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\begin{document}
\title{Descent}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
In the chapter on topologies on schemes
(see Topologies, Section \ref{topologies-section-introduction}) we introduced
Zariski, \'etale, fppf, smooth, syntomic and fpqc coverings of schemes.
In this chapter we discuss what kind of structures over schemes
can be descended through such coverings.
See for example \cite{Gr-I}, \cite{Gr-II}, \cite{Gr-III},
\cite{Gr-IV}, \cite{Gr-V}, and \cite{Gr-VI}.
This is also meant to introduce the notions of
descent, descent data, effective descent data, in the less formal
setting of descent questions for quasi-coherent sheaves, schemes, etc.
The formal notion, that of a stack over a site, is discussed in
the chapter on stacks (see Stacks, Section \ref{stacks-section-introduction}).
\section{Descent data for quasi-coherent sheaves}
\label{section-equivalence}
\noindent
In this chapter we will use the convention where
the projection maps $\text{pr}_i : X \times \ldots \times X \to X$
are labeled starting with $i = 0$. Hence we have
$\text{pr}_0, \text{pr}_1 : X \times X \to X$,
$\text{pr}_0, \text{pr}_1, \text{pr}_2 : X \times X \times X \to X$,
etc.
\begin{definition}
\label{definition-descent-datum-quasi-coherent}
Let $S$ be a scheme. Let $\{f_i : S_i \to S\}_{i \in I}$ be a family
of morphisms with target $S$.
\begin{enumerate}
\item A {\it descent datum $(\mathcal{F}_i, \varphi_{ij})$
for quasi-coherent sheaves} with respect to the given family
is given by a quasi-coherent sheaf $\mathcal{F}_i$ on $S_i$ for
each $i \in I$, an isomorphism of quasi-coherent
$\mathcal{O}_{S_i \times_S S_j}$-modules
$\varphi_{ij} : \text{pr}_0^*\mathcal{F}_i \to \text{pr}_1^*\mathcal{F}_j$
for each pair $(i, j) \in I^2$
such that for every triple of indices $(i, j, k) \in I^3$ the
diagram
$$
\xymatrix{
\text{pr}_0^*\mathcal{F}_i \ar[rd]_{\text{pr}_{01}^*\varphi_{ij}}
\ar[rr]_{\text{pr}_{02}^*\varphi_{ik}} & &
\text{pr}_2^*\mathcal{F}_k \\
& \text{pr}_1^*\mathcal{F}_j \ar[ru]_{\text{pr}_{12}^*\varphi_{jk}} &
}
$$
of $\mathcal{O}_{S_i \times_S S_j \times_S S_k}$-modules
commutes. This is called the {\it cocycle condition}.
\item A {\it morphism $\psi : (\mathcal{F}_i, \varphi_{ij}) \to
(\mathcal{F}'_i, \varphi'_{ij})$ of descent data} is given
by a family $\psi = (\psi_i)_{i\in I}$ of morphisms of
$\mathcal{O}_{S_i}$-modules $\psi_i : \mathcal{F}_i \to \mathcal{F}'_i$
such that all the diagrams
$$
\xymatrix{
\text{pr}_0^*\mathcal{F}_i \ar[r]_{\varphi_{ij}} \ar[d]_{\text{pr}_0^*\psi_i}
& \text{pr}_1^*\mathcal{F}_j \ar[d]^{\text{pr}_1^*\psi_j} \\
\text{pr}_0^*\mathcal{F}'_i \ar[r]^{\varphi'_{ij}} &
\text{pr}_1^*\mathcal{F}'_j \\
}
$$
commute.
\end{enumerate}
\end{definition}
\noindent
A good example to keep in mind is the following.
Suppose that $S = \bigcup S_i$ is an open covering.
In that case we have seen descent data for sheaves of sets in
Sheaves, Section \ref{sheaves-section-glueing-sheaves}
where we called them ``glueing data for sheaves of sets
with respect to the given covering''. Moreover, we proved
that the category of glueing data is equivalent to the category
of sheaves on $S$. We will show the analogue in the setting above when
$\{S_i \to S\}_{i\in I}$ is an fpqc covering.
\medskip\noindent
In the extreme case where the covering $\{S \to S\}$
is given by $\text{id}_S$ a descent datum is necessarily
of the form $(\mathcal{F}, \text{id}_\mathcal{F})$. The cocycle
condition guarantees that the identity on $\mathcal{F}$ is the
only permitted map in this case. The following lemma shows
in particular that to every quasi-coherent sheaf of
$\mathcal{O}_S$-modules there is associated a unique
descent datum with respect to any given family.
\begin{lemma}
\label{lemma-refine-descent-datum}
Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ and
$\mathcal{V} = \{V_j \to V\}_{j \in J}$
be families of morphisms of schemes with fixed target.
Let $(g, \alpha : I \to J, (g_i)) : \mathcal{U} \to \mathcal{V}$
be a morphism of families of maps with fixed target, see
Sites, Definition \ref{sites-definition-morphism-coverings}.
Let $(\mathcal{F}_j, \varphi_{jj'})$ be a descent
datum for quasi-coherent sheaves with respect to the
family $\{V_j \to V\}_{j \in J}$. Then
\begin{enumerate}
\item The system
$$
\left(g_i^*\mathcal{F}_{\alpha(i)},
(g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')}\right)
$$
is a descent datum with respect to the family $\{U_i \to U\}_{i \in I}$.
\item This construction is functorial in the descent datum
$(\mathcal{F}_j, \varphi_{jj'})$.
\item Given a second morphism $(g', \alpha' : I \to J, (g'_i))$
of families of maps with fixed target with $g = g'$
there exists a functorial isomorphism of descent data
$$
(g_i^*\mathcal{F}_{\alpha(i)},
(g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')})
\cong
((g'_i)^*\mathcal{F}_{\alpha'(i)},
(g'_i \times g'_{i'})^*\varphi_{\alpha'(i)\alpha'(i')}).
$$
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted. Hint: The maps
$g_i^*\mathcal{F}_{\alpha(i)} \to (g'_i)^*\mathcal{F}_{\alpha'(i)}$
which give the isomorphism of descent data in part (3)
are the pullbacks of the maps $\varphi_{\alpha(i)\alpha'(i)}$ by the
morphisms $(g_i, g'_i) : U_i \to V_{\alpha(i)} \times_V V_{\alpha'(i)}$.
\end{proof}
\noindent
Any family $\mathcal{U} = \{S_i \to S\}_{i \in I}$ is a refinement of
the trivial covering $\{S \to S\}$ in a unique way. For
a quasi-coherent sheaf $\mathcal{F}$ on $S$ we denote simply
$(\mathcal{F}|_{S_i}, can)$ the descent datum with respect to
$\mathcal{U}$ obtained by the procedure above.
\begin{definition}
\label{definition-descent-datum-effective-quasi-coherent}
Let $S$ be a scheme.
Let $\{S_i \to S\}_{i \in I}$ be a family of morphisms
with target $S$.
\begin{enumerate}
\item Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module.
We call the unique descent on $\mathcal{F}$ datum with respect to the covering
$\{S \to S\}$ the {\it trivial descent datum}.
\item The pullback of the trivial descent datum to
$\{S_i \to S\}$ is called the {\it canonical descent datum}.
Notation: $(\mathcal{F}|_{S_i}, can)$.
\item A descent datum $(\mathcal{F}_i, \varphi_{ij})$
for quasi-coherent sheaves with respect to the given covering
is said to be {\it effective} if there exists a quasi-coherent
sheaf $\mathcal{F}$ on $S$ such that $(\mathcal{F}_i, \varphi_{ij})$
is isomorphic to $(\mathcal{F}|_{S_i}, can)$.
\end{enumerate}
\end{definition}
\begin{lemma}
\label{lemma-zariski-descent-effective}
Let $S$ be a scheme.
Let $S = \bigcup U_i$ be an open covering.
Any descent datum on quasi-coherent sheaves
for the family $\mathcal{U} = \{U_i \to S\}$ is
effective. Moreover, the functor from the category of
quasi-coherent $\mathcal{O}_S$-modules to the category
of descent data with respect to $\mathcal{U}$ is fully faithful.
\end{lemma}
\begin{proof}
This follows immediately from
Sheaves, Section \ref{sheaves-section-glueing-sheaves}
and the fact that being quasi-coherent is a local property, see
Modules, Definition \ref{modules-definition-quasi-coherent}.
\end{proof}
\noindent
To prove more we first need to study the case of modules over rings.
\section{Descent for modules}
\label{section-descent-modules}
\noindent
Let $R \to A$ be a ring map.
By Simplicial, Example \ref{simplicial-example-push-outs-simplicial-object}
this gives rise to a cosimplicial $R$-algebra
$$
\xymatrix{
A
\ar@<1ex>[r]
\ar@<-1ex>[r]
&
A \otimes_R A
\ar@<0ex>[l]
\ar@<2ex>[r]
\ar@<0ex>[r]
\ar@<-2ex>[r]
&
A \otimes_R A \otimes_R A
\ar@<1ex>[l]
\ar@<-1ex>[l]
}
$$
Let us denote this $(A/R)_\bullet$ so that $(A/R)_n$ is the $(n + 1)$-fold
tensor product of $A$ over $R$. Given a map
$\varphi : [n] \to [m]$ the $R$-algebra map $(A/R)_\bullet(\varphi)$
is the map
$$
a_0 \otimes \ldots \otimes a_n
\longmapsto
\prod\nolimits_{\varphi(i) = 0} a_i
\otimes
\prod\nolimits_{\varphi(i) = 1} a_i
\otimes \ldots \otimes
\prod\nolimits_{\varphi(i) = m} a_i
$$
where we use the convention that the empty product is $1$. Thus the first
few maps, notation as in
Simplicial, Section \ref{simplicial-section-cosimplicial-object}, are
$$
\begin{matrix}
\delta^1_0 & : & a_0 & \mapsto & 1 \otimes a_0 \\
\delta^1_1 & : & a_0 & \mapsto & a_0 \otimes 1 \\
\sigma^0_0 & : & a_0 \otimes a_1 & \mapsto & a_0a_1 \\
\delta^2_0 & : & a_0 \otimes a_1 & \mapsto & 1 \otimes a_0 \otimes a_1 \\
\delta^2_1 & : & a_0 \otimes a_1 & \mapsto & a_0 \otimes 1 \otimes a_1 \\
\delta^2_2 & : & a_0 \otimes a_1 & \mapsto & a_0 \otimes a_1 \otimes 1 \\
\sigma^1_0 & : & a_0 \otimes a_1 \otimes a_2 & \mapsto & a_0a_1 \otimes a_2 \\
\sigma^1_1 & : & a_0 \otimes a_1 \otimes a_2 & \mapsto & a_0 \otimes a_1a_2
\end{matrix}
$$
and so on.
\medskip\noindent
An $R$-module $M$ gives rise to a cosimplicial $(A/R)_\bullet$-module
$(A/R)_\bullet \otimes_R M$. In other words
$M_n = (A/R)_n \otimes_R M$ and using the $R$-algebra maps
$(A/R)_n \to (A/R)_m$ to define the corresponding maps on
$M \otimes_R (A/R)_\bullet$.
\medskip\noindent
The analogue to a descent datum
for quasi-coherent sheaves in the setting of modules is the following.
\begin{definition}
\label{definition-descent-datum-modules}
Let $R \to A$ be a ring map.
\begin{enumerate}
\item A {\it descent datum $(N, \varphi)$ for modules
with respect to $R \to A$}
is given by an $A$-module $N$ and a isomorphism of
$A \otimes_R A$-modules
$$
\varphi : N \otimes_R A \to A \otimes_R N
$$
such that the {\it cocycle condition} holds: the diagram
of $A \otimes_R A \otimes_R A$-module maps
$$
\xymatrix{
N \otimes_R A \otimes_R A \ar[rr]_{\varphi_{02}}
\ar[rd]_{\varphi_{01}}
& &
A \otimes_R A \otimes_R N \\
& A \otimes_R N \otimes_R A \ar[ru]_{\varphi_{12}} &
}
$$
commutes (see below for notation).
