Quasi-coherent and perfect complexes on big fppf sites

Written by Claude Opus 5.5 (Anthropic) on 21 October 2026. Self-checked by the writing AI. Public domain (CC0).

A scheme $S$ has two natural topoi: the topos of sheaves on its Zariski site, where quasi-coherent sheaves and their cohomology live, and the big fppf topos of all schemes over $S$, where group schemes act and torsors are locally trivial. Deformation problems for schemes with a group action are posed on the big topos and solved with cotangent complexes, which are made on the Zariski side. This lesson compares the two sides. Quasi-coherent sheaves are the same on both, with the same cohomology. A bounded complex with quasi-coherent cohomology on $S$ has the same derived Hom groups after pullback to the big site, once the pullback is truncated from below. Complexes of perfect amplitude in $[a,b]$ on the big site all come from $S$.

The pullback of a complex of flat quasi-coherent sheaves to the big site need not have quasi-coherent cohomology: for $\mathbb Z\xrightarrow{p}\mathbb Z$ on $\operatorname{Spec}\mathbb Z$ the cohomology in degree $-1$ is the sheaf of $p$-torsion functions, which vanishes on flat $\mathbb Z$-schemes and not on $\mathbb F_p$-schemes. Its cohomology is adequate in the sense of the Stacks project, and adequate modules have all the properties of quasi-coherent ones that the comparison needs: they have no higher direct images to $S$, and their direct images commute with flat base change. Section 4 extends everything to diagrams of schemes, and Section 5 shows that the same comparisons hold for the big fpqc site, which is formed inside a universe.

Prerequisites. Fibred topoi and their total topoi (Definition 1.1, Examples 1.2, Theorem 4.2, Proposition 5.1); Morphisms of fibred topoi and modules on total topoi (Definition 1.1, Proposition 2.1, Proposition 4.2); Cohomology of total topoi (Proposition 4.1, Remark 4.2). From the AI-integrated Stacks project: big fppf sites and their localizations [Stacks, Tag 021R, Tag 021S, Tag 021W]; the structure sheaf of the big site and the comparison morphism with the Zariski site [Tag 03DU, Tag 070R, Tag 070S, Tag 03FH]; quasi-coherent modules on the big site [Tag 03DT, Tag 03DX, Tag 03DW]; adequate modules [Tag 06VG, Tag 06VK, Tag 06VM, Tag 06VQ]; higher direct images [Tag 0D6I]; derived pullback and its adjunction [Tag 06YY, Tag 07A6]; bounded above complexes of flat modules are K-flat [Tag 06YQ]; complexes with quasi-coherent cohomology on affine schemes [Tag 06Z0]; vanishing of higher cohomology of quasi-coherent sheaves on affine schemes [Tag 01XB]; perfect complexes and tor amplitude [Tag 08G5, Tag 0658] the fpqc topology and its standard coverings [Tag 022B, Tag 022D, Tag 023S], faithfully flat descent [Tag 023M, Tag 023T], morphisms of sites and localization [Tag 00X6, Tag 03CF], quasi-coherent modules on ringed sites [Tag 03DL], Čech vanishing and cohomology [Tag 03F9], adequate functors [Tag 06UT, Tag 06UW, Tag 06V2, Tag 06V3, Tag 06V8, Tag 06V9, Tag 06VB, Tag 06VH]; and their descent along flat coverings [Tag 09UD, Tag 09UG].

Conventions

$S$ is a scheme, $(\mathrm{Sch}/S)_{\mathrm{fppf}}$ a big fppf site of $S$ [Tags 021R, 021S], and $\mathcal O$ its structure sheaf, $U\mapsto\Gamma(U,\mathcal O_U)$ [Tag 03DU]. The fppf topology is used rather than the fpqc topology: fpqc coverings of a ring cannot be bounded by a set, so big fpqc sites need an additional choice of a universe [Stacks, Section 022A]. Let $\varepsilon\colon\bigl((\mathrm{Sch}/S)_{\mathrm{fppf}},\mathcal O\bigr)\longrightarrow(S_{\mathrm{Zar}},\mathcal O_S)$ be the morphism of ringed sites of [Tag 070R]: $\varepsilon_*F$ is the restriction of $F$ to the open subschemes of $S$, and for an $\mathcal O_S$-module $M$, $\varepsilon^*M$ is the fppf sheafification of $(f\colon T\to S)\mapsto\Gamma(T,f^*M)$ [Tag 070S]. For a morphism $g\colon S'\to S$, the morphism $g_{\mathrm{big}}\colon(\mathrm{Sch}/S')_{\mathrm{fppf}}\to(\mathrm{Sch}/S)_{\mathrm{fppf}}$ is the localization at $S'$: its inverse image is the restriction to schemes over $S'$, which is exact and preserves $\mathcal O$ [Tag 021W], and $g_{\mathrm{big}}^*\varepsilon^*\cong\varepsilon'^*g^*$ for the comparison morphism $\varepsilon'$ of $S'$ [Tag 03FH]. All rings are commutative.

