Proof. The proof of [Reference LurieLur3, Lemma 5.1.5] goes over essentially unchanged in simplicial commutative rings, but now the rings $R_n$ can be assumed to be perfect rather than merely almost perfect. This follows from the construction of the algebras denote $A(x)_n$ in [Reference LurieLur3], and the fact that in the universal example $\mathbb {Z}$ is a perfect module over $\mathbb {Z}[x]$ of Tor dimension explicitly bounded by $1$, thanks to the Koszul resolution.

Proof. Note first that the natural maps $\operatorname {Spec}(R/I^{n}) \to \operatorname {Spec}(R)$ factor uniquely through $\operatorname {Spf}(R)$ – this determines the natural morphism of the proposition. Let $R_n$ be the Koszul-type algebra killing the $n$th powers of a finite set of generators for the ideal $I \subset R$ (in dg language this would be $R[B_1,\ldots,B_r; dB_i = f_i^{n}]$) – it satisfies the conditions of Proposition 2.1.2 by the proof of [Reference LurieLur3, Lemma 5.1.5]. Note that $\operatorname {Spf}^{\mathrm {cl}} R \simeq \mathop {\varinjlim }\limits \operatorname {Spec}(\pi _0(R_n))$ since $I^{(n-1)r+1} \subset (f_1^{n}, \ldots,f_r^{n}) \subset I^{n}$. We must thus show that the natural map

\[ \mathop{\varinjlim}\limits \operatorname{Spec}(\pi_0(R_n)) \longrightarrow \mathop{\varinjlim}\limits \operatorname{Spec}(R_n) \]

is an equivalence, that is, that the map of pro-objects in (almost perfect) commutative $R$-algebras $\{R_n\} \to \{ \pi _0(R_n) \}$ is a pro-equivalence. There is almost an argument for this in [Reference LurieLur3, Lemma 5.2.17], which implies that

\[ \{ \tau_{\leqslant k} R_n \} \longrightarrow \{ \pi_0(R_n) \} \]

is an equivalence of pro-objects in (almost perfect) $R$-modules for all $k$. By the bound on the Tor amplitude in Proposition 2.1.2, taking $k>r+1$ establishes the pro-equivalence at the level of $R$-modules. Furthermore, the module-level statement does in fact imply the algebra statement (i.e. potential issues with pro-algebras versus algebras in pro-objects do not get in the way) because we may restrict to $(r+1)$-truncated algebras (cf. the proof of [Reference LurieLur3, Lemma 6.3.3]).

Proof. Let $\{R_n\}$ be a tower of perfect $R$-algebras having Tor amplitude at most $d$ as in Proposition 2.1.2. Then we may identify $\operatorname {QC}(\operatorname {Spec}(R)) \simeq R {\rm {}\hbox{-}{\rm mod}}$ and $\operatorname {QC}(\operatorname {Spf}(R)) \simeq \mathop {\varprojlim }\limits _n R_n {\rm {}\hbox{-}{\rm mod}}$. Under these identifications, the functor $i_* i^{*}$ is identified with

\[ M \mapsto i_* i^{*} (M) = \mathop{\varprojlim}\limits_n \big(R_n \otimes_R M\big). \]

If $M \in R {\rm {}\hbox{-}{\rm mod}}_{<0}$ then $R_n \otimes _R M \in R {\rm {}\hbox{-}{\rm mod}}_{< d}$ by hypothesis on $R_n$.

Proof. Note that this is proved, under the additional hypothesis that we restrict to connective objects on both sides, in [Reference LurieLur3, Lemma 5.1.10]. We will prove this stronger result by exploiting our stronger assumptions in the form of Lemma 2.2.1.

We note that $i^{*}$ has a right adjoint $i_*$. If we identify $\operatorname {QC}(\operatorname {Spec}(R)) = R {\rm {}\hbox{-}{\rm mod}}$ and $\operatorname {QC}(\operatorname {Spf}(R)) = \mathop {\varprojlim }\limits _n R_n {\rm {}\hbox{-}{\rm mod}}$, then $i_*$ is given by the inverse limit in $R$-modules

\[ i_*(\{M_n\}) = \mathop{\varprojlim}\limits_n M_n \in R\text{-mod}. \]

It is enough to prove two assertions.

(i) If $M \in R {\rm {}\hbox{-}{\rm mod}}$, then the natural map

\[ M \to i_* i^{*} M \]

identifies $i_* i^{*} M$ with the $I$-completion of $M$. If $M$ is almost connective, this follows by [Reference LurieLur3, Remark 5.1.11]. Since $R {\rm {}\hbox{-}{\rm mod}}$ is right $t$-complete we may write $M = \mathop {\varinjlim }\limits \tau _{\geqslant -k} M$, and it is enough to show that both $i_* i^{*}$ and the $I$-completion functor $M \mapsto \widehat {M}$ preserve this directed colimit. By Lemma 2.1.12, it is enough to show that both $i_* i^{*}$ and the $I$-completion functor have bounded left $t$-amplitude. For $i_* i^{*}$ this is Lemma 2.2.1, while for the $I$-completion it is [Reference LurieLur3, Remark 5.11]. This shows that $M \to i_* i^{*} M$ is the $I$-completion for all $M \in R {\rm {}\hbox{-}{\rm mod}}$.