\item A {\it morphism $(N, \varphi) \to (N', \varphi')$ of descent data}
is a morphism of $A$-modules $\psi : N \to N'$ such that
the diagram
$$
\xymatrix{
N \otimes_R A \ar[r]_\varphi \ar[d]_{\psi \otimes \text{id}_A} &
A \otimes_R N \ar[d]^{\text{id}_A \otimes \psi} \\
N' \otimes_R A \ar[r]^{\varphi'} &
A \otimes_R N'
}
$$
is commutative.
\end{enumerate}
\end{definition}
\noindent
In the definition we use the notation that
$\varphi_{01} = \varphi \otimes \text{id}_A$,
$\varphi_{12} = \text{id}_A \otimes \varphi$, and
$\varphi_{02}(n \otimes 1 \otimes 1) = \sum a_i \otimes 1 \otimes n_i$
if $\varphi(n \otimes 1) = \sum a_i \otimes n_i$. All three are
$A \otimes_R A \otimes_R A$-module homomorphisms. Equivalently we have
$$
\varphi_{ij}
=
\varphi \otimes_{(A/R)_1, \ (A/R)_\bullet(\tau^2_{ij})} (A/R)_2
$$
where $\tau^2_{ij} : [1] \to [2]$ is the map
$0 \mapsto i$, $1 \mapsto j$. Namely,
$(A/R)_{\bullet}(\tau^2_{02})(a_0 \otimes a_1) =
a_0 \otimes 1 \otimes a_1$,
and similarly for the others\footnote{Note that
$\tau^2_{ij} = \delta^2_k$, if $\{i, j, k\} = [2] = \{0, 1, 2\}$,
see Simplicial, Definition \ref{simplicial-definition-face-degeneracy}.}.
\medskip\noindent
We need some more notation to be able to state the next lemma.
Let $(N, \varphi)$ be a descent datum with respect to a ring map $R \to A$.
For $n \geq 0$ and $i \in [n]$ we set
$$
N_{n, i} =
A \otimes_R
\ldots
\otimes_R A \otimes_R N \otimes_R A \otimes_R
\ldots
\otimes_R A
$$
with the factor $N$ in the $i$th spot. It is an $(A/R)_n$-module.
If we introduce the maps $\tau^n_i : [0] \to [n]$, $0 \mapsto i$
then we see that
$$
N_{n, i} = N \otimes_{(A/R)_0, \ (A/R)_\bullet(\tau^n_i)} (A/R)_n
$$
For $0 \leq i \leq j \leq n$ we let $\tau^n_{ij} : [1] \to [n]$
be the map such that $0$ maps to $i$ and $1$ to $j$. Similarly
to the above the homomorphism $\varphi$ induces isomorphisms
$$
\varphi^n_{ij}
=
\varphi \otimes_{(A/R)_1, \ (A/R)_\bullet(\tau^n_{ij})} (A/R)_n :
N_{n, i} \longrightarrow N_{n, j}
$$
of $(A/R)_n$-modules when $i < j$. If $i = j$ we set
$\varphi^n_{ij} = \text{id}$. Since these are all isomorphisms they allow us
to move the factor $N$ to any spot we like. And the cocycle condition
exactly means that it does not matter how we do this (e.g., as a composition
of two of these or at once). Finally, for any $\beta : [n] \to [m]$
we define the morphism
$$
N_{\beta, i} : N_{n, i} \to N_{m, \beta(i)}
$$
as the unique $(A/R)_\bullet(\beta)$-semi linear map such that
$$
N_{\beta, i}(1 \otimes \ldots \otimes n \otimes \ldots \otimes 1)
=
1 \otimes \ldots \otimes n \otimes \ldots \otimes 1
$$
for all $n \in N$.
This hints at the following lemma.
\begin{lemma}
\label{lemma-descent-datum-cosimplicial}
Let $R \to A$ be a ring map.
Given a descent datum $(N, \varphi)$ we can associate to it a
cosimplicial $(A/R)_\bullet$-module $N_\bullet$\footnote{We should really
write $(N, \varphi)_\bullet$.} by the
rules $N_n = N_{n, n}$ and given $\beta : [n] \to [m]$
setting we define
$$
N_\bullet(\beta) = (\varphi^m_{\beta(n)m}) \circ N_{\beta, n} :
N_{n, n} \longrightarrow N_{m, m}.
$$
This procedure is functorial in the descent datum.
\end{lemma}
\begin{proof}
Here are the first few maps
where $\varphi(n \otimes 1) = \sum \alpha_i \otimes x_i$
$$
\begin{matrix}
\delta^1_0 & : & N & \to & A \otimes N & n & \mapsto & 1 \otimes n \\
\delta^1_1 & : & N & \to & A \otimes N & n & \mapsto &
\sum \alpha_i \otimes x_i\\
\sigma^0_0 & : & A \otimes N & \to & N & a_0 \otimes n & \mapsto & a_0n \\
\delta^2_0 & : & A \otimes N & \to & A \otimes A \otimes N &
a_0 \otimes n & \mapsto & 1 \otimes a_0 \otimes n \\
\delta^2_1 & : & A \otimes N & \to & A \otimes A \otimes N &
a_0 \otimes n & \mapsto & a_0 \otimes 1 \otimes n \\
\delta^2_2 & : & A \otimes N & \to & A \otimes A \otimes N &
a_0 \otimes n & \mapsto & \sum a_0 \otimes \alpha_i \otimes x_i \\
\sigma^1_0 & : & A \otimes A \otimes N & \to & A \otimes N &
a_0 \otimes a_1 \otimes n & \mapsto & a_0a_1 \otimes n \\
\sigma^1_1 & : & A \otimes A \otimes N & \to & A \otimes N &
a_0 \otimes a_1 \otimes n & \mapsto & a_0 \otimes a_1n
\end{matrix}
$$
with notation as in
Simplicial, Section \ref{simplicial-section-cosimplicial-object}.
We first verify the two properties $\sigma^0_0 \circ \delta^1_0 = \text{id}$
and $\sigma^0_0 \circ \delta^1_1 = \text{id}$.
The first one, $\sigma^0_0 \circ \delta^1_0 = \text{id}$, is clear from
the explicit description of the morphisms above.
To prove the second relation we have to use the cocycle condition
(because it does not holds for an arbitrary isomorphism
$\varphi : N \otimes_R A \to A \otimes_R N$). Write
$p = \sigma^0_0 \circ \delta^1_1 : N \to N$. By the description of the
maps above we deduce that $p$ is also equal to
$$
p = \varphi \otimes \text{id} :
N = (N \otimes_R A) \otimes_{(A \otimes_R A)} A
\longrightarrow
(A \otimes_R N) \otimes_{(A \otimes_R A)} A = N
$$
Since $\varphi$ is an isomorphism we see that $p$ is an isomorphism.
Write $\varphi(n \otimes 1) = \sum \alpha_i \otimes x_i$ for certain
$\alpha_i \in A$ and $x_i \in N$. Then $p(n) = \sum \alpha_ix_i$.
Next, write
$\varphi(x_i \otimes 1) = \sum \alpha_{ij} \otimes y_j$ for
certain $\alpha_{ij} \in A$ and $y_j \in N$. Then the cocycle condition
says that
$$
\sum \alpha_i \otimes \alpha_{ij} \otimes y_j
=
\sum \alpha_i \otimes 1 \otimes x_i.
$$
This means that $p(n) = \sum \alpha_ix_i = \sum \alpha_i\alpha_{ij}y_j =
\sum \alpha_i p(x_i) = p(p(n))$. Thus $p$ is a projector, and since it is
an isomorphism it is the identity.
\medskip\noindent
To prove fully that $N_\bullet$ is a cosimplicial module we have to check
all 5 types of relations of
Simplicial, Remark \ref{simplicial-remark-relations-cosimplicial}.
The relations on composing $\sigma$'s are obvious.
The relations on composing $\delta$'s come down to the
cocycle condition for $\varphi$.
In exactly the same way as above one checks the relations
$\sigma_j \circ \delta_j = \sigma_j \circ \delta_{j + 1} = \text{id}$.
Finally, the other relations on compositions of $\delta$'s and $\sigma$'s
hold for any $\varphi$ whatsoever.
\end{proof}
\noindent
Note that to an $R$-module $M$ we can associate a canonical
descent datum, namely $(M \otimes_R A, can)$ where
$can : (M \otimes_R A) \otimes_R A \to A \otimes_R (M \otimes_R A)$
is the obvious map:
$(m \otimes a) \otimes a' \mapsto a \otimes (m \otimes a')$.
\begin{lemma}
\label{lemma-canonical-descent-datum-cosimplicial}
Let $R \to A$ be a ring map.
Let $M$ be an $R$-module. The cosimplicial
$(A/R)_\bullet$-module associated to the canonical descent
datum is isomorphic to the cosimplicial module $(A/R)_\bullet \otimes_R M$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{definition}
\label{definition-descent-datum-effective-module}
Let $R \to A$ be a ring map.
We say a descent datum $(N, \varphi)$ is {\it effective}
if there exists an $R$-module $M$ and an isomorphism
of descent data from $(M \otimes_R A, can)$ to
$(N, \varphi)$.
\end{definition}
\noindent
Let $R \to A$ be a ring map.
Let $(N, \varphi)$ be a descent datum.
We may take the cochain complex $s(N_\bullet)$ associated
with $N_\bullet$ (see
Simplicial, Section \ref{simplicial-section-dold-kan-cosimplicial}).
It has the following shape:
$$
N \to A \otimes_R N \to A \otimes_R A \otimes_R N \to \ldots
$$
We can describe the maps.
The first map is the map
$$
n \longmapsto 1 \otimes n - \varphi(n \otimes 1).
$$
The second map on pure tensors has the values
$$
a \otimes n \longmapsto 1 \otimes a \otimes n
- a \otimes 1 \otimes n + a \otimes \varphi(n \otimes 1).
$$
It is clear how the pattern continues.
\medskip\noindent
In the special case
where $N = A \otimes_R M$ we see that for any $m \in M$
the element $1 \otimes m$ is in the kernel of the first map
of the cochain complex associated to the cosimplicial
module $(A/R)_\bullet \otimes_R M$. Hence we get an extended cochain complex
\begin{equation}
\label{equation-extended-complex}
0 \to M \to A \otimes_R M \to A \otimes_R A \otimes_R M \to \ldots
\end{equation}
Here we think of the $0$ as being in degree $-2$,
the module $M$ in degree $-1$, the module $A \otimes_R M$ in
degree $0$, etc. Note that this complex has the shape
$$
0 \to R \to A \to A \otimes_R A \to A \otimes_R A \otimes_R A \to \ldots
$$
when $M = R$.
\begin{lemma}
\label{lemma-with-section-exact}
Suppose that $R \to A$ has a section.
Then for any $R$-module $M$ the extended cochain complex
(\ref{equation-extended-complex}) is exact.
\end{lemma}
\begin{proof}
By
Simplicial, Lemma \ref{simplicial-lemma-push-outs-simplicial-object-w-section}
the map $R \to (A/R)_\bullet$ is a homotopy equivalence
of cosimplicial $R$-algebras
(here $R$ denotes the constant cosimplicial $R$-algebra).
Hence $M \to (A/R)_\bullet \otimes_R M$ is
a homotopy equivalence in the category of cosimplicial
$R$-modules, because $\otimes_R M$ is a
functor from the category of $R$-algebras to the category
of $R$-modules, see
Simplicial, Lemma \ref{simplicial-lemma-functorial-homotopy}.
This implies that the induced map of associated
complexes is a homotopy equivalence, see
Simplicial, Lemma \ref{simplicial-lemma-homotopy-s-Q}.