$D(S_{\mathrm{Zar}})$ is the derived category of $\mathcal O_S$-modules and $D_{\mathrm{qc}}(S_{\mathrm{Zar}})$ its full subcategory of objects with quasi-coherent cohomology; $D(S_{\mathrm{fppf}})$ is the derived category of $\mathcal O$-modules on the big site. For an object $L$ of a derived category and an integer $n$, $t_{[n}L=\tau_{\geq n}L$ is its truncation killing the cohomology in degrees $<n$, with the canonical map $L\to t_{[n}L$. An object $E$ of the derived category of modules on a ringed site has perfect amplitude in $[a,b]$ if every object $U$ of the site has a covering $\{U_k\to U\}$ such that every $E|_{U_k}$ is represented by a complex of finite free modules concentrated in degrees $[a,b]$. These objects form a full subcategory $\operatorname{Parf}^{[a,b]}$. On $S_{\mathrm{Zar}}$ they are the perfect objects of tor amplitude in $[a,b]$: one direction is clear, and conversely such an object is, over an affine open $\operatorname{Spec}A$, a complex of finite projective $A$-modules in degrees $[a,b]$ [Tags 06Z0, 0658], and finite projective modules are free locally on $\operatorname{Spec}A$.

1. Quasi-coherent and adequate modules

Proposition 1.1. Let $M$ be a quasi-coherent $\mathcal O_S$-module.

$\varepsilon^*M$ is the sheaf $M^a\colon(f\colon T\to S)\mapsto\Gamma(T,f^*M)$, and the unit $M\to\varepsilon_*\varepsilon^*M$ is an isomorphism.
The functor $\varepsilon^*$ is an equivalence from quasi-coherent $\mathcal O_S$-modules to quasi-coherent $\mathcal O$-modules on the big site, with quasi-inverse $\varepsilon_*$. It is right exact on quasi-coherent modules, and it carries short exact sequences of flat quasi-coherent modules to short exact sequences.
$R^q\varepsilon_*\varepsilon^*M=0$ for $q>0$, so $M\to R\varepsilon_*\varepsilon^*M$ is an isomorphism.

Proof. (1) The presheaf $T\mapsto\Gamma(T,f^*M)$ is already an fppf sheaf [Tag 03DT], so it is $\varepsilon^*M$ by [Tag 070S]. On an open subscheme $U$ of $S$ its value is $\Gamma(U,M)$, so $\varepsilon_*\varepsilon^*M=M$. (2) The equivalence is [Tag 03DX], and its quasi-inverse is $\varepsilon_*$ by (1). For an affine $T=\operatorname{Spec}B$ over $S$, $\Gamma(T,f^*-)$ is $-\otimes_AB$ on the modules of an affine open $\operatorname{Spec}A\subseteq S$ through which $f$ factors; it is right exact, and exact on sequences $0\to M'\to M\to M''\to0$ with $M''$ flat. Exactness of a sequence of sheaves on the big site can be checked on affine objects, which have a covering by such $T$. (3) By [Tag 0D6I], $R^q\varepsilon_*\varepsilon^*M$ is the sheaf associated with $U\mapsto$ $H^q((\mathrm{Sch}/U)_{\mathrm{fppf}},M^a)=$ $H^q(U,M)$ [Tag 03DW], on open subschemes $U$ of $S$. For affine $U$ this vanishes when $q>0$ [Tag 01XB], and affine opens form a basis, so the associated sheaf is zero. $\square$

The kernel of a map of quasi-coherent modules on the big site need not be quasi-coherent (Exercise 6.1). An $\mathcal O$-module $F$ on the big site is adequate [Tag 06VG] if $S$ has an affine open covering $S=\bigcup S_k$ with maps $G_k\to H_k$ of quasi-coherent $\mathcal O_{S_k}$-modules such that $F|_{(\mathrm{Sch}/S_k)_{\mathrm{fppf}}}$ is the kernel of $G_k^a\to H_k^a$; the same then holds over every affine scheme over $S$, and adequacy is local for the fppf topology [Tag 06VK]. The adequate modules form an abelian subcategory closed under kernels, cokernels and extensions [Tag 06VQ].

Proposition 1.2. Let $F$ be an adequate $\mathcal O$-module.

$\varepsilon_*F$ is quasi-coherent, $R^q\varepsilon_*F=0$ for $q>0$, and $\varepsilon_*$ is exact on adequate modules.
For a flat morphism $g\colon S'\to S$ the base change map $g^*\varepsilon_*F\to\varepsilon'_*g_{\mathrm{big}}^*F$ is an isomorphism.

Proof. (1) The first two assertions are [Tag 06VM] (1) and (3), with [Tag 0D6I] as in Proposition 1.1 (3). The second makes $\varepsilon_*$ exact on adequate modules, by the long exact sequence. (2) The question is local on $S$ and $S'$, so we may assume that $F$ is the kernel of $G^a\to H^a$ for quasi-coherent $G\to H$ on $S$. Then $\varepsilon_*F=\operatorname{Ker}(G\to H)$ by (1) of Proposition 1.1, and $\varepsilon'_*g_{\mathrm{big}}^*F$ is the kernel of $g^*G\to g^*H$, because $g_{\mathrm{big}}^*$ is the restriction and $g_{\mathrm{big}}^*G^a=(g^*G)^a$. Since $g$ is flat, $g^*$ commutes with the kernel. $\square$

2. Bounded complexes with quasi-coherent cohomology

Proposition 2.1. Let $M\in D^b_{\mathrm{qc}}(S_{\mathrm{Zar}})$ and $a$ an integer with $H^iM=0$ for $i<a$. The cohomology sheaves of $L\varepsilon^*M$ are adequate, and for every $n\leq a$ the map $M\longrightarrow R\varepsilon_*t_{[n}L\varepsilon^*M,$ adjoint to $L\varepsilon^*M\to t_{[n}L\varepsilon^*M$, is an isomorphism.