(ii) Let $\{R_n\}$ be as in Proposition 2.1.2. We must show that if $\{M_n\} \in \mathop {\varprojlim }\limits _n R_n {\rm {}\hbox{-}{\rm mod}}$ then the unit

\[ i^{*} i_* \{M_n\} \longrightarrow \{M_n\} \]

is an equivalence. More concretely, we must show that for each $k$ the natural map

\[ R_k \otimes_R \mathop{\varprojlim}\limits_n M_n \longrightarrow M_k \]

is an equivalence. Since $R_k$ is perfect over $R$, $R_k \otimes _R$ commutes with limits, so we may identify this with the map

\[ \mathop{\varprojlim}\limits_n\big( R_k \otimes_R M_n\big) \simeq \mathop{\varprojlim}\limits_n \big( (R_k \otimes_R R_n) \otimes_{R_n} M_n \big) \longrightarrow M_k. \]

Recall that $\operatorname {Spf}(\widehat {R}) \simeq \mathop {\varinjlim }\limits _n \operatorname {Spec}(R_n)$ and that fiber products preserve filtered colimits of (pre)sheaves, so that we obtain

\[ \operatorname{Spec}(R_k) \times_{\operatorname{Spec}(R)} \operatorname{Spf}(\widehat{R}) \simeq \mathop{\varinjlim}\limits_n \operatorname{Spec}(R_k \otimes_R R_n). \]

Furthermore, $\operatorname {Spec}(R_k) \times _{\operatorname {Spec}(R)} \operatorname {Spf}(\widehat {R}) \simeq \operatorname {Spec}(R_k)$ since $\operatorname {Spec}(R_k) \to \operatorname {Spec}(R)$ factors through the monomorphism $\operatorname {Spf}(R) \to \operatorname {Spec}(R)$. Thus, the natural map

\[ M_k \longrightarrow \mathop{\varprojlim}\limits_n \big( (R_k \otimes_R R_n) \otimes_{R_n} M_n \big) \]

is an equivalence by (i). This completes the proof.

Proof. It is enough to prove the claims when $\mathscr {X}$ is affine, for the claims are local by construction (i.e. because we required the formation of $i_+$ to be local).

Suppose now that $\mathscr {X} = \operatorname {Spec}(R)$ and $Z$ is cut out by a finitely generated ideal $I \subset \pi _0(R)$. Let $(R {\rm {}\hbox{-}{\rm mod}})^{I\text{-nil}} \subset R {\rm {}\hbox{-}{\rm mod}}$ denote the full subcategory of locally $I$-nilpotent modules. By [Reference LurieLur3, Proposition 4.2.5] these are equivalent via the $I$-completion functor and the ‘local cohomology’ functor $\Gamma _I$. Notice that $i^{*}$ vanishes on $I$-local modules (i.e. those supported away from $Z$), so that

\[ i^{*}(M) \simeq i^{*}(\widehat{M}) \simeq i^{*}(\Gamma_I(M)). \]

It thus follows from Lemma 2.2.2 and the above mentioned equivalence that the composite

\[ (R{-{\rm mod}})^{I\text{-nil}} \subset R\text{-mod} \stackrel{i^{*}}\longrightarrow \operatorname{QC}(\operatorname{Spf}(R)) \]

is also an equivalence. Let $i_+$ be the composite of an inverse to this functor with the inclusion $(R {\rm {}\hbox{-}{\rm mod}})^{I\text{-nil}} \subset R {\rm {}\hbox{-}{\rm mod}}$ – more explicitly, $i_+ = \Gamma _I \circ i_*$. Since the inclusion is left adjoint to $\Gamma _I \colon R {\rm {}\hbox{-}{\rm mod}} \to (R {\rm {}\hbox{-}{\rm mod}})^{I\text{-nil}}$ it follows that $i_+$ is left adjoint to $i^{*}$.

It is easy to check that $i_+$, so defined, is local on $R$: given a cartesian diagram

we wish to check that

\[ (i')_+ \widehat{\pi}^{*} (\mathscr{F}) \longrightarrow \pi^{*} i_+ (\mathscr{F}) \]

is an equivalence for all $\mathscr {F} \in \operatorname {QC}(\operatorname {Spf}(R))$. But by the above we may suppose that $\mathscr {F} = i^{*} \mathscr {F}'$ for some $\mathscr {F} \in (R {\rm {}\hbox{-}{\rm mod}})^{I\text{-nil}}$, and then notice that $\pi ^{*} \mathscr {F}' \in (R' {\rm {}\hbox{-}{\rm mod}})^{I'\text{-nil}}$. The result is then immediate from the equivalence applied upstairs and downstairs.