Since the complex associated to the constant cosimplicial
$R$-module $M$ is the complex
$$
\xymatrix{
M \ar[r]^0 & M \ar[r]^1 & M \ar[r]^0 & M \ar[r]^1 & M \ldots
}
$$
we win (since the extended version simply puts an extra $M$ at
the beginning).
\end{proof}
\begin{lemma}
\label{lemma-ff-exact}
Suppose that $R \to A$ is faithfully flat, see
Algebra, Definition \ref{algebra-definition-flat}.
Then for any $R$-module $M$ the extended cochain complex
(\ref{equation-extended-complex}) is exact.
\end{lemma}
\begin{proof}
Suppose we can show there exists a faithfully flat ring map
$R \to R'$ such that the result holds for the ring map
$R' \to A' = R' \otimes_R A$. Then the result follows for
$R \to A$. Namely, for any $R$-module $M$ the cosimplicial
module $(M \otimes_R R') \otimes_{R'} (A'/R')_\bullet$ is
just the cosimplicial module $R' \otimes_R (M \otimes_R (A/R)_\bullet)$.
Hence the vanishing of cohomology of the complex associated to
$(M \otimes_R R') \otimes_{R'} (A'/R')_\bullet$ implies the
vanishing of the cohomology of the complex associated to
$M \otimes_R (A/R)_\bullet$ by faithful flatness of $R \to R'$.
Similarly for the vanishing of cohomology groups in degrees
$-1$ and $0$ of the extended complex (proof omitted).
\medskip\noindent
But we have such a faithful flat extension. Namely $R' = A$ works
because the ring map $R' = A \to A' = A \otimes_R A$ has a section
$a \otimes a' \mapsto aa'$ and
Lemma \ref{lemma-with-section-exact}
applies.
\end{proof}
\noindent
Here is how the complex relates to the question of effectivity.
\begin{lemma}
\label{lemma-recognize-effective}
Let $R \to A$ be a faithfully flat ring map.
Let $(N, \varphi)$ be a descent datum.
Then $(N, \varphi)$ is effective if and only if the canonical
map
$$
A \otimes_R H^0(s(N_\bullet)) \longrightarrow N
$$
is an isomorphism.
\end{lemma}
\begin{proof}
If $(N, \varphi)$ is effective, then we may write $N = A \otimes_R M$
with $\varphi = can$. It follows that $H^0(s(N_\bullet)) = M$ by
Lemmas \ref{lemma-canonical-descent-datum-cosimplicial}
and \ref{lemma-ff-exact}. Conversely, suppose the map of the lemma
is an isomorphism. In this case set $M = H^0(s(N_\bullet))$.
This is an $R$-submodule of $N$,
namely $M = \{n \in N \mid 1 \otimes n = \varphi(n \otimes 1)\}$.
The only thing to check is that via the isomorphism
$A \otimes_R M \to N$
the canonical descent data agrees with $\varphi$.
We omit the verification.
\end{proof}
\begin{lemma}
\label{lemma-descent-descends}
Let $R \to A$ be a ring map, and let $R \to R'$ be faithfully flat.
Set $A' = R' \otimes_R A$. If all descent data for $R' \to A'$ are
effective, then so are all descent data for $R \to A$.
\end{lemma}
\begin{proof}
Let $(N, \varphi)$ be a descent datum for $R \to A$.
Set $N' = R' \otimes_R N = A' \otimes_A N$, and denote
$\varphi' = \text{id}_{R'} \otimes \varphi$ the base change
of the descent datum $\varphi$. Then $(N', \varphi')$ is
a descent datum for $R' \to A'$ and
$H^0(s(N'_\bullet)) = R' \otimes_R H^0(s(N_\bullet))$.
Moreover, the map
$A' \otimes_{R'} H^0(s(N'_\bullet)) \to N'$ is identified
with the base change of the $A$-module map
$A \otimes_R H^0(s(N)) \to N$ via the faithfully flat map
$A \to A'$. Hence we conclude by Lemma \ref{lemma-recognize-effective}.
\end{proof}
\noindent
Here is the main result of this section.
Its proof may seem a little clumsy; for a more highbrow approach see
Remark \ref{remark-homotopy-equivalent-cosimplicial-algebras} below.
\begin{proposition}
\label{proposition-descent-module}
\begin{slogan}
Effective descent for modules along faithfully flat ring maps.
\end{slogan}
Let $R \to A$ be a faithfully flat ring map.
Then
\begin{enumerate}
\item any descent datum on modules with respect to $R \to A$
is effective,
\item the functor $M \mapsto (A \otimes_R M, can)$ from $R$-modules
to the category of descent data is an equivalence, and
\item the inverse functor is given by $(N, \varphi) \mapsto H^0(s(N_\bullet))$.
\end{enumerate}
\end{proposition}
\begin{proof}
We only prove (1) and omit the proofs of (2) and (3).
As $R \to A$ is faithfully flat, there exists a faithfully flat
base change $R \to R'$ such that $R' \to A' = R' \otimes_R A$ has
a section (namely take $R' = A$ as in the proof of
Lemma \ref{lemma-ff-exact}). Hence, using
Lemma \ref{lemma-descent-descends}
we may assume that $R \to A$ as a section, say $\sigma : A \to R$.
Let $(N, \varphi)$ be a descent datum relative to $R \to A$.
Set
$$
M = H^0(s(N_\bullet)) = \{n \in N \mid 1 \otimes n = \varphi(n \otimes 1)\}
\subset
N
$$
By Lemma \ref{lemma-recognize-effective} it suffices to show that
$A \otimes_R M \to N$ is an isomorphism.
\medskip\noindent
Take an element $n \in N$. Write
$\varphi(n \otimes 1) = \sum a_i \otimes x_i$ for certain
$a_i \in A$ and $x_i \in N$. By Lemma \ref{lemma-descent-datum-cosimplicial}
we have $n = \sum a_i x_i$ in $N$ (because
$\sigma^0_0 \circ \delta^1_0 = \text{id}$ in any cosimplicial object).
Next, write $\varphi(x_i \otimes 1) = \sum a_{ij} \otimes y_j$ for
certain $a_{ij} \in A$ and $y_j \in N$.
The cocycle condition means that
$$
\sum a_i \otimes a_{ij} \otimes y_j = \sum a_i \otimes 1 \otimes x_i
$$
in $A \otimes_R A \otimes_R N$. We conclude two things from this.
First, by applying $\sigma$ to the first $A$ we conclude that
$\sum \sigma(a_i) \varphi(x_i \otimes 1) = \sum \sigma(a_i) \otimes x_i$
which means that $\sum \sigma(a_i) x_i \in M$. Next, by applying
$\sigma$ to the middle $A$ and multiplying out we conclude that
$\sum_i a_i (\sum_j \sigma(a_{ij}) y_j) = \sum a_i x_i = n$. Hence
by the first conclusion we see that $A \otimes_R M \to N$ is
surjective. Finally, suppose that $m_i \in M$ and
$\sum a_i m_i = 0$. Then we see by applying $\varphi$ to
$\sum a_im_i \otimes 1$ that $\sum a_i \otimes m_i = 0$.
In other words $A \otimes_R M \to N$ is injective and we win.
\end{proof}
\begin{remark}
\label{remark-standard-covering}
Let $R$ be a ring. Let $f_1, \ldots, f_n\in R$ generate the
unit ideal. The ring $A = \prod_i R_{f_i}$ is a faithfully flat
$R$-algebra. We remark that the cosimplicial ring $(A/R)_\bullet$
has the following ring in degree $n$:
$$
\prod\nolimits_{i_0, \ldots, i_n} R_{f_{i_0}\ldots f_{i_n}}
$$
Hence the results above recover
Algebra, Lemmas \ref{algebra-lemma-standard-covering},
\ref{algebra-lemma-cover-module} and \ref{algebra-lemma-glue-modules}.
But the results above actually say more because of exactness
in higher degrees. Namely, it implies that {\v C}ech cohomology of
quasi-coherent sheaves on affines is trivial. Thus we get a second
proof of Cohomology of Schemes, Lemma
\ref{coherent-lemma-cech-cohomology-quasi-coherent-trivial}.
\end{remark}
\begin{remark}
\label{remark-homotopy-equivalent-cosimplicial-algebras}
Let $R$ be a ring. Let $A_\bullet$ be a cosimplicial $R$-algebra.
In this setting a descent datum corresponds to an cosimplicial
$A_\bullet$-module $M_\bullet$ with the property that for
every $n, m \geq 0$ and every $\varphi : [n] \to [m]$ the
map $M(\varphi) : M_n \to M_m$ induces an isomorphism
$$
M_n \otimes_{A_n, A(\varphi)} A_m \longrightarrow M_m.
$$
Let us call such a cosimplicial module a {\it cartesian module}.
In this setting, the proof of Proposition \ref{proposition-descent-module}
can be split in the following steps
\begin{enumerate}
\item If $R \to R'$ is faithfully flat, $R \to A$ any ring map,
then descent data for $A/R$ are effective if
descent data for $(R' \otimes_R A)/R'$ are effective.
\item Let $A$ be an $R$-algebra. Descent data for $A/R$ correspond
to cartesian $(A/R)_\bullet$-modules.
\item If $R \to A$ has a section then $(A/R)_\bullet$ is homotopy
equivalent to $R$, the constant cosimplicial
$R$-algebra with value $R$.
\item If $A_\bullet \to B_\bullet$ is a homotopy equivalence of
cosimplicial $R$-algebras then the functor
$M_\bullet \mapsto M_\bullet \otimes_{A_\bullet} B_\bullet$
induces an equivalence of categories between cartesian
$A_\bullet$-modules and cartesian $B_\bullet$-modules.
\end{enumerate}
For (1) see Lemma \ref{lemma-descent-descends}.
Part (2) uses Lemma \ref{lemma-descent-datum-cosimplicial}.
Part (3) we have seen in the proof of Lemma \ref{lemma-with-section-exact}
(it relies on Simplicial,
Lemma \ref{simplicial-lemma-push-outs-simplicial-object-w-section}).
Moreover, part (4) is a triviality if you think about it right!
\end{remark}
\section{Descent for universally injective morphisms}
\label{section-descent-universally-injective}
\noindent
Numerous constructions in algebraic geometry are made using techniques of
{\it descent}, such as constructing objects over a given space by first
working over a somewhat larger space which projects down to the given space,
or verifying a property of a space or a morphism by pulling back along a
covering map. The utility of such techniques is of course dependent on
identification of a wide class of {\it effective descent morphisms}.
Early in the Grothendieckian development of modern algebraic geometry,
the class of morphisms which are {\it quasi-compact} and {\it faithfully flat}
was shown to be effective for descending objects, morphisms, and many
properties thereof.
\medskip\noindent
As usual, this statement comes down to a property of rings and modules.
For a homomorphism $f: R \to S$ to be an effective descent morphism for
modules, Grothendieck showed that it is sufficient for $f$ to be
faithfully flat. However, this excludes many natural examples: for instance,
any split ring homomorphism is an effective descent morphism. One natural
example of this even arises in the proof of faithfully flat descent: for
$f: R \to S$ any ring homomorphism, $1_S \otimes f: S \to S \otimes_R S$
is split by the multiplication map whether or not it is flat.
\medskip\noindent
One may then ask whether there is a natural ring-theoretic condition
implying effective descent for modules which includes both the case of a
faithfully flat morphism and that of a split ring homomorphism. It may
surprise the reader (at least it surprised this author) to learn that a
complete answer to this question has been known since around 1970! Namely,
it is not hard to check that a necessary condition for $f: R \to S$ to be
an effective descent morphism for modules is that $f$ must be
{\it universally injective} in the category of $R$-modules, that is, for
any $R$-module $M$, the map $1_M \otimes f: M \to M \otimes_R S$
must be injective. This then turns out to be a sufficient condition as well.
For example, if $f$ is split in the category of $R$-modules (but not
necessarily in the category of rings), then $f$ is an effective descent
morphism for modules.