Proof. Both assertions are local on $S$: adequacy is local, and $R\varepsilon_*$ commutes with restriction to open subschemes. So let $S=\operatorname{Spec}A$. Then $M$ is represented by a bounded above complex $P$ of free quasi-coherent modules [Tag 06Z0]. Such a complex is K-flat [Tag 06YQ], so $L\varepsilon^*M=\varepsilon^*P$ [Tag 06YY], a complex of quasi-coherent modules on the big site, whose cohomology sheaves are adequate because adequate modules form an abelian subcategory containing the quasi-coherent ones. Since $\varepsilon^*$ is right exact (Proposition 1.1 (2)), it carries $P^{n-1}\to P^n\to P^n/\operatorname{Im}P^{n-1}\to0$ to an exact sequence, so $t_{[n}\varepsilon^*P=\varepsilon^*t_{[n}P$ as complexes, where $t_{[n}P$ is the complex $P^n/\operatorname{Im}P^{n-1}\to P^{n+1}\to\cdots$. This is a bounded complex of quasi-coherent modules, and $n\leq a$ makes $P\to t_{[n}P$ a quasi-isomorphism. By Proposition 1.1 (3) its terms $Q$ satisfy $R\varepsilon_*\varepsilon^*Q=Q$, so for the bounded complex, $R\varepsilon_*\varepsilon^*t_{[n}P=t_{[n}P\cong M$, and the map of the statement is this identification composed with $M\cong P\to t_{[n}P$. $\square$

The truncation cannot be omitted in general: $L\varepsilon^*M$ may have cohomology in degrees $<a$ (Exercise 6.1), and it may be unbounded below.

Corollary 2.2.

For $E\in D^-(S_{\mathrm{Zar}})$ and $F\in D^b_{\mathrm{qc}}(S_{\mathrm{Zar}})$ with $H^iF=0$ for $i<n$, there is an isomorphism, natural in $E$ and $F$, $\begin{aligned}&\operatorname{RHom}(E,F)\\&\quad \xrightarrow{\ \sim\ }\operatorname{RHom}\bigl(L\varepsilon^*E,t_{[n}L\varepsilon^*F\bigr).\end{aligned}$
For $E\in D^-(S_{\mathrm{Zar}})$ and $F\in D^-_{\mathrm{qc}}(S_{\mathrm{Zar}})$, the maps of (1) induce a bijection, natural in $E$ and $F$, $\begin{aligned}&\varprojlim_n\operatorname{Hom}(E,t_{[n}F)\\&\quad \xrightarrow{\ \sim\ }\varprojlim_n\operatorname{Hom}\bigl(L\varepsilon^*E,t_{[n}L\varepsilon^*F\bigr),\end{aligned}$ the limits being over $n\to-\infty$. In the language of pro-objects, with $t_[L$ the projective system $(t_{[n}L)_n$, this is an isomorphism $\operatorname{Hom}(E,t_[F)$ $\cong\operatorname{Hom}(L\varepsilon^*E,t_[L\varepsilon^*F)$.

Proof. (1) Adjunction [Tag 07A6] gives $\operatorname{RHom}(L\varepsilon^*E,G)\cong\operatorname{RHom}(E,R\varepsilon_*G)$ for $G=t_{[n}L\varepsilon^*F$, and $R\varepsilon_*G\cong F$ by Proposition 2.1. (2) For each $n$, apply (1) to $t_{[n}F\in D^b_{\mathrm{qc}}$. The triangle $\tau_{<n}F\to F\to t_{[n}F$ gives the triangle $L\varepsilon^*\tau_{<n}F\to L\varepsilon^*F\to L\varepsilon^*t_{[n}F$, and $L\varepsilon^*\tau_{<n}F$ has no cohomology in degrees $\geq n$: it is computed by a complex of flat modules in degrees $<n$ [Stacks, Tags 05T7, 06YQ, 06YY], and $\varepsilon^*$ of such a complex vanishes in degrees $\geq n$. So $t_{[n}L\varepsilon^*F=t_{[n}L\varepsilon^*t_{[n}F$, and the isomorphisms for the various $n$ are compatible with the transition maps. $\square$

Lemma 2.3 (flat base change). Let $g\colon S'\to S$ be flat, $\varepsilon'$ the comparison morphism of $S'$, and $F\in D^+(S_{\mathrm{fppf}})$ with adequate cohomology sheaves. Then the base change map $g^*R\varepsilon_*F\longrightarrow R\varepsilon'_*g_{\mathrm{big}}^*F$ is an isomorphism.

Proof. The inverse image $g_{\mathrm{big}}^*$ is exact and preserves adequacy [Tag 06VK]. By Proposition 1.2 (1), applied over $S$ and over $S'$, both sides have the cohomology sheaves $g^*\varepsilon_*H^iF$ and $\varepsilon'_*g_{\mathrm{big}}^*H^iF$: for a complex with $\varepsilon_*$-acyclic cohomology in $D^+$, the spectral sequence of $R\varepsilon_*$ degenerates, and $g^*$ is exact. The base change map is compatible with these identifications, and on each $H^i$ it is the isomorphism of Proposition 1.2 (2). $\square$

3. Perfect complexes

Theorem 3.1. For every interval $[a,b]$, the functor $L\varepsilon^*$ induces an equivalence $\operatorname{Parf}^{[a,b]}(S_{\mathrm{Zar}})\xrightarrow{\ \sim\ }\operatorname{Parf}^{[a,b]}(S_{\mathrm{fppf}}),$ with quasi-inverse $R\varepsilon_*$.