Finally, the projection formula assertion follows from [Reference LurieLur3, 4.1.22, 4.2.6].

Proof. By general non-sense, it is enough to prove that the base-change map of left adjoints

\[ (i')_+ \widehat{\pi}^{*} \longrightarrow \pi^{*} i_+ \]

is an equivalence. By Theorem 2.2.3 the assertion is local on $\mathscr {Y}$ so that we may suppose it is affine. Also, by Theorem 2.2.3 we know that $i^{*}$ is essentially surjective so that it is enough to show that map

\[ (i')_+ (i')^{*} \pi^{*} \simeq (i')_+ \widehat{\pi}^{*} i^{*} \longrightarrow \pi^{*} i_+ i^{*} \]

is an equivalence. By another application of Theorem 2.2.3 we see that this is true (since $i_+ i^{*} \simeq \mathop {{\rm id}}\nolimits$ and similarly for $i'$).

The affine statement follows by applying Lemma 2.2.2.

Proof. The proof consists of four steps.

Step 1: reducing to $F, G$ bounded coherent. It suffices to prove the assertion under the additional hypothesis that $G \in \operatorname {DCoh}(\mathscr {X})$ is bounded, because $G = \mathop {\varprojlim }\limits \tau _{\leqslant n} G$, the limit being taken in $\operatorname {QC}(\mathscr {X})^{\mathrm {acn}}$, and Theorem 2.2.3 implies that $i^{\ast }$ preserves limits. Suppose for concreteness, shifting as needed, that $G \in \operatorname {APerf}(\mathscr {X})^{\mathrm {cn}}_{\leqslant k}$. Let $\mathcal {C} \subset \operatorname {QC}(\mathscr {X})^{\mathrm {acn}}$ denote the full subcategory consisting of those $F$ for which the map

\[ \operatorname{Map}_\mathscr{X}(F,G) \longrightarrow \operatorname{Map}_{\widehat{\mathscr{X}}}(i^{*} F, i^{*} G) \]

is an equivalence for our fixed $G$. Note that $\mathcal {C}$ is closed under extensions and arbitrary colimits which exist in $\operatorname {QC}(\mathscr {X})^{\mathrm {acn}}$. We wish to show that $\mathcal {C} = \operatorname {QC}(\mathscr {X})^{\mathrm {acn}}$.

Note that Theorem 2.2.3 allows us to identify the right-hand side with

\[ \operatorname{Map}_{\widehat{\mathscr{X}}}(i^{*} F, i^{*} G) \simeq \operatorname{Map}_{\mathscr{X}}(R\Gamma_{Z'} F, G) \simeq \operatorname{Map}_{\mathscr{X}}(F \otimes_{\mathcal{O}_X} R\Gamma_{Z'} \mathcal{O}_X, G). \]

Furthermore, $R\Gamma _{Z'} \mathcal {O}_X$ is almost connective, so that tensoring with it is right $t$-exact up to a shift. So $\mathcal {C} \subset \operatorname {QC}(\mathscr {X})$ is the subcategory for which the map to this is an equivalence. We see that $\mathcal {C}$ is closed under arbitrary colimits and contains $\operatorname {QC}(\mathscr {X})_{\geqslant d}$ for some $d$ – since $G$ is bounded and tensoring by $R\Gamma _{Z'} \mathcal {O}_X$ is right $t$-exact up to a shift. By Theorem A.2.1 it is thus enough to show that $\operatorname {APerf}(\mathscr {X})_{< d} \subset \mathcal {C}$, that is, that the map

\[ \operatorname{Map}_\mathscr{X}(F, G) \longrightarrow \operatorname{Map}_\mathscr{X}(F \otimes_{\mathcal{O}_X} R\Gamma_{Z'} \mathcal{O}_X, G) \]

is an equivalence for all $F \in \operatorname {APerf}(\mathscr {X})_{< d}$.

In particular, we are reduced to proving this if both $F, G \in \operatorname {DCoh}(\mathscr {X})$.

Step 2: reducing to a fact on inner Homs. Taking shifts, we may reduce to the case $G \in \operatorname {DCoh}(\mathscr {X})_{\leqslant 0}$ and $F \in \operatorname {DCoh}(\mathscr {X})_{\geqslant 0}$. Let

\[ H = \operatorname{RHom}^{\otimes_{\operatorname{QC}(X)}}_\mathscr{X}(F, G) \in \operatorname{QC}(\mathscr{X}) \]

be the inner Hom for the monoidal structure on $\operatorname {QC}(\mathscr {X})$. Using the definition of inner Hom, it is enough to show that

(2)\begin{equation} R\Gamma(X, H) \longrightarrow \operatorname{RHom}_X(R\Gamma_{Z'} \mathcal{O}_X, H) \end{equation}

is an equivalence. Note that $H$ is coconnective with coherent homology sheaves by Lemma 2.3.2. Thus, it is enough to prove this assertion for any such $H$, not necessarily arising as an inner Hom.