\medskip\noindent
The history of this result is a bit involved: it was originally asserted
by Olivier \cite{olivier}, who called universally injective morphisms
{\it pure}, but without a clear indication of proof. One can extract the
result from the work of Joyal and Tierney \cite{joyal-tierney}, but to the
best of our knowledge, the first free-standing proof to appear in the
literature is that of Mesablishvili \cite{mesablishvili1}. The first purpose
of this section is to expose Mesablishvili's proof; this requires little
modification of his original presentation aside from correcting typos, with
the one exception that we make explicit the relationship between the
customary definition of a descent datum in algebraic geometry and the one
used in \cite{mesablishvili1}. The proof turns to be entirely
category-theoretic, and consequently can be put in the language of monads
(and thus applied in other contexts); see \cite{janelidze-tholen}.
\medskip\noindent
The second purpose of this section is to collect some information about which
properties of modules, algebras, and morphisms can be descended along
universally injective ring homomorphisms. The cases of finite modules
and flat modules were treated by Mesablishvili \cite{mesablishvili2}.
\subsection{Category-theoretic preliminaries}
\label{subsection-category-prelims}
\noindent
We start by recalling a few basic notions from category theory which will
simplify the exposition. In this subsection, fix an ambient category.
\medskip\noindent
For two morphisms $g_1, g_2: B \to C$, recall that an {\it equalizer}
of $g_1$ and $g_2$ is a morphism $f: A \to B$ which satisfies
$g_1 \circ f = g_2 \circ f$ and is universal for this property.
This second statement means that any commutative diagram
$$
\xymatrix{A' \ar[rd]^e \ar@/^1.5pc/[rrd] \ar@{-->}[d] & & \\
A \ar[r]^f & B \ar@<1ex>[r]^{g_1} \ar@<-1ex>[r]_{g_2} &
C
}
$$
without the dashed arrow can be uniquely completed. We also say in this
situation that the diagram
\begin{equation}
\label{equation-equalizer}
\xymatrix{
A \ar[r]^f & B \ar@<1ex>[r]^{g_1} \ar@<-1ex>[r]_{g_2} & C
}
\end{equation}
is an equalizer. Reversing arrows gives the definition of a {\it coequalizer}.
See Categories, Sections \ref{categories-section-equalizers} and
\ref{categories-section-coequalizers}.
\medskip\noindent
Since it involves a universal property, the property of being an equalizer is
typically not stable under applying a covariant functor. Just as for
monomorphisms and epimorphisms, one can get around this in some
cases by exhibiting splittings.
\begin{definition}
\label{definition-split-equalizer}
A {\it split equalizer} is a diagram (\ref{equation-equalizer}) with
$g_1 \circ f = g_2 \circ f$ for which there exist auxiliary morphisms
$h : B \to A$ and $i : C \to B$ such that
\begin{equation}
\label{equation-split-equalizer-conditions}
h \circ f = 1_A, \quad f \circ h = i \circ g_1, \quad i \circ g_2 = 1_B.
\end{equation}
\end{definition}
\noindent
The point is that the equalities among arrows force (\ref{equation-equalizer})
to be an equalizer: the map $e$ factors uniquely through $f$ by writing
$e = f \circ (h \circ e)$. Consequently, applying a covariant functor
to a split equalizer gives a split equalizer; applying a contravariant functor
gives a {\it split coequalizer}, whose definition is apparent.
\subsection{Universally injective morphisms}
\label{subsection-universally-injective}
\noindent
Recall that $\textit{Rings}$ denotes the category of commutative rings
with $1$. For an object $R$ of $\textit{Rings}$ we denote $\text{Mod}_R$
the category of $R$-modules.
\begin{remark}
\label{remark-reflects}
Any functor $F : \mathcal{A} \to \mathcal{B}$ of abelian categories
which is exact and takes nonzero objects to nonzero objects reflects
injections and surjections. Namely, exactness implies that
$F$ preserves kernels and cokernels (compare with
Homology, Section \ref{homology-section-functors}).
For example, if $f : R \to S$ is a
faithfully flat ring homomorphism, then
$\bullet \otimes_R S: \text{Mod}_R \to \text{Mod}_S$ has these properties.
\end{remark}
\noindent
Let $R$ be a ring. Recall that a morphism $f : M \to N$ in $\text{Mod}_R$
is {\it universally injective} if for all $P \in \text{Mod}_R$,
the morphism $f \otimes 1_P: M \otimes_R P \to N \otimes_R P$ is injective.
See Algebra, Definition \ref{algebra-definition-universally-injective}.
\begin{definition}
\label{definition-universally-injective}
A ring map $f: R \to S$ is {\it universally injective}
if it is universally injective as a morphism in $\text{Mod}_R$.
\end{definition}
\begin{example}
\label{example-split-injection-universally-injective}
Any split injection in $\text{Mod}_R$ is universally injective. In particular,
any split injection in $\textit{Rings}$ is universally injective.
\end{example}
\begin{example}
\label{example-cover-universally-injective}
For a ring $R$ and $f_1, \ldots, f_n \in R$ generating the unit
ideal, the morphism $R \to R_{f_1} \oplus \ldots \oplus R_{f_n}$ is
universally injective. Although this is immediate from
Lemma \ref{lemma-faithfully-flat-universally-injective},
it is instructive to check it directly: we immediately reduce to the case
where $R$ is local, in which case some $f_i$ must be a unit and so the map
$R \to R_{f_i}$ is an isomorphism.
\end{example}
\begin{lemma}
\label{lemma-faithfully-flat-universally-injective}
Any faithfully flat ring map is universally injective.
\end{lemma}
\begin{proof}
This is a reformulation of Algebra, Lemma
\ref{algebra-lemma-faithfully-flat-universally-injective}.
\end{proof}
\noindent
The key observation from \cite{mesablishvili1} is that universal injectivity
can be usefully reformulated in terms of a splitting, using the usual
construction of an injective cogenerator in $\text{Mod}_R$.
\begin{definition}
\label{definition-C}
Let $R$ be a ring. Define the contravariant functor
{\it $C$} $ : \text{Mod}_R \to \text{Mod}_R$ by setting
$$
C(M) = \Hom_{\textit{Ab}}(M, \mathbf{Q}/\mathbf{Z}),
$$
with the $R$-action on $C(M)$ given by $rf(s) = f(rs)$.
\end{definition}
\noindent
This functor was denoted $M \mapsto M^\vee$ in
More on Algebra, Section \ref{more-algebra-section-injectives-modules}.
\begin{lemma}
\label{lemma-C-is-faithful}
For a ring $R$, the functor $C : \text{Mod}_R \to \text{Mod}_R$ is
exact and reflects injections and surjections.
\end{lemma}
\begin{proof}
Exactness is More on Algebra, Lemma \ref{more-algebra-lemma-vee-exact}
and the other properties follow from this, see
Remark \ref{remark-reflects}.
\end{proof}
\begin{remark}
\label{remark-adjunction}
We will use frequently the standard adjunction between $\Hom$ and tensor
product, in the form of the natural isomorphism of contravariant functors
\begin{equation}
\label{equation-adjunction}
C(\bullet_1 \otimes_R \bullet_2) \cong \Hom_R(\bullet_1, C(\bullet_2)):
\text{Mod}_R \times \text{Mod}_R \to \text{Mod}_R
\end{equation}
taking $f: M_1 \otimes_R M_2 \to \mathbf{Q}/\mathbf{Z}$ to the map $m_1 \mapsto
(m_2 \mapsto f(m_1 \otimes m_2))$. See
Algebra, Lemma \ref{algebra-lemma-hom-from-tensor-product-variant}.
A corollary of this observation is that if
$$
\xymatrix@C=9pc{
C(M) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N) \ar[r] & C(P)
}
$$
is a split coequalizer diagram in $\text{Mod}_R$, then so is
$$
\xymatrix@C=9pc{
C(M \otimes_R Q) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N \otimes_R Q) \ar[r] & C(P
\otimes_R Q)
}
$$
for any $Q \in \text{Mod}_R$.
\end{remark}
\begin{lemma}
\label{lemma-split-surjection}
Let $R$ be a ring. A morphism $f: M \to N$ in $\text{Mod}_R$ is universally
injective if and only if $C(f): C(N) \to C(M)$ is a split surjection.
\end{lemma}
\begin{proof}
By (\ref{equation-adjunction}), for any $P \in \text{Mod}_R$ we have a
commutative diagram
$$
\xymatrix@C=9pc{
\Hom_R( P, C(N)) \ar[r]_{\Hom_R(P,C(f))} \ar[d]^{\cong} &
\Hom_R(P,C(M)) \ar[d]^{\cong} \\
C(P \otimes_R N ) \ar[r]^{C(1_{P} \otimes f)} & C(P \otimes_R M ).
}
$$
If $f$ is universally injective, then $1_{C(M)} \otimes f: C(M) \otimes_R M \to
C(M) \otimes_R N$ is injective,
so both rows in the above diagram are surjective for $P = C(M)$. We may thus
lift
$1_{C(M)} \in \Hom_R(C(M), C(M))$ to some $g \in \Hom_R(C(N), C(M))$ splitting
$C(f)$.
Conversely, if $C(f)$ is a split surjection, then
both rows in the above diagram are surjective,
so by Lemma \ref{lemma-C-is-faithful}, $1_{P} \otimes f$ is injective.
\end{proof}
\begin{remark}
\label{remark-functorial-splitting}
Let $f: M \to N$ be a universally injective morphism in $\text{Mod}_R$. By
choosing a splitting
$g$ of $C(f)$, we may construct a functorial splitting of $C(1_P \otimes f)$
for each $P \in \text{Mod}_R$.
Namely, by (\ref{equation-adjunction}) this amounts to splitting $\Hom_R(P,
C(f))$ functorially in $P$,
and this is achieved by the map $g \circ \bullet$.
\end{remark}
\subsection{Descent for modules and their morphisms}
\label{subsection-descent-modules-morphisms}
\noindent
Throughout this subsection, fix a ring map $f: R \to S$. As seen in
Section \ref{section-descent-modules} we can use the language of cosimplicial
algebras to talk about descent data for modules, but in this
subsection we prefer a more down to earth terminology.
\medskip\noindent
For $i = 1, 2, 3$, let $S_i$ be the $i$-fold tensor product of $S$ over $R$.
Define the ring homomorphisms $\delta_0^1, \delta_1^1: S_1 \to S_2$,
$\delta_{01}^1, \delta_{02}^1, \delta_{12}^1: S_1 \to S_3$, and
$\delta_0^2, \delta_1^2, \delta_2^2: S_2 \to S_3$ by the formulas
\begin{align*}
\delta^1_0 (a_0) & = 1 \otimes a_0 \\
\delta^1_1 (a_0) & = a_0 \otimes 1 \\
\delta^2_0 (a_0 \otimes a_1) & = 1 \otimes a_0 \otimes a_1 \\
\delta^2_1 (a_0 \otimes a_1) & = a_0 \otimes 1 \otimes a_1 \\
\delta^2_2 (a_0 \otimes a_1) & = a_0 \otimes a_1 \otimes 1 \\
\delta_{01}^1(a_0) & = 1 \otimes 1 \otimes a_0 \\
\delta_{02}^1(a_0) & = 1 \otimes a_0 \otimes 1 \\
\delta_{12}^1(a_0) & = a_0 \otimes 1 \otimes 1.
\end{align*}
In other words, the upper index indicates the source ring, while the lower
index indicates where to insert factors of 1. (This notation is compatible
with the notation introduced in Section \ref{section-descent-modules}.)
\medskip\noindent
Recall\footnote{To be precise, our $\theta$ here is the inverse of
$\varphi$ from Definition \ref{definition-descent-datum-modules}.}
from Definition \ref{definition-descent-datum-modules} that for
$M \in \text{Mod}_S$, a {\it descent datum} on $M$ relative to $f$ is
an isomorphism
$$
\theta :
M \otimes_{S,\delta^1_0} S_2
\longrightarrow
M \otimes_{S,\delta^1_1} S_2
$$
of $S_2$-modules satisfying the {\it cocycle condition}
\begin{equation}
\label{equation-cocycle-condition}
(\theta \otimes \delta_2^2) \circ (\theta \otimes \delta_2^0) = (\theta \otimes
\delta_2^1):
M \otimes_{S, \delta^1_{01}} S_3 \to M \otimes_{S,\delta^1_{12}} S_3.