Proof. The functor. On an open $U\subseteq S$ where $E$ is a complex of finite free modules in degrees $[a,b]$, $L\varepsilon^*E$ is the same complex of free $\mathcal O$-modules on $(\mathrm{Sch}/U)_{\mathrm{fppf}}$, so $L\varepsilon^*E$ has perfect amplitude in $[a,b]$; in particular $L\varepsilon^*E\in D^{[a,b]}$.

Full faithfulness. For $E,E'\in\operatorname{Parf}^{[a,b]}(S_{\mathrm{Zar}})$, Corollary 2.2 (1) with $n=a$ gives $\operatorname{Hom}(E,E')\cong$ $\operatorname{Hom}(L\varepsilon^*E,t_{[a}L\varepsilon^*E')=$ $\operatorname{Hom}(L\varepsilon^*E,L\varepsilon^*E')$, and this bijection is induced by $L\varepsilon^*$.

Essential surjectivity. Let $F\in\operatorname{Parf}^{[a,b]}(S_{\mathrm{fppf}})$. Choose an fppf covering $\{g_k\colon S_k\to S\}$ and complexes $P_k$ of finite free $\mathcal O_{S_k}$-modules in degrees $[a,b]$ with $g_{k,\mathrm{big}}^*F\cong\varepsilon_k^*P_k$; a complex of finite free $\mathcal O$-modules on the big site of $S_k$ is $\varepsilon_k^*$ of the complex of free $\mathcal O_{S_k}$-modules given by the same matrices of global functions. The cohomology sheaves of $F$ are adequate: those of $\varepsilon_k^*P_k$ are, and adequacy is fppf local. So Lemma 2.3 and Proposition 2.1 give $g_k^*R\varepsilon_*F\cong R\varepsilon_{k*}\varepsilon_k^*P_k\cong P_k .$ The morphism $\coprod_kS_k\to S$ is flat and surjective, so $R\varepsilon_*F$ is perfect of tor amplitude in $[a,b]$ [Tags 09UD, 09UG], that is, it has perfect amplitude in $[a,b]$. The counit $L\varepsilon^*R\varepsilon_*F\to F$ becomes, after restriction to $S_k$ and the identifications above, the counit $L\varepsilon_k^*R\varepsilon_{k*}\varepsilon_k^*P_k\to\varepsilon_k^*P_k$, which is an isomorphism because the unit $P_k\to R\varepsilon_{k*}\varepsilon_k^*P_k$ is one. An isomorphism is detected on a covering, so the counit is an isomorphism, and $F\cong L\varepsilon^*R\varepsilon_*F$. $\square$

Remark 3.2 (non-flat base change). For an arbitrary morphism $g\colon S'\to S$ and $F\in D^b(S_{\mathrm{fppf}})$ there is a base change map $Lg^*R\varepsilon_*F\to R\varepsilon'_*g_{\mathrm{big}}^*F$. If $F$ has perfect amplitude in $[a,b]$, it is an isomorphism: by Theorem 3.1, $F\cong L\varepsilon^*E$ with $E$ perfect, so the source is $Lg^*E$, and the target is $R\varepsilon'_*L\varepsilon'^*Lg^*E\cong Lg^*E$ by Theorem 3.1 over $S'$, since $Lg^*E$ is perfect of the same amplitude. In general it is not an isomorphism (Exercise 6.2).

4. Diagrams of schemes

Let $I$ be a small category and $X\colon I\to(\mathrm{Sch}/S)$ a diagram of schemes over $S$, with the morphisms $X_\alpha\colon X_i\to X_j$ for the arrows $\alpha\colon i\to j$.

Construction 4.1. The Zariski topoi $(X_i)_{\mathrm{Zar}}$, with the morphisms of topoi defined by the $X_\alpha$, form a topos fibred over $I$ (Definition 1.1 of the lesson on fibred topoi), ringed by the $\mathcal O_{X_i}$. Its total topos is the Zariski topos $\operatorname{Top}(X_{\mathrm{Zar}})$ of $X$, ringed by $\mathcal O$. Representable functors are fppf sheaves [Stacks, Tag 023Q], so by the Yoneda embedding $X$ is a diagram in the topos of sheaves on $(\mathrm{Sch}/S)_{\mathrm{fppf}}$, and it defines a topos fibred over $I$ with levels $\operatorname{Sh}((\mathrm{Sch}/X_i)_{\mathrm{fppf}})$ (Examples 1.2 (c) of that lesson and [Tag 021W]); its total topoi are the fppf topoi $\operatorname{Top}(X_{\mathrm{fppf}})$ and $\operatorname{Top}^{\mathrm o}(X_{\mathrm{fppf}})$ of $X$. The comparison morphisms $\varepsilon_i$ of the $X_i$ are compatible with the transition morphisms, through the isomorphisms $X_{\alpha,\mathrm{big}}^*\varepsilon_j^*\cong\varepsilon_i^*X_\alpha^*$ [Tag 03FH], so they form a morphism of fibred topoi over the identity of $I$, and a morphism of ringed topoi $\varepsilon\colon\operatorname{Top}(X_{\mathrm{fppf}})\longrightarrow\operatorname{Top}(X_{\mathrm{Zar}})$ (Proposition 2.1 of the lesson on morphisms of fibred topoi).