Step 3: reducing to a fact on $\mathscr {S}$. Consider the fiber sequence of quasi-coherent sheaves on $\mathscr {S}$, $R\Gamma _Z \mathcal {O}_\mathscr {S} \longrightarrow \mathcal {O}_\mathscr {S} \longrightarrow j_* \mathcal {O}_\mathscr {U}$, where $j$ is the open immersion $j \colon \mathscr {U} = \mathscr {S} \setminus Z \to \mathscr {S}$. Applying $\pi ^{\ast }$ to this sequence, we can use the base-change formula to identify the third term with $(j')_\ast \mathcal {O}_{\mathscr {U}'}$, where $j' \colon \mathscr {U}' = \mathscr {X} \setminus Z' \to \mathscr {X}$ is the base-changed open immersion. It follows that $R\Gamma _{Z'} \mathcal {O}_\mathscr {X} \simeq \pi ^{*}(R\Gamma _Z \mathcal {O}_\mathscr {S})$.

This lets us identify the morphism of (2) with the morphism

\[ R\Gamma(\mathscr{S}, \pi_* H) \longrightarrow \operatorname{RHom}_\mathscr{S}(R\Gamma_Z \mathcal{O}_\mathscr{S}, \pi_* H). \]

Since $H$ was left-bounded with coherent homology, the assumption on $\pi$ ensures that has $\pi _* H$ has left-bounded, coherent, homology sheaves on $\mathscr {S}$ (see Remark 2.4.2 below).

It is thus enough to show that for any $H' \in \operatorname {QC}(\mathscr {S})$ having coherent homology sheaves the natural map $R\Gamma (\mathscr {S}, H') \to \operatorname {RHom}_\mathscr {S}(R\Gamma _Z \mathcal {O}_\mathscr {S}, H')$ is an equivalence.

Step 4: completing the proof. Note first that, since $R\Gamma _Z \mathcal {O}_\mathscr {S}$ is almost connective, we can reduce as above to the case where $H'$ is right-bounded and hence almost perfect. Next, note that $i^{*} R\Gamma _Z \mathcal {O}_\mathscr {S} = \mathcal {O}_{\widehat {\mathscr {S}}}$ so that another application of Theorem 2.2.3 yields

\[ \operatorname{RHom}_\mathscr{S}(R\Gamma_Z \mathcal{O}_\mathscr{S}, H') \simeq \operatorname{RHom}_{\operatorname{APerf}(\widehat{\mathscr{S}})}(i^{*} \mathcal{O}_\mathscr{S}, i^{*} H'). \]

Thus, the desired assertion follows by our assumption that $\operatorname {APerf}(\mathscr {S}) \to \operatorname {APerf}(\widehat {\mathscr {S}})$ is an equivalence.

Proof. Note that [Reference PreygelPre11, A.1.1] guarantees that the formation of $H$ is fppf local since $F$ is almost perfect and $G$ is bounded above. As explained in the proof there, this does not require condition $(\ast )$ from [Reference PreygelPre11], since the proof reduces to the affine case. Now the claim that $H$ is coconnective with coherent homology sheaves is flat local, so in proving it we may suppose that $\mathscr {X} = \operatorname {Spec}(A)$ is affine with $A$ noetherian. In this case, the proof in [Reference PreygelPre11] shows that there is a third-quadrant spectral sequence converging to the homology groups of $H$ whose starting term consists of finite sums of the homology of $G$. This shows both that the homology of $H$ is appropriately bounded above and that each homology sheaf is coherent.

Proof. It is clear that $(\mathrm {i}) \Rightarrow (\mathrm {ii}) \Rightarrow (\mathrm {iii}) \Rightarrow (\mathrm {iv})$. Now assume (iv). For any $\widehat {F} \in {\rm Coh} (\widehat {\mathscr {X}})$, one can find $G_0,G_1 \in {\rm Coh} (\mathscr {X})$ along with surjections $i^{\ast } G_0 \to \widehat {F}$ and $i^{\ast } G_1 \to \ker (i^{\ast } G_0 \to \widehat {F})$. It follows that $\tau _{\leqslant 0} \operatorname {cone}(i^{\ast } G_1 \to i^{\ast } G_0) \simeq \widehat {F}$. But by full faithfulness the morphism $i^{\ast } G_1 \to i^{\ast } G_0$ is induced by a morphism $G_1 \to G_0$ and we have $\widehat {F} \simeq i^{\ast } \tau _{\leqslant 0} \operatorname {cone}(G_1 \to G_0)$. Hence, we have shown (iii), and (ii) follows from this by a simple inductive argument using the $t$-structure and Proposition 2.1.11. Finally, for $(\mathrm {ii}) \Rightarrow (\mathrm {i})$, given $\widehat {F} \in \operatorname {APerf}(\widehat {\mathscr {X}})$, full faithfulness of $i^{\ast }$ and essential surjectivity on $\operatorname {APerf}(\widehat {\mathscr {X}})_{\geqslant n}$ allow one to lift $\{\tau _{\leqslant n} \widehat {F}\}$ to a tower $\cdots \to F_1 \to F_0$ in $\operatorname {APerf}(\mathscr {X})$ with $i^{\ast }(F_n) \simeq \tau _{\leqslant n} \widehat {F}$. Then part (iv) of Proposition 2.1.11 and part (ii) of Lemma 2.1.12 guarantee that