\end{equation}
Let $DD_{S/R}$ be the category of $S$-modules equipped with descent data
relative to $f$.
\medskip\noindent
For example, for $M_0 \in \text{Mod}_R$ and a choice of isomorphism
$M \cong M_0 \otimes_R S$ gives rise to a descent datum by identifying
$M \otimes_{S,\delta^1_0} S_2$ and $M \otimes_{S,\delta^1_1} S_2$
naturally with $M_0 \otimes_R S_2$. This construction in particular
defines a functor $f^*: \text{Mod}_R \to DD_{S/R}$.
\begin{definition}
\label{definition-effective-descent}
The functor $f^*: \text{Mod}_R \to DD_{S/R}$
is called {\it base extension along $f$}. We say that $f$ is a
{\it descent morphism for modules} if $f^*$ is fully
faithful. We say that $f$ is an {\it effective descent morphism for modules}
if $f^*$ is an equivalence of categories.
\end{definition}
\noindent
Our goal is to show that for $f$ universally injective, we can use $\theta$ to
locate $M_0$ within $M$. This process makes crucial use of some equalizer
diagrams.
\begin{lemma}
\label{lemma-equalizer-M}
For $(M,\theta) \in DD_{S/R}$, the diagram
\begin{equation}
\label{equation-equalizer-M}
\xymatrix@C=8pc{
M \ar[r]^{\theta \circ (1_M \otimes \delta_0^1)} &
M \otimes_{S, \delta_1^1} S_2
\ar@<1ex>[r]^{(\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0)}
\ar@<-1ex>[r]_{1_{M \otimes S_2} \otimes \delta^2_1} &
M \otimes_{S, \delta_{12}^1} S_3
}
\end{equation}
is a split equalizer.
\end{lemma}
\begin{proof}
Define the ring homomorphisms $\sigma^0_0: S_2 \to S_1$ and $\sigma_0^1,
\sigma_1^1: S_3 \to S_2$ by the formulas
\begin{align*}
\sigma^0_0 (a_0 \otimes a_1) & = a_0a_1 \\
\sigma^1_0 (a_0 \otimes a_1 \otimes a_2) & = a_0a_1 \otimes a_2 \\
\sigma^1_1 (a_0 \otimes a_1 \otimes a_2) & = a_0 \otimes a_1a_2.
\end{align*}
We then take the auxiliary morphisms to be
$1_M \otimes \sigma_0^0: M \otimes_{S, \delta_1^1} S_2 \to M$
and $1_M \otimes \sigma_0^1: M \otimes_{S,\delta_{12}^1} S_3 \to M \otimes_{S,
\delta_1^1} S_2$.
Of the compatibilities required in (\ref{equation-split-equalizer-conditions}),
the first follows from tensoring the cocycle condition
(\ref{equation-cocycle-condition}) with $\sigma_1^1$
and the others are immediate.
\end{proof}
\begin{lemma}
\label{lemma-equalizer-CM}
For $(M, \theta) \in DD_{S/R}$, the diagram
\begin{equation}
\label{equation-coequalizer-CM}
\xymatrix@C=8pc{
C(M \otimes_{S, \delta_{12}^1} S_3)
\ar@<1ex>[r]^{C((\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0))}
\ar@<-1ex>[r]_{C(1_{M \otimes S_2} \otimes \delta^2_1)} &
C(M \otimes_{S, \delta_1^1} S_2 )
\ar[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} & C(M).
}
\end{equation}
obtained by applying $C$ to (\ref{equation-equalizer-M}) is a split
coequalizer.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-equalizer-S}
The diagram
\begin{equation}
\label{equation-equalizer-S}
\xymatrix@C=8pc{
S_1 \ar[r]^{\delta^1_1} &
S_2 \ar@<1ex>[r]^{\delta^2_2} \ar@<-1ex>[r]_{\delta^2_1} &
S_3
}
\end{equation}
is a split equalizer.
\end{lemma}
\begin{proof}
In Lemma \ref{lemma-equalizer-M}, take $(M, \theta) = f^*(S)$.
\end{proof}
\noindent
This suggests a definition of a potential quasi-inverse functor for $f^*$.
\begin{definition}
\label{definition-pushforward}
Define the functor {\it $f_*$} $: DD_{S/R} \to \text{Mod}_R$ by taking
$f_*(M, \theta)$ to be the $R$-submodule of $M$ for which the diagram
\begin{equation}
\label{equation-equalizer-f}
\xymatrix@C=8pc{f_*(M,\theta) \ar[r] & M \ar@<1ex>^{\theta \circ (1_M \otimes
\delta_0^1)}[r] \ar@<-1ex>_{1_M \otimes \delta_1^1}[r] &
M \otimes_{S, \delta_1^1} S_2
}
\end{equation}
is an equalizer.
\end{definition}
\noindent
Using Lemma \ref{lemma-equalizer-M} and the fact that the restriction functor
$\text{Mod}_S \to \text{Mod}_R$ is right adjoint to the base extension
functor $\bullet \otimes_R S: \text{Mod}_R \to \text{Mod}_S$,
we deduce that $f_*$ is right adjoint to $f^*$.
\medskip\noindent
We are ready for the key lemma. In the faithfully flat case this is a
triviality (see Remark \ref{remark-descent-lemma}),
but in the general case some argument is needed.
\begin{lemma}
\label{lemma-descent-lemma}
If $f$ is universally injective, then the diagram
\begin{equation}
\label{equation-equalizer-f2}
\xymatrix@C=8pc{
f_*(M, \theta) \otimes_R S
\ar[r]^{\theta \circ (1_M \otimes \delta_0^1)} &
M \otimes_{S, \delta_1^1} S_2
\ar@<1ex>[r]^{(\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0)}
\ar@<-1ex>[r]_{1_{M \otimes S_2} \otimes \delta^2_1} &
M \otimes_{S, \delta_{12}^1} S_3
}
\end{equation}
obtained by tensoring (\ref{equation-equalizer-f}) over $R$ with $S$ is an
equalizer.
\end{lemma}
\begin{proof}
By
Lemma \ref{lemma-split-surjection} and
Remark \ref{remark-functorial-splitting},
the map $C(1_N \otimes f): C(N \otimes_R S) \to C(N)$ can be split functorially
in $N$. This gives the upper vertical arrows in the commutative diagram
$$
\xymatrix@C=8pc{
C(M \otimes_{S, \delta_1^1} S_2)
\ar@<1ex>^{C(\theta \circ (1_M \otimes \delta_0^1))}[r]
\ar@<-1ex>_{C(1_M \otimes \delta_1^1)}[r] \ar[d] &
C(M) \ar[r]\ar[d] & C(f_*(M,\theta)) \ar@{-->}[d] \\
C(M \otimes_{S,\delta_{12}^1} S_3)
\ar@<1ex>^{C((\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0))}[r]
\ar@<-1ex>_{C(1_{M \otimes S_2} \otimes \delta^2_1)}[r] \ar[d] &
C(M \otimes_{S, \delta_1^1} S_2 )
\ar[r]^{C(\theta \circ (1_M \otimes \delta_0^1))}
\ar[d]^{C(1_M \otimes \delta_1^1)} &
C(M) \ar[d] \ar@{=}[dl] \\
C(M \otimes_{S, \delta_1^1} S_2)
\ar@<1ex>[r]^{C(\theta \circ (1_M \otimes \delta_0^1))}
\ar@<-1ex>[r]_{C(1_M \otimes \delta_1^1)} &
C(M) \ar[r] &
C(f_*(M,\theta))
}
$$
in which the compositions along the columns are identity morphisms.
The second row is the coequalizer diagram
(\ref{equation-coequalizer-CM}); this produces the dashed arrow.
From the top right square, we obtain auxiliary morphisms $C(f_*(M,\theta)) \to
C(M)$
and $C(M) \to C(M\otimes_{S,\delta_1^1} S_2)$ which imply that the first row is
a split coequalizer diagram.
By Remark \ref{remark-adjunction}, we may tensor with $S$ inside $C$ to obtain
the split coequalizer diagram
$$
\xymatrix@C=8pc{
C(M \otimes_{S,\delta_2^2 \circ \delta_1^1} S_3)
\ar@<1ex>^{C((\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0))}[r]
\ar@<-1ex>_{C(1_{M \otimes S_2} \otimes \delta^2_1)}[r] &
C(M \otimes_{S, \delta_1^1} S_2 )
\ar[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} &
C(f_*(M,\theta) \otimes_R S).
}
$$
By Lemma \ref{lemma-C-is-faithful}, we conclude
(\ref{equation-equalizer-f2}) must also be an equalizer.
\end{proof}
\begin{remark}
\label{remark-descent-lemma}
If $f$ is a split injection in $\text{Mod}_R$, one can simplify the argument by
splitting $f$ directly,
without using $C$. Things are even simpler if $f$ is faithfully flat; in this
case,
the conclusion of Lemma \ref{lemma-descent-lemma}
is immediate because tensoring over $R$ with $S$ preserves all equalizers.
\end{remark}
\begin{theorem}
\label{theorem-descent}
The following conditions are equivalent.
\begin{enumerate}
\item[(a)] The morphism $f$ is a descent morphism for modules.
\item[(b)] The morphism $f$ is an effective descent morphism for modules.
\item[(c)] The morphism $f$ is universally injective.
\end{enumerate}
\end{theorem}
\begin{proof}
It is clear that (b) implies (a). We now check that (a) implies (c). If $f$ is
not universally injective, we can find $M \in \text{Mod}_R$ such that the map
$1_M \otimes f: M \to M \otimes_R S$ has nontrivial kernel $N$.
The natural projection $M \to M/N$ is not an isomorphism, but its image in
$DD_{S/R}$ is an isomorphism.
Hence $f^*$ is not fully faithful.
\medskip\noindent
We finally check that (c) implies (b). By Lemma \ref{lemma-descent-lemma}, for
$(M, \theta) \in DD_{S/R}$,
the natural map $f^* f_*(M,\theta) \to M$ is an isomorphism of $S$-modules. On
the other hand, for $M_0 \in \text{Mod}_R$,
we may tensor (\ref{equation-equalizer-S}) with $M_0$ over $R$ to obtain an
equalizer sequence,
so $M_0 \to f_* f^* M$ is an isomorphism. Consequently, $f_*$ and $f^*$ are
quasi-inverse functors, proving the claim.
\end{proof}
\subsection{Descent for properties of modules}
\label{subsection-descent-properties-modules}
\noindent
Throughout this subsection, fix a universally injective ring map $f : R \to S$,
an object $M \in \text{Mod}_R$, and a ring map $R \to A$. We now investigate
the question of which properties of $M$ or $A$ can be checked after base
extension along $f$. We start with some results from
\cite{mesablishvili2}.
\begin{lemma}
\label{lemma-flat-to-injective}
If $M \in \text{Mod}_R$ is flat, then $C(M)$ is an injective $R$-module.
\end{lemma}
\begin{proof}
Let $0 \to N \to P \to Q \to 0$ be an exact sequence in $\text{Mod}_R$. Since
$M$ is flat,
$$
0 \to N \otimes_R M \to P \otimes_R M \to Q \otimes_R M \to 0
$$
is exact.
By Lemma \ref{lemma-C-is-faithful},
$$
0 \to C(Q \otimes_R M) \to C(P \otimes_R M) \to C(N \otimes_R M) \to 0
$$
is exact. By (\ref{equation-adjunction}), this last sequence can be rewritten
as
$$
0 \to \Hom_R(Q, C(M)) \to \Hom_R(P, C(M)) \to \Hom_R(N, C(M)) \to 0.
$$
Hence $C(M)$ is an injective object of $\text{Mod}_R$.