A module $M$ on $\operatorname{Top}(X_{\mathrm{Zar}})$ is a family of $\mathcal O_{X_i}$-modules $M_i$ with maps $M_\alpha\colon X_\alpha^*M_j\to M_i$ (Proposition 4.2 of the lesson on morphisms of fibred topoi). It is termwise quasi-coherent if every $M_i$ is quasi-coherent, and cartesian if every $M_\alpha$ is an isomorphism; the same terms are used on the fppf side. $D_{\mathrm{qc}}(X_{\mathrm{Zar}})$ is the full subcategory of $D(X_{\mathrm{Zar}})$ of objects with termwise quasi-coherent cohomology. An object $E$ of $D^-(X_{\mathrm{Zar}})$ or of $D^-(X_{\mathrm{fppf}})$ is quasi-cartesian if for every arrow $\alpha$ the map $LX_\alpha^*E_j\to E_i$ is an isomorphism.

Lemma 4.2.

For $E\in D^-(X_{\mathrm{Zar}})$ and every $i$, $(L\varepsilon^*E)_i\cong L\varepsilon_i^*(E_i)$, and for $G\in D^+(X_{\mathrm{fppf}})$, $(R\varepsilon_*G)_i\cong R\varepsilon_{i*}(G_i)$.
Proposition 2.1 and Corollary 2.2 hold with $S$ replaced by $X$: for $M\in D^b_{\mathrm{qc}}(X_{\mathrm{Zar}})$ with $H^iM=0$ for $i<a$ and $n\leq a$, $M\cong R\varepsilon_*t_{[n}L\varepsilon^*M$, and for $E\in D^-(X_{\mathrm{Zar}})$ the corresponding isomorphisms of $\operatorname{RHom}$ and of $\varprojlim_n\operatorname{Hom}$ hold.
$L\varepsilon^*$ carries quasi-cartesian objects to quasi-cartesian objects. An object $E$ of $D^-(X_{\mathrm{fppf}})$ is quasi-cartesian if and only if every cohomology module $H^nE$ is cartesian.

Proof. (1) The restriction to the level $i$ has the left adjoint $i_!$ with $i_!(L)_j=\bigoplus_{\beta\colon j\to i}X_\beta^*L$, and tensor products are computed levelwise (Proposition 4.2 (b), (c) of the lesson on morphisms of fibred topoi). A free module $\mathcal O_{X_i}^{(U)}$ on an object $U$ of $(X_i)_{\mathrm{Zar}}$ has flat inverse images, so $i_!\mathcal O_{X_i}^{(U)}$ has flat components; every module on $\operatorname{Top}(X_{\mathrm{Zar}})$ is a quotient of a sum of such modules, since maps out of $i_!\mathcal O_{X_i}^{(U)}$ are sections over $U$ at the level $i$. So every $E\in D^-(X_{\mathrm{Zar}})$ is represented by a bounded above complex $P$ of modules with flat components [Stacks, Tag 05T7]. A module with flat components is flat, tensor products being levelwise, so $P$ is K-flat [Tag 06YQ] and $L\varepsilon^*E=\varepsilon^*P$ [Tag 06YY]. The inverse image is computed levelwise, so $(\varepsilon^*P)_i=\varepsilon_i^*P_i$, and $P_i$ is a bounded above complex of flat modules, which computes $L\varepsilon_i^*E_i$. The second assertion is Proposition 4.1 and Remark 4.2 of the lesson on cohomology of total topoi.

(2) By (1), the map $M\to R\varepsilon_*t_{[n}L\varepsilon^*M$ is, at the level $i$, the map of Proposition 2.1 for $M_i$ on $X_i$, truncations being levelwise; an isomorphism of modules on the total topos is detected levelwise. The rest follows as in Corollary 2.2, with the adjunction [Tag 07A6] for the ringed total topoi (topoi by Theorem 4.2 of the lesson on fibred topoi).

(3) For an arrow $\alpha\colon i\to j$, $LX_\alpha^*$ and $L\varepsilon^*$ commute up to isomorphism, by the isomorphism $X_{\alpha,\mathrm{big}}^*\varepsilon_j^*\cong\varepsilon_i^*X_\alpha^*$ applied to complexes of flat modules; the map $LX_\alpha^*(L\varepsilon^*E)_j\to(L\varepsilon^*E)_i$ is $L\varepsilon_i^*$ of $LX_\alpha^*E_j\to E_i$, by (1). On the fppf side $X_{\alpha,\mathrm{big}}^*$ is the restriction to the localization, an exact functor, so $LX_{\alpha,\mathrm{big}}^*=X_{\alpha,\mathrm{big}}^*$ commutes with cohomology sheaves, and $X_{\alpha,\mathrm{big}}^*E_j\to E_i$ is an isomorphism if and only if it is on every $H^n$. $\square$

On the Zariski side the second statement of (3) fails: a quasi-cartesian object can have cohomology modules that are not cartesian (Exercise 6.3).