\[ i^{\ast} (\mathop{\varprojlim}\limits F_n) \simeq \mathop{\varprojlim}\limits i^{\ast} F_n \simeq \widehat{F}. \]

Proof. Let us denote $S = \operatorname {Spec}(R[\![t]\!])$ and $R_n = R[\![t]\!]/(t^{n})$. Let $F \in \operatorname {DCoh}(\mathscr {X})$, and consider the $R[\![t]\!]$-module $E:= (\pi _S)_\ast (F|_{\mathscr {X}_S}) \in \operatorname {QC}(S)_{\leqslant 0}$. Applying the base change Lemma A.1.3 to the finite Tor amplitude inclusions $\operatorname {Spec}(R_n) \hookrightarrow S$ and the full-faithfulness hypothesis of the lemma, we have

\[ E \simeq \operatorname{RHom}_{\mathscr{X}_S}(\mathcal{O}_{\mathscr{X}_S},F|_{\mathscr{X}_S}) \xrightarrow{\simeq} \operatorname{RHom}_{\widehat{\mathscr{X}}_S}(\mathcal{O}_{\widehat{\mathscr{X}}_S},F|_{\widehat{\mathscr{X}}_S}) \simeq \mathop{\varprojlim}\limits R_n \otimes_{R[\![t]\!]} E, \]

so $E$ is complete as an $R[\![t]\!]$-module, that is, $E \xrightarrow {\simeq } \widehat {E}$ is an equivalence. Applying base change to the flat map $S \to \operatorname {Spec}(R)$, we have $E \simeq R[\![t]\!] \otimes _R \pi _\ast (F)$.

It therefore suffices to show that a complex $M \in R {\rm {}\hbox{-}{\rm mod}}_{\leqslant 0}$ such that $R[\![t]\!] \otimes _R M$ is complete must have coherent homology. By [Reference LurieLur3, Theorem 4.2.13] it suffices to prove this for $M \in \operatorname {QC}(R)^{\heartsuit }$, for which we may assume that $R$ is classical. Express $M$ as a filtered union of coherent submodules $M = \bigcup M_\alpha$. If $M$ is not coherent, then this union does not stabilize, so we can find a sequence $m_0,m_1,\ldots \in M$ which is not contained in any $M_\alpha$. Then $R[[t]] \otimes _R M = \bigcup R[[t]] \otimes _R M_\alpha$, but the element $\sum t^{i} m_i \in H_0( (R[[t]] \otimes _R M)^{\wedge })$ does not lie in the image of any of the $R[[t]]\otimes _R M_\alpha$, and so $R[[t]] \otimes _R M$ does not surject onto its completion. This contradicts the hypothesis that $R[\![t]\!] \otimes M$ is complete.

Proof. The property of a morphism being closed is fppf local on the target, so it suffices to assume that $\mathscr {Y} = \operatorname {Spec}(R)$ is affine, and in (ii) it suffices to assume that $\mathscr {Y}' = \operatorname {Spec}(A)$ is affine. We have immediate implications $(\mathrm {i}) \Rightarrow (\mathrm {ii}) \Rightarrow (\mathrm {iii})$, and $(\mathrm {iii}) \Rightarrow (\mathrm {ii})$ because we can write $A' = A[x_1,\ldots,x_n] / I$ for some ideal $I$ and thus realize $Y'$ as a closed subscheme of $Y \times \mathbf {A}^{n}$. So we must show that if $\mathscr {X}_{A} \to \operatorname {Spec}(A)$ is closed for any finitely generated $R$-algebra $A$, then it is closed for any $R$-algebra.

By relative noetherian approximation, $\mathscr {X}$ can be obtained by base change from a finitely presented algebraic stack over $\mathscr {X}'$ a noetherian subring $R' \subset R$. By Theorem A.2.1, $\operatorname {QC}(\mathscr {X}')^{\heartsuit }$ is a locally noetherian $R'$-linear abelian category in the sense of [Reference PopescuPop73, § 5.8]. For any $R'$-algebra $A$, $\operatorname {QC}(\mathscr {X}_A)^{\heartsuit }$ is the category of $A$-module objects in $\operatorname {QC}(\mathscr {X}')^{\heartsuit }$, and it is a locally finitely presentable (i.e. compactly generated) abelian category. In fact, it is generated by the pullbacks of coherent sheaves on $\mathscr {X}'$, which are finitely presented objects of $\operatorname {QC}(\mathscr {X}_A)^{\heartsuit }$ [Reference Artin and ZhangAZ01, B3.17, B5.1i].