\end{proof}
\begin{theorem}
\label{theorem-descend-module-properties}
If $M \otimes_R S$ has one of the following properties as an $S$-module
\begin{enumerate}
\item[(a)]
finitely generated;
\item[(b)]
finitely presented;
\item[(c)]
flat;
\item[(d)]
faithfully flat;
\item[(e)]
finite projective;
\end{enumerate}
then so does $M$ as an $R$-module (and conversely).
\end{theorem}
\begin{proof}
To prove (a), choose a finite set $\{n_i\}$ of generators of $M \otimes_R S$
in $\text{Mod}_S$. Write each $n_i$ as $\sum_j m_{ij} \otimes s_{ij}$ with
$m_{ij} \in M$ and $s_{ij} \in S$. Let $F$ be the finite free $R$-module with
basis $e_{ij}$ and let $F \to M$ be the $R$-module map sending $e_{ij}$ to
$m_{ij}$. Then $F \otimes_R S\to M \otimes_R S$ is surjective, so
$\Coker(F \to M) \otimes_R S$ is zero and hence $\Coker(F \to M)$
is zero. This proves (a).
\medskip\noindent
To see (b) assume $M \otimes_R S$ is finitely presented. Then $M$ is finitely
generated by (a). Choose a surjection $R^{\oplus n} \to M$ with kernel $K$.
Then $K \otimes_R S \to S^{\oplus r} \to M \otimes_R S \to 0$ is exact.
By Algebra, Lemma \ref{algebra-lemma-extension}
the kernel of $S^{\oplus r} \to M \otimes_R S$
is a finite $S$-module. Thus we can find finitely many elements
$k_1, \ldots, k_t \in K$ such that the images of $k_i \otimes 1$ in
$S^{\oplus r}$ generate the kernel of $S^{\oplus r} \to M \otimes_R S$.
Let $K' \subset K$ be the submodule generated by $k_1, \ldots, k_t$.
Then $M' = R^{\oplus r}/K'$ is a finitely presented $R$-module
with a morphism $M' \to M$ such that $M' \otimes_R S \to M \otimes_R S$
is an isomorphism. Thus $M' \cong M$ as desired.
\medskip\noindent
To prove (c), let $0 \to M' \to M'' \to M \to 0$ be a short exact sequence in
$\text{Mod}_R$. Since $\bullet \otimes_R S$ is a right exact functor,
$M'' \otimes_R S \to M \otimes_R S$ is surjective. So by
Lemma \ref{lemma-C-is-faithful} the map
$C(M \otimes_R S) \to C(M'' \otimes_R S)$ is injective.
If $M \otimes_R S$ is flat, then
Lemma \ref{lemma-flat-to-injective} shows
$C(M \otimes_R S)$ is an injective object of $\text{Mod}_S$, so the injection
$C(M \otimes_R S) \to C(M'' \otimes_R S)$
is split in $\text{Mod}_S$ and hence also in $\text{Mod}_R$.
Since $C(M \otimes_R S) \to C(M)$ is a split surjection by
Lemma \ref{lemma-split-surjection}, it follows that
$C(M) \to C(M'')$ is a split injection in $\text{Mod}_R$. That is, the sequence
$$
0 \to C(M) \to C(M'') \to C(M') \to 0
$$
is split exact.
For $N \in \text{Mod}_R$, by (\ref{equation-adjunction}) we see that
$$
0 \to C(M \otimes_R N) \to C(M'' \otimes_R N) \to C(M' \otimes_R N) \to 0
$$
is split exact. By Lemma \ref{lemma-C-is-faithful},
$$
0 \to M' \otimes_R N \to M'' \otimes_R N \to M \otimes_R N \to 0
$$
is exact. This implies $M$ is flat over $R$. Namely, taking
$M'$ a free module surjecting onto $M$ we conclude that
$\text{Tor}_1^R(M, N) = 0$ for all modules $N$ and we can use
Algebra, Lemma \ref{algebra-lemma-characterize-flat}.
This proves (c).
\medskip\noindent
To deduce (d) from (c), note that if $N \in \text{Mod}_R$ and $M \otimes_R N$
is zero,
then $M \otimes_R S \otimes_S (N \otimes_R S) \cong (M \otimes_R N) \otimes_R
S$ is zero,
so $N \otimes_R S$ is zero and hence $N$ is zero.
\medskip\noindent
To deduce (e) at this point, it suffices to recall that $M$ is finitely
generated and projective if and only if it is finitely presented and flat.
See Algebra, Lemma \ref{algebra-lemma-finite-projective}.
\end{proof}
\noindent
There is a variant for $R$-algebras.
\begin{theorem}
\label{theorem-descend-algebra-properties}
If $A \otimes_R S$ has one of the following properties as an $S$-algebra
\begin{enumerate}
\item[(a)]
of finite type;
\item[(b)]
of finite presentation;
\item[(c)]
formally unramified;
\item[(d)]
unramified;
\item[(e)]
\'etale;
\end{enumerate}
then so does $A$ as an $R$-algebra (and of course conversely).
\end{theorem}
\begin{proof}
To prove (a), choose a finite set $\{x_i\}$ of generators of $A \otimes_R S$
over $S$. Write each $x_i$ as $\sum_j y_{ij} \otimes s_{ij}$ with
$y_{ij} \in A$ and $s_{ij} \in S$. Let $F$ be the polynomial $R$-algebra
on variables $e_{ij}$ and let $F \to M$ be the $R$-algebra map sending
$e_{ij}$ to $y_{ij}$. Then $F \otimes_R S\to A \otimes_R S$ is surjective, so
$\Coker(F \to A) \otimes_R S$ is zero and hence $\Coker(F \to A)$
is zero. This proves (a).
\medskip\noindent
To see (b) assume $A \otimes_R S$ is a finitely presented $S$-algebra.
Then $A$ is finite type over $R$ by (a). Choose a surjection
$R[x_1, \ldots, x_n] \to A$ with kernel $I$.
Then $I \otimes_R S \to S[x_1, \ldots, x_n] \to A \otimes_R S \to 0$ is exact.
By Algebra, Lemma \ref{algebra-lemma-finite-presentation-independent}
the kernel of $S[x_1, \ldots, x_n] \to A \otimes_R S$
is a finitely generated ideal. Thus we can find finitely many elements
$y_1, \ldots, y_t \in I$ such that the images of $y_i \otimes 1$ in
$S[x_1, \ldots, x_n]$ generate the kernel of
$S[x_1, \ldots, x_n] \to A \otimes_R S$.
Let $I' \subset I$ be the ideal generated by $y_1, \ldots, y_t$.
Then $A' = R[x_1, \ldots, x_n]/I'$ is a finitely presented $R$-algebra
with a morphism $A' \to A$ such that $A' \otimes_R S \to A \otimes_R S$
is an isomorphism. Thus $A' \cong A$ as desired.
\medskip\noindent
To prove (c), recall that $A$ is formally unramified over $R$ if and only
if the module of relative differentials $\Omega_{A/R}$ vanishes, see
Algebra, Lemma \ref{algebra-lemma-characterize-formally-unramified} or
\cite[Proposition~17.2.1]{EGA4}.
Since $\Omega_{(A \otimes_R S)/S} = \Omega_{A/R} \otimes_R S$,
the vanishing descends by Theorem \ref{theorem-descent}.
\medskip\noindent
To deduce (d) from the previous cases, recall that $A$ is unramified
over $R$ if and only if $A$ is formally unramified and of finite type
over $R$, see
Algebra, Lemma \ref{algebra-lemma-formally-unramified-unramified}.
\medskip\noindent
To prove (e), recall that by
Algebra, Lemma \ref{algebra-lemma-etale-flat-unramified-finite-presentation}
or \cite[Th\'eor\`eme~17.6.1]{EGA4} the algebra
$A$ is \'etale over $R$ if and only if
$A$ is flat, unramified, and of finite presentation over $R$.
\end{proof}
\begin{remark}
\label{remark-when-locally-split}
It would make things easier to have a faithfully
flat ring homomorphism $g: R \to T$ for which $T \to S \otimes_R T$ has some
extra structure.
For instance, if one could ensure that $T \to S \otimes_R T$ is split in
$\textit{Rings}$,
then it would follow that every property of a module or algebra which is stable
under base extension
and which descends along faithfully flat morphisms also descends along
universally injective morphisms.
An obvious guess would be to find $g$ for which $T$ is not only faithfully flat
but also injective in $\text{Mod}_R$,
but even for $R = \mathbf{Z}$ no such homomorphism can exist.
\end{remark}
\section{Fpqc descent of quasi-coherent sheaves}
\label{section-fpqc-descent-quasi-coherent}
\noindent
The main application of flat descent for modules is
the corresponding descent statement for quasi-coherent
sheaves with respect to fpqc-coverings.
\begin{lemma}
\label{lemma-standard-fpqc-covering}
Let $S$ be an affine scheme.
Let $\mathcal{U} = \{f_i : U_i \to S\}_{i = 1, \ldots, n}$
be a standard fpqc covering of $S$, see
Topologies, Definition \ref{topologies-definition-standard-fpqc}.
Any descent datum on quasi-coherent sheaves
for $\mathcal{U} = \{U_i \to S\}$ is effective.
Moreover, the functor from the category of
quasi-coherent $\mathcal{O}_S$-modules to the category
of descent data with respect to $\mathcal{U}$ is fully faithful.
\end{lemma}
\begin{proof}
This is a restatement of Proposition \ref{proposition-descent-module}
in terms of schemes. First, note that a descent datum $\xi$
for quasi-coherent sheaves with respect to $\mathcal{U}$
is exactly the same as a descent datum $\xi'$ for quasi-coherent sheaves
with respect to the covering
$\mathcal{U}' = \{\coprod_{i = 1, \ldots, n} U_i \to S\}$.
Moreover, effectivity for $\xi$ is the same as effectivity for $\xi'$.
Hence we may assume $n = 1$, i.e., $\mathcal{U} = \{U \to S\}$
where $U$ and $S$ are affine. In this case descent data
correspond to descent data on modules with respect to the ring map
$$
\Gamma(S, \mathcal{O})
\longrightarrow
\Gamma(U, \mathcal{O}).
$$
Since $U \to S$ is surjective and flat, we see that this ring map
is faithfully flat. In other words,
Proposition \ref{proposition-descent-module} applies and we win.
\end{proof}
\begin{proposition}
\label{proposition-fpqc-descent-quasi-coherent}
Let $S$ be a scheme.
Let $\mathcal{U} = \{\varphi_i : U_i \to S\}$ be an fpqc covering, see
Topologies, Definition \ref{topologies-definition-fpqc-covering}.
Any descent datum on quasi-coherent sheaves
for $\mathcal{U} = \{U_i \to S\}$ is effective.
Moreover, the functor from the category of
quasi-coherent $\mathcal{O}_S$-modules to the category
of descent data with respect to $\mathcal{U}$ is fully faithful.
\end{proposition}
\begin{proof}
Let $S = \bigcup_{j \in J} V_j$ be an affine open covering.
For $j, j' \in J$ we denote $V_{jj'} = V_j \cap V_{j'}$ the intersection
(which need not be affine). For $V \subset S$ open we denote
$\mathcal{U}_V = \{V \times_S U_i \to V\}_{i \in I}$ which is a
fpqc-covering (Topologies, Lemma \ref{topologies-lemma-fpqc}).