5. The big fpqc topos

The coverings of the faithfully flat quasi-compact topology cannot be bounded by a set [Stacks, Section 022A], so the big fpqc site is formed in a universe: a set $\mathbf U$ that contains the elements of its elements, the pairs and the power sets of its elements, the unions of families of its elements indexed by elements of $\mathbf U$, and the set of natural numbers.

5.1. The site. Let $\mathbf U\in\mathbf V$ be universes with $S\in\mathbf U$. Let $(\mathrm{Sch}/S)_{\mathrm{fpqc}}$ be the category of schemes over $S$ that are elements of $\mathbf U$, with the fpqc coverings $\{T_k\to T\}_{k\in K}$ [Tag 022B] with $K\in\mathbf U$. Fibre products can be formed in $\mathbf U$, and these coverings satisfy the axioms of a site [Tag 022D]; the category is an element of $\mathbf V$. The big fpqc topos of $S$ is the category of sheaves of sets on it that are elements of $\mathbf V$, ringed by $\mathcal O\colon T\mapsto\Gamma(T,\mathcal O_T)$, which is a sheaf by [Tag 03DT]. Open coverings are fpqc coverings (open immersions are flat, and an affine open of $T$ is covered by finitely many affine opens contained in the members), so the inclusion of the small Zariski site $S_{\mathrm{Zar}}$ is continuous; it commutes with finite limits, and defines a morphism of ringed sites $\varepsilon\colon((\mathrm{Sch}/S)_{\mathrm{fpqc}},\mathcal O)\to(S_{\mathrm{Zar}},\mathcal O_S)$ with $\varepsilon_*F$ the restriction of $F$ to the open subschemes [Tag 00X6]. For $g\colon S'\to S$ with $S'\in\mathbf U$, $g_{\mathrm{big}}$ is the localization at $S'$ [Tag 03CF], whose inverse image is the restriction to schemes over $S'$. For an $\mathcal O_S$-module $M$, $W(M)$ denotes the presheaf $(f\colon T\to S)\mapsto\Gamma(T,f^*M)$. Quasi-coherent $\mathcal O$-modules are those that are locally cokernels of maps of free modules [Tag 03DL], and an $\mathcal O$-module is adequate if $S$ has an affine open covering $S=\bigcup S_k$ with maps $G_k\to H_k$ of quasi-coherent modules such that $F|_{(\mathrm{Sch}/S_k)_{\mathrm{fpqc}}}$ is the kernel of $W(G_k)\to W(H_k)$, as in Section 1.

Lemma 5.2. On the big fpqc site of $S$:

Let $T=\operatorname{Spec}A$ be affine. Every fpqc covering of $T$ is refined by a standard one, a finite family $\{\operatorname{Spec}A_j\to T\}$ of affine schemes, each mapping into a member, with $A\to\prod_jA_j$ faithfully flat [Tag 023S]. If $C$ is an adequate functor on $A$-algebras [Tag 06UT] and $A\to B$ is faithfully flat, the extended Čech complex $0\to C(A)\to C(B)\to$ $C(B\otimes_AB)\to\cdots$ is exact.
For a quasi-coherent $\mathcal O_S$-module $M$, $W(M)$ is a sheaf, $\varepsilon^*M=W(M)$ and $\varepsilon_*W(M)=M$. The functor $W$ is an equivalence from quasi-coherent $\mathcal O_S$-modules to quasi-coherent $\mathcal O$-modules.
For an adequate $F$ and an affine $T=\operatorname{Spec}A$ over $S$, the functor $F_A\colon B\mapsto F(\operatorname{Spec}B)$ on $A$-algebras is adequate.
An $\mathcal O$-module that is adequate on the members of an fpqc covering of $S$ is adequate. Kernels, cokernels and extensions of adequate modules are adequate, and so is $W(M)$ for quasi-coherent $M$.
For an adequate $F$, $H^q(T,F)=0$ for every affine $T$ over $S$ and $q>0$; $\varepsilon_*F$ is quasi-coherent, $R^q\varepsilon_*F=0$ for $q>0$, and for flat $g\colon S'\to S$ the base change map $g^*\varepsilon_*F\to\varepsilon'_*g_{\mathrm{big}}^*F$ is an isomorphism.

Proof. (1) The first assertion is [Tag 023S]. For the second, $A\to B^{\otimes(n+1)}$ is flat, so $C(B^{\otimes(n+1)})=C(A)\otimes_AB^{\otimes(n+1)}$ [Tag 06UW], and the extended Čech complex is $C(A)\otimes_A$ the extended Amitsur complex of $A\to B$, which is exact [Tag 023M].