Now consider an $R$-algebra $A$, and write $A = \bigcup _\alpha A_\alpha$ as a filtered union of finitely generated $R$-algebras $A_\alpha$. Then pullback induces an equivalence of categories of finitely presented objects

\begin{align*} (\operatorname{QC}(\mathscr{X}_A)^{\heartsuit})^{\mathrm{fp}} \xrightarrow{\simeq} \varinjlim_\alpha (\operatorname{QC}(\mathscr{X}_{A_\alpha})^{\heartsuit})^{\mathrm{fp}}. \end{align*}

If $\mathscr {Z} \hookrightarrow \mathscr {X}_A$ is a finitely presented closed immersion, then $\mathcal {O}_\mathscr {Z} = \!\!\mathop {\rm ~coker}\nolimits (M \to \mathcal {O}_{\mathscr {X}_A})$ for some finitely presented quasi-coherent sheaf $M$. Then the map $M \to \mathcal {O}_{\mathscr {X}_A}$ is the pullback of a map $M_\alpha \to \mathcal {O}_{\mathscr {X}_{A_\alpha }}$ in $\operatorname {QC}(\mathscr {X}_{A_\alpha })^{\heartsuit }$ with $M_\alpha$ finitely presented. In particular, $\mathscr {Z}$ is the preimage of a closed substack $\mathscr {Z}_\alpha \hookrightarrow \mathscr {X}_{A_\alpha }$. This gives a diagram of topological spaces, in which the square is cartesian:

It follows that $\pi (|\mathscr {Z}|) = f^{-1}(\pi _\alpha (|\mathscr {Z}_\alpha |))$, and hence $\pi (\mathscr {Z})$ is closed, because by hypothesis $\pi _\alpha$ is a closed map.

So we have shown that the image under $\pi$ of the underlying subset of a finitely presented closed immersion is closed in $|\!\operatorname {Spec}(A)|$, and we must show that the same is true for the subset underlying any closed immersion $\mathscr {Z} \hookrightarrow \mathscr {X}_A$. We know that $\mathcal {O}_\mathscr {Z} = \!\!\mathop {\rm ~coker}\nolimits (M \to \mathcal {O}_{\mathscr {X}_A})$ for some $M \in \operatorname {QC}(\mathscr {X}_A)^{\heartsuit }$. Writing $M = \varinjlim _\alpha M_\alpha$ as a colimit of finitely presented quasi-coherent sheaves, we can write $\mathscr {Z}$ as a cofiltered intersection of schemes $\mathscr {Z} = \cap _\alpha \mathscr {Z}_\alpha$, where $\mathcal {O}_{\mathscr {Z}_\alpha } = \!\!\mathop {\rm ~coker}\nolimits (M_\alpha \to \mathcal {O}_{\mathscr {X}_A})$, hence $\mathscr {Z}_\alpha \to \mathscr {X}_A$ is a finitely presented closed immersion.

We claim that $\pi (|\mathscr {Z}|) = \cap _\alpha \pi (|\mathscr {Z}_\alpha |)$. The inclusion $\pi (|\mathscr {Z}|) \subset \cap _\alpha \pi (|\mathscr {Z}_\alpha |)$ is clear. If $p \in \pi (|\mathscr {Z}|) \subset |\!\operatorname {Spec}(A)|$ is a point, represented by a map $\operatorname {Spec}(K) \to \operatorname {Spec}(A)$ for some field $K$, then $\mathscr {Z}_K \subset \mathscr {X}_K$ is non-empty. Because $\mathscr {X}_K$ is noetherian, the cofiltered system $(\mathscr {Z}_\alpha )_K$ is eventually constant, that is, $\mathscr {Z}_K = (\mathscr {Z}_\alpha )_K$ for $\alpha$ sufficiently large. It follows that $p \in \cap _\alpha \pi (|\mathscr {Z}_\alpha |)$, hence $\pi (|\mathscr {Z}|) = \cap _\alpha \pi (|\mathscr {Z}_\alpha |)$. This shows that $\pi (|\mathscr {Z}|)$ is closed, because we have already shown that $\pi (|\mathscr {Z}_\alpha |)$ is closed for all $\alpha$.

