X)$ is $S$-perfect. If $K, M \in D(\mathcal{O}_ X)$, then $K \otimes _{\mathcal{O}_ X}^\mathbf {L} M$ is $S$-perfect if $K$ is perfect and $M$ is $S$-perfect.

Proof. First proof: reduce to the affine case using Lemma 36.35.3 and then apply More on Algebra, Lemma 15.82.3. $\square$

Lemma 36.35.6. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $g : S' \to S$ be a morphism of schemes. Set $X' = S' \times _ S X$ and denote $g' : X' \to X$ the projection. If $K \in D(\mathcal{O}_ X)$ is $S$-perfect, then $L(g')^*K$ is $S'$-perfect.

Proof. First proof: reduce to the affine case using Lemma 36.35.3 and then apply More on Algebra, Lemma 15.82.5.

Second proof: $L(g')^*K$ is pseudo-coherent by Cohomology, Lemma 20.44.3 and the bounded tor dimension property follows from Lemma 36.22.8. $\square$

Situation 36.35.7. Let $S = \mathop{\mathrm{lim}}\nolimits _{i \in I} S_ i$ be a limit of a directed system of schemes with affine transition morphisms $g_{i'i} : S_{i'} \to S_ i$. We assume that $S_ i$ is quasi-compact and quasi-separated for all $i \in I$. We denote $g_ i : S \to S_ i$ the projection. We fix an element $0 \in I$ and a flat morphism of finite presentation $X_0 \to S_0$. We set $X_ i = S_ i \times _{S_0} X_0$ and $X = S \times _{S_0} X_0$ and we denote the transition morphisms $f_{i'i} : X_{i'} \to X_ i$ and $f_ i : X \to X_ i$ the projections.

Lemma 36.35.8. In Situation 36.35.7. 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

\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{X_ i})}(K_ i, L_ i) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, L) \]

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 $D((X_0 \to S_0)^{-1}\mathcal{O}_{S_0})$

Proof. For every quasi-compact open $U_0 \subset X_0$ consider the condition $P$ that

\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{U_ i})}(K_ i|_{U_ i}, L_ i|_{U_ i}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(K|_ U, L|_ U) \]

is an isomorphism where $U = f_0^{-1}(U_0)$ and $U_ i = f_{i0}^{-1}(U_0)$. If $P$ holds for $U_0$, $V_0$ and $U_0 \cap V_0$, then it holds for $U_0 \cup V_0$ by Mayer-Vietoris for hom in the derived category, see Cohomology, Lemma 20.33.3.

Denote $\pi _0 : X_0 \to S_0$ the given morphism. Then we can first consider $U_0 = \pi _0^{-1}(W_0)$ with $W_0 \subset S_0$ quasi-compact open. By the induction principle of Cohomology of Schemes, Lemma 30.4.1 applied to quasi-compact opens of $S_0$ and the remark above, we find that it is enough to prove $P$ for $U_0 = \pi _0^{-1}(W_0)$ with $W_0$ affine. In other words, we have reduced to the case where $S_0$ 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 $S_0$ are affine, the result follows from More on Algebra, Lemma 15.82.7. Namely, by Lemmas 36.10.1 and 36.3.5 the statement is translated into computations of homs in the derived categories of modules. Then Lemma 36.10.2 shows that the complex of modules corresponding to $K_0$ is pseudo-coherent. And Lemma 36.10.5 shows that the complex of modules corresponding to $L_0$ has finite tor dimension over $\mathcal{O}_{S_0}(S_0)$. Thus the assumptions of More on Algebra, Lemma 15.82.7 are satisfied and we win. $\square$

Lemma 36.35.9. In Situation 36.35.7 the category of $S$-perfect objects of $D(\mathcal{O}_ X)$ is the colimit of the categories of $S_ i$-perfect objects of $D(\mathcal{O}_{X_ i})$.