(2) $W(M)$ satisfies the sheaf condition for fpqc coverings [Tag 03DT]. On open subschemes it is $M$, so $\varepsilon_*W(M)=M$. For an $\mathcal O$-module $G$, restricting a map $W(M)\to G$ to the open subschemes gives a map $\operatorname{Hom}(W(M),G)\to\operatorname{Hom}(M,\varepsilon_*G)$, and it is bijective. Indeed, let $\varphi\colon M\to\varepsilon_*G$. For an affine $T=\operatorname{Spec}B$ whose map $f$ to $S$ factors through an affine open $U=\operatorname{Spec}R$, $\Gamma(T,f^*M)=M(U)\otimes_RB$, and $m\otimes b\mapsto b\,\varphi(m)|_T$ is the only $\mathcal O$-linear map to $G(T)$ extending $\varphi$. For two such opens these maps agree locally on $T$, on affine opens of $T$ mapping into affine opens contained in both, hence agree, $G$ being a sheaf; they are compatible with restriction, and the affine schemes of this kind cover every object, so they glue uniquely to a map $W(M)\to G$. Hence $W(M)$ has the universal property of $\varepsilon^*M$, the left adjoint of $\varepsilon_*$ on modules, and $\varepsilon^*M=W(M)$; by adjunction $\operatorname{Hom}(W(M),W(N))=\operatorname{Hom}(M,N)$, so $W$ is fully faithful. A quasi-coherent $\mathcal O$-module $F$ is, on the members of an fpqc covering $\{S_i\to S\}$, the cokernel of a map of free modules, that is, of $W$ of a map of free modules, and $W$ preserves cokernels, as it does on affine schemes; so $F|_{S_i}\cong W(M_i)$ with $M_i$ quasi-coherent. By full faithfulness over the $S_i\times_SS_j$, the restrictions of $F$ give isomorphisms between the pullbacks of the $M_i$ satisfying the cocycle condition, and fpqc descent [Tag 023T] gives a quasi-coherent $M$ with $W(M)\cong F$, both being sheaves isomorphic on the covering compatibly with the gluing.

(3) Cover $T$ by finitely many affine opens $T_j=\operatorname{Spec}A_j$, each mapping into some $S_k=\operatorname{Spec}R_k$ of the definition. On $R_k$-algebras $F$ is the kernel of $G_k\otimes_{R_k}-\to H_k\otimes_{R_k}-$, an adequate functor, so $F_{A_j}$ is adequate [Tag 06V9], and so is $F_{A'}$ for $A'=\prod_jA_j$ [Tag 06VB], $F$ being a Zariski sheaf. For every $A$-algebra $B$, $F(B)\to F(B\otimes_AA')\to$ $F(B\otimes_AA'\otimes_AA')$ is exact by the sheaf property for the open covering of $\operatorname{Spec}B$ by the $\operatorname{Spec}(B\otimes_AA_j)$. So $F_A$ is adequate [Tag 06VH].

(4) Let $F$ be adequate on the members of an fpqc covering $\{S_i\to S\}$, and $T=\operatorname{Spec}A$ an affine open of $S$. Refine the covering of $T$ by a standard one $\{\operatorname{Spec}A_j\to T\}$ (1), with each $\operatorname{Spec}A_j$ mapping into some $S_i$. By (3) for $F|_{S_i}$, each $F_{A_j}$ is adequate, hence so is $F_{A'}$, $A'=\prod_jA_j$ [Tag 06VB], and the sequence of [Tag 06VH] (2) is exact by the sheaf property for the fpqc covering $\{\operatorname{Spec}(B\otimes_AA_j)\to\operatorname{Spec}B\}$. So $F_A$ is the kernel of $\underline M\to\underline N$ for a map of $A$-modules $M\to N$ [Tags 06VH, 06UT], and $F|_T$ is the kernel of $W(\tilde M)\to W(\tilde N)$, both sides being sheaves that agree on affine schemes over $T$, which cover every object over $T$. For a map $F\to G$ of adequate modules, the kernels and cokernels $K_A$, $C_A$ of the adequate functors $F_A\to G_A$ are adequate [Tags 06V3, 06V2]. The kernel of $F\to G$ has the values $K_A$ on affine schemes. By (1), the $C_A$ satisfy the sheaf condition for standard coverings, in particular for finite coverings by standard open subsets, so the Zariski sheafification $C$ of the presheaf cokernel has the values $C_A$ on affine schemes and satisfies the sheaf condition for Zariski coverings and for standard fpqc coverings; hence it is an fpqc sheaf [Tag 022H], and it is the cokernel of $F\to G$. Both are adequate by the argument just given. Extensions are treated in the same way with [Tag 06V8], and $W(M)$ is the kernel of $W(M)\to W(0)$.

(5) Apply [Tag 03F9] to the affine schemes over $S$ and their standard coverings, which are stable under fibre products and refine all coverings of affine schemes: the Čech complexes of $F$ for these coverings are the extended Čech complexes of (1), with the term in degree $-1$ removed, because $F(\operatorname{Spec}\prod_jA_j)=\prod_jF(\operatorname{Spec}A_j)$; they have no cohomology in positive degrees, so $H^q(T,F)=0$ for affine $T$ and $q>0$. Then $R^q\varepsilon_*F$, the sheaf associated with $U\mapsto H^q(U,F)$ on the open subschemes of $S$ [Tag 0D6I], vanishes for $q>0$, affine opens forming a basis. On an affine open $S_k$ of the definition, $\varepsilon_*F$ is the kernel of $G_k\to H_k$, hence quasi-coherent; base change along a flat $g$ commutes with this kernel, which gives the last assertion as in Proposition 1.2 (2). $\square$

Proposition 5.3. Propositions 1.1, 1.2 and 2.1, Corollary 2.2, Lemma 2.3, Theorem 3.1, Remark 3.2, Construction 4.1 and Lemma 4.2 hold with the big fpqc site of 5.1 in place of the big fppf site, and fpqc coverings in place of fppf coverings.