By definition of an fpqc covering, we can find for each $j \in J$ a
finite set $K_j$, a map $\underline{i} : K_j \to I$,
affine opens $U_{\underline{i}(k), k} \subset U_{\underline{i}(k)}$,
$k \in K_j$ such that
$\mathcal{V}_j = \{U_{\underline{i}(k), k} \to V_j\}_{k \in K_j}$ is
a standard fpqc covering of $V_j$. And of course, $\mathcal{V}_j$
is a refinement of $\mathcal{U}_{V_j}$. Picture
$$
\xymatrix{
\mathcal{V}_j \ar[r] \ar@{~>}[d] &
\mathcal{U}_{V_j} \ar[r] \ar@{~>}[d] &
\mathcal{U} \ar@{~>}[d] \\
V_j \ar@{=}[r] & V_j \ar[r] & S
}
$$
where the top horizontal arrows are morphisms of families of
morphisms with fixed target (see
Sites, Definition \ref{sites-definition-morphism-coverings}).
\medskip\noindent
To prove the proposition you show successively the
faithfulness, fullness, and essential surjectivity of the
functor from quasi-coherent sheaves to descent data.
\medskip\noindent
Faithfulness. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent
sheaves on $S$ and let $a, b : \mathcal{F} \to \mathcal{G}$ be
homomorphisms of $\mathcal{O}_S$-modules.
Suppose $\varphi_i^*(a) = \varphi^*(b)$ for all $i$.
Pick $s \in S$. Then $s = \varphi_i(u)$ for some $i \in I$ and
$u \in U_i$. Since $\mathcal{O}_{S, s} \to \mathcal{O}_{U_i, u}$
is flat, hence faithfully flat
(Algebra, Lemma \ref{algebra-lemma-local-flat-ff}) we see
that $a_s = b_s : \mathcal{F}_s \to \mathcal{G}_s$. Hence $a = b$.
\medskip\noindent
Fully faithfulness. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent
sheaves on $S$ and let
$a_i : \varphi_i^*\mathcal{F} \to \varphi_i^*\mathcal{G}$ be
homomorphisms of $\mathcal{O}_{U_i}$-modules such that
$\text{pr}_0^*a_i = \text{pr}_1^*a_j$ on $U_i \times_U U_j$.
We can pull back these morphisms to get morphisms
$$
a_k :
\mathcal{F}|_{U_{\underline{i}(k), k}}
\longrightarrow
\mathcal{G}|_{U_{\underline{i}(k), k}}
$$
$k \in K_j$ with notation as above. Moreover,
Lemma \ref{lemma-refine-descent-datum} assures us
that these define a morphism between (canonical) descent data on
$\mathcal{V}_j$. Hence, by
Lemma \ref{lemma-standard-fpqc-covering}, we get correspondingly
unique morphisms $a_j : \mathcal{F}|_{V_j} \to \mathcal{G}|_{V_j}$.
To see that $a_j|_{V_{jj'}} = a_{j'}|_{V_{jj'}}$ we use that
both $a_j$ and $a_{j'}$ agree with the pullback of the morphism
$(a_i)_{i \in I}$ of (canonical) descent data to any covering
refining both $\mathcal{V}_{j, V_{jj'}}$ and
$\mathcal{V}_{j', V_{jj'}}$, and using the faithfulness already
shown. For example the covering
$\mathcal{V}_{jj'} =
\{V_k \times_S V_{k'} \to V_{jj'}\}_{k \in K_j, k' \in K_{j'}}$
will do.
\medskip\noindent
Essential surjectivity. Let $\xi = (\mathcal{F}_i, \varphi_{ii'})$
be a descent datum for quasi-coherent sheaves relative to the covering
$\mathcal{U}$. Pull back this descent datum to get descent data
$\xi_j$ for quasi-coherent sheaves relative to the coverings
$\mathcal{V}_j$ of $V_j$. By Lemma \ref{lemma-standard-fpqc-covering}
once again there exist
quasi-coherent sheaves $\mathcal{F}_j$ on $V_j$ whose associated
canonical descent datum is isomorphic to $\xi_j$. By fully faithfulness
(proved above) we see there are isomorphisms
$$
\phi_{jj'} :
\mathcal{F}_j|_{V_{jj'}}
\longrightarrow
\mathcal{F}_{j'}|_{V_{jj'}}
$$
corresponding to the isomorphism of descent data between the pullback
of $\xi_j$ and $\xi_{j'}$ to $\mathcal{V}_{jj'}$. To see that these
maps $\phi_{jj'}$ satisfy the cocycle condition we use faithfulness
(proved above) over the triple intersections $V_{jj'j''}$. Hence, by
Lemma \ref{lemma-zariski-descent-effective}
we see that the sheaves $\mathcal{F}_j$
glue to a quasi-coherent sheaf $\mathcal{F}$ as desired.
We still have to verify that the canonical descent datum relative to
$\mathcal{U}$ associated to $\mathcal{F}$ is isomorphic to the descent
datum we started out with. This verification is omitted.
\end{proof}
\section{Galois descent for quasi-coherent sheaves}
\label{section-galois-descent}
\noindent
Galois descent for quasi-coherent sheaves is just a special
case of fpqc descent for quasi-coherent sheaves. In this section
we will explain how to translate from a Galois descent to
an fpqc descent and then apply earlier results to conclude.
\medskip\noindent
Let $k'/k$ be a field extension. Then $\{\Spec(k') \to \Spec(k)\}$
is an fpqc covering. Let $X$ be a scheme over $k$. For a $k$-algebra
$A$ we set $X_A = X \times_{\Spec(k)} \Spec(A)$. By
Topologies, Lemma \ref{topologies-lemma-fpqc}
we see that $\{X_{k'} \to X\}$ is an fpqc covering. Observe that
$$
X_{k'} \times_X X_{k'} = X_{k' \otimes_k k'}
\quad\text{and}\quad
X_{k'} \times_X X_{k'} \times_X X_{k'} = X_{k' \otimes_k k' \otimes_k k'}
$$
Thus a descent datum for quasi-coherent sheaves with respect to
$\{X_{k'} \to X\}$ is given by a quasi-coherent sheaf $\mathcal{F}$
on $X_{k'}$, an isomorphism
$\varphi : \text{pr}_0^*\mathcal{F} \to \text{pr}_1^*\mathcal{F}$
on $X_{k' \otimes_k k'}$
which satisfies an obvious cocycle condition on
$X_{k' \otimes_k k' \otimes_k k'}$.
We will work out what this means in the case of a Galois extension below.
\medskip\noindent
Let $k'/k$ be a finite Galois extension with Galois group
$G = \text{Gal}(k'/k)$. Then there are $k$-algebra isomorphisms
$$
k' \otimes_k k' \longrightarrow \prod\nolimits_{\sigma \in G} k',\quad
a \otimes b \longrightarrow \prod a\sigma(b)
$$
and
$$
k' \otimes_k k' \otimes_k k' \longrightarrow
\prod\nolimits_{(\sigma, \tau) \in G \times G} k',\quad
a \otimes b \otimes c \longrightarrow \prod a\sigma(b)\sigma(\tau(c))
$$
The reason for choosing here $a\sigma(b)\sigma(\tau(c))$
and not $a\sigma(b)\tau(c)$ is that the formulas below simplify but
it isn't strictly necessary. Given $\sigma \in G$ we denote
$$
f_\sigma = \text{id}_X \times \Spec(\sigma) :
X_{k'} \longrightarrow X_{k'}
$$
Please keep in mind that because $\Spec(-)$ is a contravariant functor we have
$f_{\sigma \tau} = f_\tau \circ f_\sigma$ and not the other way around.
Using the first isomorphism above we obtain an identification
$$
X_{k' \otimes_k k'} = \coprod\nolimits_{\sigma \in G} X_{k'}
$$
such that $\text{pr}_0$ corresponds to the map
$$
\coprod\nolimits_{\sigma \in G} X_{k'}
\xrightarrow{\coprod \text{id}}
X_{k'}
$$
and such that $\text{pr}_1$ corresponds to the map
$$
\coprod\nolimits_{\sigma \in G} X_{k'}
\xrightarrow{\coprod f_\sigma}
X_{k'}
$$
Thus we see that a descent datum $\varphi$ on $\mathcal{F}$ over $X_{k'}$
corresponds to a family of isomorphisms
$\varphi_\sigma : \mathcal{F} \to f_\sigma^*\mathcal{F}$.
To work out the cocycle condition we use the identification
$$
X_{k' \otimes_k k' \otimes_k k'} =
\coprod\nolimits_{(\sigma, \tau) \in G \times G} X_{k'}.
$$
we get from our isomorphism of algebras above.
Via this identification the map $\text{pr}_{01}$ corresponds to
the map
$$
\coprod\nolimits_{(\sigma, \tau) \in G \times G} X_{k'}
\longrightarrow
\coprod\nolimits_{\sigma \in G} X_{k'}
$$
which maps the summand with index $(\sigma, \tau)$ to the summand
with index $\sigma$ via the identity morphism. The map $\text{pr}_{12}$
corresponds to the map
$$
\coprod\nolimits_{(\sigma, \tau) \in G \times G} X_{k'}
\longrightarrow
\coprod\nolimits_{\sigma \in G} X_{k'}
$$
which maps the summand with index $(\sigma, \tau)$ to the summand
with index $\tau$ via the morphism $f_\sigma$. Finally, the map
$\text{pr}_{02}$ corresponds to the map
$$
\coprod\nolimits_{(\sigma, \tau) \in G \times G} X_{k'}
\longrightarrow
\coprod\nolimits_{\sigma \in G} X_{k'}
$$
which maps the summand with index $(\sigma, \tau)$ to the summand
with index $\sigma\tau$ via the identity morphism.
Thus the cocycle condition
$$
\text{pr}_{02}^*\varphi = \text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi
$$
translates into one condition for each pair $(\sigma, \tau)$, namely
$$
\varphi_{\sigma\tau} = f_\sigma^*\varphi_\tau \circ \varphi_\sigma
$$
as maps $\mathcal{F} \to f_{\sigma\tau}^*\mathcal{F}$.
(Everything works out beautifully; for example the target of
$\varphi_\sigma$ is $f_\sigma^*\mathcal{F}$ and the
source of $f_\sigma^*\varphi_\tau$ is $f_\sigma^*\mathcal{F}$ as well.)
\begin{lemma}
\label{lemma-galois-descent}
Let $k'/k$ be a (finite) Galois extension with Galois group $G$.
Let $X$ be a scheme over $k$. The category of quasi-coherent
$\mathcal{O}_X$-modules is equivalent to the category of systems
$(\mathcal{F}, (\varphi_\sigma)_{\sigma \in G})$ where
\begin{enumerate}
\item $\mathcal{F}$ is a quasi-coherent module on $X_{k'}$,
\item $\varphi_\sigma : \mathcal{F} \to f_\sigma^*\mathcal{F}$
is an isomorphism of modules,
\item $\varphi_{\sigma\tau} = f_\sigma^*\varphi_\tau \circ \varphi_\sigma$
for all $\sigma, \tau \in G$.
\end{enumerate}
Here $f_\sigma = \text{id}_X \times \Spec(\sigma) : X_{k'} \to X_{k'}$.
\end{lemma}
\begin{proof}
As seen above a datum $(\mathcal{F}, (\varphi_\sigma)_{\sigma \in G})$
as in the lemma is the same thing as a descent datum for the
fpqc covering $\{X_{k'} \to X\}$. Thus the lemma follows from
Proposition \ref{proposition-fpqc-descent-quasi-coherent}.
\end{proof}
\noindent
A slightly more general case of the above is the following.
Suppose we have a surjective finite \'etale morphism $X \to Y$
and a finite group $G$ together with a group homomorphism
$G^{opp} \to \text{Aut}_Y(X), \sigma \mapsto f_\sigma$
such that the map
$$
G \times X \longrightarrow X \times_Y X,\quad
(\sigma, x) \longmapsto (x, f_\sigma(x))
$$
is an isomorphism. Then the same result as above holds.
\begin{lemma}
\label{lemma-galois-descent-more-general}
Let $X \to Y$, $G$, and $f_\sigma : X \to X$ be as above.