Proof. For every quasi-compact open $U_0 \subset X_0$ consider the condition $P$ that the functor

\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} D_{S_ i\text{-perfect}}(\mathcal{O}_{U_ i}) \longrightarrow D_{S\text{-perfect}}(\mathcal{O}_ U) \]

is an equivalence where $U = f_0^{-1}(U_0)$ and $U_ i = f_{i0}^{-1}(U_0)$. We observe that we already know this functor is fully faithful by Lemma 36.35.8. Thus it suffices to prove essential surjectivity.

Suppose that $P$ holds for quasi-compact opens $U_0$, $V_0$ of $X_0$. We claim that $P$ holds for $U_0 \cup V_0$. We will use the notation $U_ i = f_{i0}^{-1}U_0$, $U = f_0^{-1}U_0$, $V_ i = f_{i0}^{-1}V_0$, and $V = f_0^{-1}V_0$ and we will abusively use the symbol $f_ i$ for all the morphisms $U \to U_ i$, $V \to V_ i$, $U \cap V \to U_ i \cap V_ i$, and $U \cup V \to U_ i \cup V_ i$. Suppose $E$ is an $S$-perfect object of $D(\mathcal{O}_{U \cup V})$. Goal: show $E$ is in the essential image of the functor. By assumption, we can find $i \geq 0$, an $S_ i$-perfect object $E_{U, i}$ on $U_ i$, an $S_ i$-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

\[ a : E_{U, i} \to (Rf_{i, *}E)|_{U_ i} \quad \text{and}\quad b : E_{V, i} \to (Rf_{i, *}E)|_{V_ i} \]

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 $c : E_{U, i}|_{U_ i \cap V_ i} \to E_{V, i}|_{U_ i \cap V_ i}$ which pulls back to the identifications

\[ Lf_ i^*E_{U, i}|_{U \cap V} \to E|_{U \cap V} \to Lf_ i^*E_{V, i}|_{U \cap V}. \]

Apply Cohomology, Lemma 20.42.1 to get an object $E_ i$ on $U_ i \cup V_ i$ 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 $S_ i$-perfect (because being relatively perfect is a 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 36.35.8 using the induction principle (Cohomology of Schemes, Lemma 30.4.1) we reduce to the case where both $X_0$ and $S_0$ are affine. (First work with opens in $S_0$ to reduce to $S_0$ affine, then work with opens in $X_0$ to reduce to $X_0$ affine.) In the affine case the result follows from More on Algebra, Lemma 15.82.7. The translation into algebra is done by Lemma 36.35.3. $\square$

Lemma 36.35.10. Let $f : X \to S$ be a morphism of schemes which is flat, proper, and of finite presentation. Let $E \in D(\mathcal{O}_ X)$ be $S$-perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ S)$ and its formation commutes with arbitrary base change.

Proof. The statement on base change is Lemma 36.22.5. Thus it suffices to show that $Rf_*E$ is a perfect object. We will reduce to the case where $S$ is Noetherian affine by a limit argument.

The question is local on $S$, hence we may assume $S$ is affine. Say $S = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits, Lemma 32.10.1 there exists an $i$ and a scheme $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits, Lemmas 32.13.1 and 32.8.7 we may assume $X_ i$ is proper and flat over $R_ i$. By Lemma 36.35.9 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 Lemma 36.27.1 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 36.35.11. Let $f : X \to S$ be a morphism of schemes. Let $E, K \in D(\mathcal{O}_ X)$. Assume

  1. $S$ is quasi-compact and quasi-separated,

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

  3. $E$ is $S$-perfect,

  4. $K$ is pseudo-coherent.