Proof. Let $S \to \mathscr {S}$ be a morphism from a noetherian affine derived scheme, and let $\mathscr {X}^{\prime } := \mathscr {X} \times _\mathscr {S} S$. By taking shifts it suffices to show that (CP) ${}_{\mathscr {S}}$ implies that $\pi ^{\prime }_\ast (\operatorname {APerf}(\mathscr {X}^{\prime })_{\geqslant d}) \subset \operatorname {APerf}(S)$ for any fixed $d$. The base-change formula, Proposition A.1.5, shows that this holds on the full subcategory of $\operatorname {APerf}(\mathscr {X}')$ spanned by derived pullbacks from $\operatorname {APerf}(\mathscr {X})$. Since $\mathscr {X}' \to \mathscr {X}$ is affine, any $F \in \operatorname {APerf}(\mathscr {X}^{\prime })_{\geqslant d}$ admits the cobar simplicial resolution by pullbacks of objects in $\operatorname {APerf}(\mathscr {X})_{\geqslant d}$. Let $d$ be bigger than the universal cohomological dimension of $\pi$, so that each term in the pushforward simplicial resolution is connective. The result follows by noting that $\operatorname {APerf}(S)_{\geqslant 0}$ is preserved by geometric realizations [Reference LurieLur17, Proposition 7.2.4.11(4)].

Proof. Recall that $\operatorname {Spf}(R) \simeq \mathop {\varinjlim }\limits \operatorname {Spec}(R_n)$ for $\{R_n\}$ as in Proposition 2.1.2 (we do not need the finite Tor amplitude assumption here, so this works in the spectral setting as well by Remark 2.1.3), so that

\[ \mathscr{F}(\operatorname{Spf}(R)) \simeq \mathop{\varprojlim}\limits_n \mathscr{F}(R_n), \]

and applying Lemma 3.1.1 to the base change of $\pi$ to each $\operatorname {Spec}(R_n)$, we obtain that

\[ \mathscr{F}(R_n) \simeq \operatorname{Tot}\{\mathscr{F}({\rm Cech}(\pi \times_{\operatorname{Spec}(R)} \operatorname{Spec}(R_n)))\}. \]

Taking the inverse limit of these equivalences we complete the proof, since

\[ \mathop{\varprojlim}\limits_n \mathscr{F}(R'^{\otimes_R \bullet+1} \otimes_R R_n) \simeq \mathscr{F}(\operatorname{Spec}(R'^{\otimes_R \bullet+1}) \times_{\operatorname{Spec}(R)} \operatorname{Spf}(R)) \simeq \mathscr{F}(R'^{\otimes_R \bullet+1}) \]

as $\operatorname {Spec}(R'^{\otimes _R \bullet +1}) \to \operatorname {Spec}(R)$ factors through the monomorphism $\operatorname {Spf}(R) \to \operatorname {Spec}(R)$.

Proof. Let $\mathcal {D}$ denote the class of all almost finitely presented surjections $\pi \colon \mathscr {X}' \to \mathscr {X}$ with locally noetherian target. Let $\mathcal {C}\subset \mathcal {D}$ consist of those $\pi$ satisfying, additionally, the condition that for every almost finitely presented affine map $\mathscr {Y} \to \mathscr {X}$, $\mathscr {F}$ has descent for the base change $\pi _\mathscr {Y}$, that is, restriction gives an equivalence $\mathscr {F}(\mathscr {Y}) \to \operatorname {Tot}\{\mathscr {F}({\rm Cech}(\pi _\mathscr {Y}))\}$. We wish to show that $\mathcal {C}$ contains all $h$-covers.

Note that if $\pi \in \mathcal {C}$, then $\mathscr {F}$ has descent for $\pi _\mathscr {Y}$ for any almost finitely presented morphism $\mathscr {Y} \to \mathscr {X}$: it suffices to consider $\mathscr {X}$ quasi-compact (i.e. noetherian). We can choose an étale surjection $p : \mathscr {V} \to \mathscr {Y}$ with $\mathscr {V}$ affine, and by étale descent and commuting limits it suffices to show that $\pi _{\mathscr {V}_n} \in \mathcal {C}$ for each level $\mathscr {V}_n = \mathscr {V}^{\times _\mathscr {Y} n}$ in the Čech nerve of $p$. Thus, we can reduce to the case where $\mathscr {Y} \to \mathscr {X}$ is representable by derived algebraic spaces. Repeating this trick three more times, we reduce to the case where $\mathscr {Y} \to \mathscr {X}$ is representable by derived schemes, then by separated derived schemes, and finally to where $\mathscr {Y} \to \mathscr {X}$ is affine.

The proof of Lemma 3.1.1, Step 2, applies verbatim to show that $\mathcal {C}$ is closed under composition and refinements, and to show locality along maps in $\mathcal {C}$.

Step 0: reduction to $\mathscr {X}$ an affine scheme. Since $\mathscr {F}$ is an fppf sheaf, the assertion is fppf local on $\mathscr {X}$. Thus, we may suppose $\mathscr {X}$ is affine.

Step 1: reduction to $\mathscr {X}$ classical and reduced, $\mathscr {X}'$ classical. Since $\operatorname {Spec}(\pi _0(R)_{\mathrm {red}}) \to \operatorname {Spec}(R)$ is a surjective closed immersion, it is in $\mathcal {C}$. So by ‘locality’ to show that $\pi \in \mathcal {C}$ it is enough to show that the base change $\pi _{\pi _0(R)_{\mathrm {red}}} \in \mathcal {C}$. That is, we may suppose that $\mathscr {X}$ is classical and reduced. Similarly, by ‘refinements’ we may suppose that $\mathscr {X}'$ is classical.