Proof. The proofs of these statements use the following facts about the big fppf site, and Lemma 5.2 supplies them for the big fpqc site: the sheaf property of $W(M)$, $\varepsilon^*M=W(M)$ and the equivalence of quasi-coherent modules [Tags 03DT, 070S, 03DX], by Lemma 5.2 (2); the vanishing of the higher cohomology of quasi-coherent modules on affine schemes [Tag 03DW], by Lemma 5.2 (5) for $F=W(M)$; the properties of adequate modules [Tags 06VK, 06VM, 06VQ], by Lemma 5.2 (3)–(5); and the localization $g_{\mathrm{big}}$ with $g_{\mathrm{big}}^*W(M)=W(g^*M)$ [Tags 021W, 03FH], by 5.1 and the definition of $W$. Representable functors are sheaves for the fpqc topology [Tag 023Q], which gives Construction 4.1. The other inputs, on complexes of modules on ringed sites and on perfect complexes on schemes [Tags 05T7, 06YQ, 06YY, 07A6, 0D6I, 06Z0, 01XB, 0658, 09UD, 09UG], hold on every ringed site, respectively concern fpqc descent on schemes. With these replacements the proofs apply word for word. $\square$

6. Exercises

6.1. Let $S=\operatorname{Spec}\mathbb Z$, $p$ a prime, and $K$ the kernel of multiplication by $p$ on $\mathcal O$ on the big site. Show that $K(T)=0$ for every flat $\mathbb Z$-scheme $T$ and $K(T)=\Gamma(T,\mathcal O_T)$ for every $\mathbb F_p$-scheme $T$. Deduce that $K$ is adequate but not quasi-coherent, and that $L\varepsilon^*(\mathcal O_S/p)$ has the nonzero cohomology sheaf $K$ in degree $-1$, although $\mathcal O_S/p$ is a quasi-coherent module in degree $0$.

6.2. Let $S=\operatorname{Spec}\mathbb Z$, $S'=\operatorname{Spec}\mathbb F_p$ with $g\colon S'\to S$, and $F=\varepsilon^*(\mathcal O_S/p)$, a quasi-coherent module in degree $0$. Compute both sides of the base change map of Remark 3.2 and show that it is not an isomorphism.

6.3. Let $I=\{0\to1\}$ and $X$ the diagram $X_\alpha\colon\operatorname{Spec}\mathbb F_p\to\operatorname{Spec}\mathbb Z$. Let $P$ be the complex $\mathcal O\xrightarrow{p}\mathcal O$ on $\operatorname{Spec}\mathbb Z$, in degrees $-1$ and $0$, and $E$ the complex of modules on $\operatorname{Top}(X_{\mathrm{Zar}})$ with $E_1=P$, $E_0=X_\alpha^*P$ and the identity as structure map. Show that $E$ is quasi-cartesian but that $H^{-1}E$ is not cartesian.

7. Complete solutions

6.1. For an affine $T=\operatorname{Spec}B$, $K(T)=\{x\in B:px=0\}$. If $B$ is flat over $\mathbb Z$ it is torsion free, so $K(T)=0$; if $pB=0$, $K(T)=B$. Both properties pass to non-affine $T$ by the sheaf property. $K$ is the kernel of $p\colon\mathcal O_S^a\to\mathcal O_S^a$, hence adequate. If $K$ were quasi-coherent, $K=N^a$ with $N=\varepsilon_*K$ (Proposition 1.1 (2)), and $N(S)=K(\operatorname{Spec}\mathbb Z)=0$, so $N=0$ and $K=0$; but $K(\operatorname{Spec}\mathbb F_p)=\mathbb F_p$. The complex $\mathbb Z\xrightarrow{p}\mathbb Z$ in degrees $-1,0$ is a flat resolution of $\mathcal O_S/p$, so $L\varepsilon^*(\mathcal O_S/p)$ is $\mathcal O\xrightarrow{p}\mathcal O$, whose $H^{-1}$ is $K$.

6.2. By Proposition 1.1, $R\varepsilon_*F=\mathcal O_S/p$, so the source is $Lg^*(\mathbb Z/p)=\mathbb Z/p\otimes^{\mathbf L}_{\mathbb Z}\mathbb F_p$, computed by $\mathbb F_p\xrightarrow{0}\mathbb F_p$: it has $\mathbb F_p$ in degrees $-1$ and $0$. The target is $R\varepsilon'_*g_{\mathrm{big}}^*F$, and $g_{\mathrm{big}}^*F=(g^*(\mathcal O_S/p))^a=\mathcal O_{S'}^a$, whose $R\varepsilon'_*$ is $\mathbb F_p$ in degree $0$ (Proposition 1.1 (3)). The map cannot be an isomorphism, since the source has $H^{-1}\neq0$.

6.3. $P$ is a complex of flat modules, so $LX_\alpha^*E_1=X_\alpha^*P=E_0$ and $E$ is quasi-cartesian. $H^{-1}E_1=0$, since multiplication by $p$ is injective on $\mathbb Z$, while $E_0$ is $\mathbb F_p\xrightarrow{0}\mathbb F_p$, with $H^{-1}E_0=\mathbb F_p$. The map $X_\alpha^*H^{-1}E_1=0\to H^{-1}E_0=\mathbb F_p$ is not an isomorphism, so $H^{-1}E$ is not cartesian.

Sources

The comparison of the Zariski and the big flat topoi of a scheme for complexes with quasi-coherent cohomology and for perfect complexes, and its use for diagrams of schemes, are due to Illusie, in his work on the cotangent complex. Adequate modules and their properties are taken from the Stacks project, whose treatment of the fpqc topology and of faithfully flat descent gives Section 5.