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

\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, E) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, \mathcal{O}_ S) \]

and the same is true after arbitrary base change: given

\[ \vcenter { \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } } \quad \quad \begin{matrix} \text{cartesian, then we have } \\ Rf'_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L(g')^*K, L(g')^*E) \\ = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (Lg^*L, \mathcal{O}_{S'}) \end{matrix} \]

Proof. Since $S$ is quasi-compact and quasi-separated, the same is true for $X$. By Lemma 36.19.1 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, Lemma 20.47.5). We obtain an inverse system $\ldots \to K_3^\vee \to K_2^\vee \to K_1^\vee $ of perfect objects. By Lemma 36.35.5 we see that $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ is $S$-perfect. Thus we may apply Lemma 36.35.10 to $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ and we obtain an inverse system

\[ \ldots \to M_3 \to M_2 \to M_1 \]

of perfect complexes on $S$ with

\[ M_ n = Rf_*(K_ n^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} E) = Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, E) \]

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 Lemma 36.22.8. 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

\[ M_{n + 1} \to M_ n \to C_ n \to M_{n + 1}[1] \]

Take $S'$ equal to the spectrum of the residue field at a point $s \in S$ and pull back to see that $C_ n \otimes _{\mathcal{O}_ S}^\mathbf {L} \kappa (s)$ has nonzero cohomology only in degrees $\geq n + a$. By More on Algebra, Lemma 15.74.6 we see that the perfect complex $C_ n$ has tor amplitude in $[n + a, m_ n]$ for some integer $m_ n$. 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 Lemma 36.19.1. Since we have

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, E) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\text{hocolim} K_ n, E) = R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, E) = R\mathop{\mathrm{lim}}\nolimits K_ n^\vee \otimes _{\mathcal{O}_ X} E \]

(Cohomology, Lemma 20.48.1) and since $R\mathop{\mathrm{lim}}\nolimits $ commutes with $Rf_*$ we find that

\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, E) = R\mathop{\mathrm{lim}}\nolimits M_ n = R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L_ n, \mathcal{O}_ S) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, \mathcal{O}_ S) \]

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

Remark 36.35.12. The reader may have noticed the similarity between Lemma 36.35.11 and Lemma 36.28.3. Indeed, the pseudo-coherent complex $L$ of Lemma 36.35.11 may be characterized as the unique pseudo-coherent complex on $S$ such that there are functorial isomorphisms

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ S}(L, \mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(K, E \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{F}) \]

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

Lemma 36.35.13. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $E$ be a pseudo-coherent object of $D(\mathcal{O}_ X)$. The following are equivalent

  1. $E$ is $S$-perfect, and

  2. $E$ 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 36.35.3. Say $X \to S$ corresponds to $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(R)$ and $E$ corresponds to $K$ in $D(A)$. If $s$ corresponds to the prime $\mathfrak p \subset R$, then $L(X_ s \to X)^*E$ corresponds to $K \otimes _ R^\mathbf {L} \kappa (\mathfrak p)$ as $R \to A$ is flat, see for example Lemma 36.22.5. Thus we see that our lemma follows from the corresponding algebra result, see More on Algebra, Lemma 15.82.10. $\square$

Remark 36.35.14. Our Definition 36.35.1 of a relatively perfect complex is equivalent to the one given in [lieblich-complexes] whenever our definition applies2. Next, suppose that $f : X \to S$ is only assumed to be locally of finite type (not necessarily flat, nor locally of finite presentation). The definition in the paper cited above is that $E \in D(\mathcal{O}_ X)$ is relatively perfect if

  1. locally on $X$ the object $E$ should be quasi-isomorphic to a finite complex of $S$-flat, finitely presented $\mathcal{O}_ X$-modules.

On the other hand, the natural generalization of our Definition 36.35.1 is

  1. $E$ is pseudo-coherent relative to $S$ (More on Morphisms, Definition 37.53.2) and $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_ S)$ (Cohomology, Definition 20.45.1).

The advantage of condition (B) is that it clearly defines a triangulated subcategory of $D(\mathcal{O}_ X)$, whereas we suspect this is not the case for condition (A). The advantage of condition (A) is that it is easier to work with in particular in regards to limits.

[1] Derived Categories, Definition 13.6.1.
[2] To see this, use Lemma 36.35.3 and More on Algebra, Lemma 15.82.4.

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