Step 2: a general descent argument for (classical) $h$-covers. We will prove, by noetherian induction, that for every closed subscheme $Z \subset \mathscr {X}$ it is the case that all $h$-coverings having target $Z$ belong to $\mathcal {C}$. Thus, we may suppose that this holds for all (classical) proper closed subschemes of $\mathscr {X}$, and we must show that every $h$-cover $\pi \colon \mathscr {X}' \to \mathscr {X}$ belongs to $\mathcal {C}$. We can choose a smooth cover $\operatorname {Spec}(A) \to \mathscr {X}'$, and the composition $\operatorname {Spec}(A) \to \mathscr {X}$ is again an $h$-cover, so it suffices by refinement to consider the case where $\pi$ is representable by derived algebraic spaces.

Since $\mathscr {X}$ is reduced, we may apply ‘generic flatness’ to $\pi$ to deduce the existence of a dense open $U \subset \mathscr {X}$ such that $\pi _U$ is flat. By ‘platification par éclatement’, there exists a closed subscheme $Z \subset \mathscr {X}$ disjoint from $U$ such that the strict transform $\widetilde {p_Z}(\mathscr {X}')$ is flat over $\operatorname {Bl}_Z(\mathscr {Y}')$. More precisely, there is a commutative diagram

such that $\pi ' \colon \mathscr {Y}' \to \mathscr {Y}$ is flat. In order to show that $\pi \in \mathcal {C}$ it is enough, by refinement, to show that $p_Z \circ \pi ' \in \mathcal {C}$. Note that $\pi '$ is surjective by the argument of [Reference VoevodskyVoe96, Proposition 3.1.3], so that it is a flat cover and thus in $\mathcal {C}$. It thus suffices to show that the blowup morphism $p_Z \in \mathcal {C}$.

We now simplify our notation by renaming $p_Z$ to just $\pi \colon \mathscr {X}' = \operatorname {Bl}_Z(\mathscr {X}) \longrightarrow \mathscr {X}$. We consider an almost finitely presented map $\mathscr {Y} = \operatorname {Spec}(R') \to \mathscr {X}$, and we wish to show that $\mathscr {F}$ has descent for the base change $\pi _{\mathscr {Y}} : \mathscr {Y}' \to \mathscr {Y}$ of $\pi$.

$\pi _{Z} \in \mathcal {C}$ by our inductive hypothesis, so by Step 1, $\pi _{R/I^{n}} \in \mathcal {C}$ for all $n \geqslant 1$, where $I \subset R$ is the ideal defining $Z$. We may summarize this with the slight abuse of notation $\pi _{\operatorname {Spf}(R)} \in \mathcal {C}$. It follows that if $\widehat {\mathscr {Y}}$ is the formal completion of $\mathscr {Y}$ along the preimage of $Z$, then $\mathscr {F}(\widehat {\mathscr {Y}}) \to \operatorname {Tot}\{\mathscr {F}({\rm Cech}(\pi _{\widehat {\mathscr {Y}}}))\}$ is an equivalence. Now consider the cartesian diagram of simplicial derived schemes

where the bottom row consists of constant simplicial schemes. This diagram is levelwise an abstract blowup square (3) with affine base, so applying our hypothesis (iii) and commuting limits, we see that

\[ \mathscr{F}(\mathscr{Y}) \to \operatorname{Tot}\{\mathscr{F}({\rm Cech}(\pi_\mathscr{Y}))\} \times_{\operatorname{Tot}\{\mathscr{F}({\rm Cech}(\pi_{\widehat{\mathscr{Y}}}))\}} \mathscr{F}(\widehat{\mathscr{Y}}) \]

is an equivalence, and hence $\mathscr {F}(\mathscr {Y}) \to \operatorname {Tot}\{\mathscr {F}({\rm Cech}(\pi _\mathscr {Y}))\}$ is an equivalence.

Proof. This follows from Theorem 2.2.3 in the derived setting, and from [Reference LurieLur3, Theorem 5.1.9] in the spectral setting for categories of almost connective modules. For the stronger statement in the spectral setting, note that the claim is local on $\mathscr {X}$, so we may assume $\mathscr {X} = \operatorname {Spec}(R)$. The claim is equivalent to the canonical maps $M \to p_\ast (p^{\ast }(M))$ and $p^{\ast } (p_\ast (F)) \to F$ being equivalences for all $M \in \operatorname {QC}_Z(\mathscr {X})$ and $F \in \operatorname {QC}_{p^{-1}(Z)}(\mathscr {X}')$. This follows from the case of almost connective objects, because the $t$ structures on these categories are right $t$-complete and $p_\ast$ and $p^{\ast }$ commute with filtered colimits by Proposition A.1.5.