The category of quasi-coherent
$\mathcal{O}_Y$-modules is equivalent to the category of systems
$(\mathcal{F}, (\varphi_\sigma)_{\sigma \in G})$ where
\begin{enumerate}
\item $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module,
\item $\varphi_\sigma : \mathcal{F} \to f_\sigma^*\mathcal{F}$
is an isomorphism of modules,
\item $\varphi_{\sigma\tau} = f_\sigma^*\varphi_\tau \circ \varphi_\sigma$
for all $\sigma, \tau \in G$.
\end{enumerate}
\end{lemma}
\begin{proof}
Since $X \to Y$ is surjective finite \'etale $\{X \to Y\}$ is
an fpqc covering. Since
$G \times X \to X \times_Y X$, $(\sigma, x) \mapsto (x, f_\sigma(x))$
is an isomorphism, we see that
$G \times G \times X \to X \times_Y X \times_Y X$,
$(\sigma, \tau, x) \mapsto (x, f_\sigma(x), f_{\sigma\tau}(x))$
is an isomorphism too. Using these identifications, the category of
data as in the lemma is the same as the category of descent data
for quasi-coherent sheaves for the covering $\{x \to Y\}$.
Thus the lemma follows from
Proposition \ref{proposition-fpqc-descent-quasi-coherent}.
\end{proof}
\section{Descent of finiteness properties of modules}
\label{section-descent-finiteness}
\noindent
In this section we prove that one can check quasi-coherent module
has a certain finiteness conditions by checking on the members of
a covering.
\begin{lemma}
\label{lemma-finite-type-descends}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that
each $f_i^*\mathcal{F}$ is a finite type $\mathcal{O}_{X_i}$-module.
Then $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module.
\end{lemma}
\begin{proof}
Omitted. For the affine case, see
Algebra, Lemma \ref{algebra-lemma-descend-properties-modules}.
\end{proof}
\begin{lemma}
\label{lemma-finite-type-descends-fppf}
Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of
locally ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_Y$-modules.
If
\begin{enumerate}
\item $f$ is open as a map of topological spaces,
\item $f$ is surjective and flat, and
\item $f^*\mathcal{F}$ is of finite type,
\end{enumerate}
then $\mathcal{F}$ is of finite type.
\end{lemma}
\begin{proof}
Let $y \in Y$ be a point. Choose a point $x \in X$ mapping to $y$.
Choose an open $x \in U \subset X$ and elements $s_1, \ldots, s_n$
of $f^*\mathcal{F}(U)$ which generate $f^*\mathcal{F}$ over $U$.
Since $f^*\mathcal{F} =
f^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X$
we can after shrinking $U$ assume $s_i = \sum t_{ij} \otimes a_{ij}$
with $t_{ij} \in f^{-1}\mathcal{F}(U)$ and $a_{ij} \in \mathcal{O}_X(U)$.
After shrinking $U$ further we may assume that $t_{ij}$ comes from
a section $s_{ij} \in \mathcal{F}(V)$ for some $V \subset Y$ open
with $f(U) \subset V$. Let $N$ be the number of sections $s_{ij}$ and
consider the map
$$
\sigma = (s_{ij}) : \mathcal{O}_V^{\oplus N} \to \mathcal{F}|_V
$$
By our choice of the sections we see that $f^*\sigma|_U$ is surjective.
Hence for every $u \in U$ the map
$$
\sigma_{f(u)} \otimes_{\mathcal{O}_{Y, f(u)}} \mathcal{O}_{X, u} :
\mathcal{O}_{X, u}^{\oplus N}
\longrightarrow
\mathcal{F}_{f(u)} \otimes_{\mathcal{O}_{Y, f(u)}} \mathcal{O}_{X, u}
$$
is surjective. As $f$ is flat, the local ring map
$\mathcal{O}_{Y, f(u)} \to \mathcal{O}_{X, u}$ is flat, hence
faithfully flat (Algebra, Lemma \ref{algebra-lemma-local-flat-ff}).
Hence $\sigma_{f(u)}$ is surjective. Since $f$ is open, $f(U)$ is
an open neighbourhood of $y$ and the proof is done.
\end{proof}
\begin{lemma}
\label{lemma-finite-presentation-descends}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that
each $f_i^*\mathcal{F}$ is an $\mathcal{O}_{X_i}$-module of finite
presentation. Then $\mathcal{F}$ is an $\mathcal{O}_X$-module
of finite presentation.
\end{lemma}
\begin{proof}
Omitted. For the affine case, see
Algebra, Lemma \ref{algebra-lemma-descend-properties-modules}.
\end{proof}
\begin{lemma}
\label{lemma-locally-generated-by-r-sections-descends}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that
each $f_i^*\mathcal{F}$ is locally generated by $r$ sections as an
$\mathcal{O}_{X_i}$-module. Then $\mathcal{F}$ is locally generated by
$r$ sections as an $\mathcal{O}_X$-module.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-finite-type-descends} we see that $\mathcal{F}$
is of finite type. Hence Nakayama's lemma
(Algebra, Lemma \ref{algebra-lemma-NAK}) implies that $\mathcal{F}$
is generated by $r$ sections in the neighbourhood of a point $x \in X$
if and only if $\dim_{\kappa(x)} \mathcal{F}_x \otimes \kappa(x) \leq r$.
Choose an $i$ and a point $x_i \in X_i$ mapping to $x$. Then
$\dim_{\kappa(x)} \mathcal{F}_x \otimes \kappa(x) =
\dim_{\kappa(x_i)} (f_i^*\mathcal{F})_{x_i} \otimes \kappa(x_i)$
which is $\leq r$ as $f_i^*\mathcal{F}$ is locally generated by $r$
sections.
\end{proof}
\begin{lemma}
\label{lemma-flat-descends}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that
each $f_i^*\mathcal{F}$ is a flat $\mathcal{O}_{X_i}$-module.
Then $\mathcal{F}$ is a flat $\mathcal{O}_X$-module.
\end{lemma}
\begin{proof}
Omitted. For the affine case, see
Algebra, Lemma \ref{algebra-lemma-descend-properties-modules}.
\end{proof}
\begin{lemma}
\label{lemma-finite-locally-free-descends}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that
each $f_i^*\mathcal{F}$ is a finite locally free $\mathcal{O}_{X_i}$-module.
Then $\mathcal{F}$ is a finite locally free $\mathcal{O}_X$-module.
\end{lemma}
\begin{proof}
This follows from the fact that a quasi-coherent sheaf is finite locally
free if and only if it is of finite presentation and flat, see
Algebra, Lemma \ref{algebra-lemma-finite-projective}.
Namely, if each $f_i^*\mathcal{F}$ is flat and of finite presentation,
then so is $\mathcal{F}$ by
Lemmas \ref{lemma-flat-descends} and
\ref{lemma-finite-presentation-descends}.
\end{proof}
\noindent
The definition of a locally projective quasi-coherent sheaf can be found in
Properties, Section \ref{properties-section-locally-projective}.
\begin{lemma}
\label{lemma-locally-projective-descends}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that
each $f_i^*\mathcal{F}$ is a locally projective $\mathcal{O}_{X_i}$-module.
Then $\mathcal{F}$ is a locally projective $\mathcal{O}_X$-module.
\end{lemma}
\begin{proof}
Omitted. For Zariski coverings this is
Properties, Lemma \ref{properties-lemma-locally-projective}.
For the affine case this is
Algebra, Theorem \ref{algebra-theorem-ffdescent-projectivity}.
\end{proof}
\begin{remark}
\label{remark-locally-free-descends}
Being locally free is a property of quasi-coherent modules which
does not descend in the fpqc topology. Namely, suppose that
$R$ is a ring and that $M$ is a projective $R$-module which is
a countable direct sum $M = \bigoplus L_n$ of rank 1 locally
free modules, but not locally free, see
Examples, Lemma \ref{examples-lemma-projective-not-locally-free}.
Then $M$ becomes free on making the faithfully flat base change
$$
R \longrightarrow
\bigoplus\nolimits_{m \geq 1}
\bigoplus\nolimits_{(i_1, \ldots, i_m) \in \mathbf{Z}^{\oplus m}}
L_1^{\otimes i_1} \otimes_R \ldots \otimes_R L_m^{\otimes i_m}
$$
But we don't know what happens for fppf coverings. In other words,
we don't know the answer to the following question:
Suppose $A \to B$ is a faithfully
flat ring map of finite presentation. Let $M$ be an $A$-module
such that $M \otimes_A B$ is free. Is $M$ a locally free
$A$-module? It turns out that if $A$ is Noetherian, then the answer
is yes. This follows from the results of \cite{Bass}. But in general
we don't know the answer. If you know the answer, or have a reference,
please email
\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.
\end{remark}
\noindent
We also add here two results which are related to the results above, but
are of a slightly different nature.
\begin{lemma}
\label{lemma-finite-over-finite-module}
Let $f : X \to Y$ be a morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Assume $f$ is a finite morphism.
Then $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite type
if and only if $f_*\mathcal{F}$ is an $\mathcal{O}_Y$-module of finite
type.
\end{lemma}
\begin{proof}
As $f$ is finite it is affine. This reduces us to the case where
$f$ is the morphism $\Spec(B) \to \Spec(A)$ given
by a finite ring map $A \to B$.
Moreover, then $\mathcal{F} = \widetilde{M}$ is the sheaf of modules
associated to the $B$-module $M$.
Note that $M$ is finite as a $B$-module if and only if
$M$ is finite as an $A$-module, see
Algebra, Lemma \ref{algebra-lemma-finite-module-over-finite-extension}.
Combined with
Properties, Lemma \ref{properties-lemma-finite-type-module}
this proves the lemma.
\end{proof}
\begin{lemma}
\label{lemma-finite-finitely-presented-module}
Let $f : X \to Y$ be a morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Assume $f$ is finite and of finite presentation.
Then $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation
if and only if $f_*\mathcal{F}$ is an $\mathcal{O}_Y$-module of finite
presentation.
\end{lemma}
\begin{proof}
As $f$ is finite it is affine. This reduces us to the case where
$f$ is the morphism $\Spec(B) \to \Spec(A)$ given
by a finite and finitely presented ring map $A \to B$.
Moreover, then $\mathcal{F} = \widetilde{M}$ is the sheaf of modules
associated to the $B$-module $M$.
Note that $M$ is finitely presented as a $B$-module if and only if
$M$ is finitely presented as an $A$-module, see
Algebra, Lemma \ref{algebra-lemma-finite-finitely-presented-extension}.
Combined with
Properties, Lemma \ref{properties-lemma-finite-presentation-module}
this proves the lemma.
\end{proof}
\section{Quasi-coherent sheaves and topologies}
\label{section-quasi-coherent-sheaves}
\noindent
Let $S$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module.
Consider the functor
\begin{equation}
\label{equation-quasi-coherent-presheaf}
(\Sch/S)^{opp} \longrightarrow \textit{Ab},
\quad
(f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F}).
\end{equation}
\begin{lemma}
\label{lemma-sheaf-condition-holds}
Let $S$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module.
Let $\tau \in \{Zariski, \linebreak[0] fpqc, \linebreak[0] fppf, \linebreak[0]
\etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
The functor defined in (\ref{equation-quasi-coherent-presheaf})
satisfies the sheaf condition with respect to any $\tau$-covering
$\{T_i \to T\}_{i \in I}$ of any scheme $T$ over $S$.
\end{lemma}
\begin{proof}
For $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale,
\linebreak[0] smooth, \linebreak[0] syntomic\}$ a $\tau$-covering
is also a fpqc-covering, see the results in
Topologies, Lemmas
\ref{topologies-lemma-zariski-etale},
\ref{topologies-lemma-zariski-etale-smooth},
\ref{topologies-lemma-zariski-etale-smooth-syntomic},
\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf}, and
\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}.
Hence it suffices to prove the theorem
for a fpqc covering. Assume that $\{f_i : T_i \to T\}_{i \in I}$
is an fpqc covering where $f : T \to S$ is given. Suppose that
we have a family of sections $s_i \in \Gamma(T_i , f_i^*f^*\mathcal{F})$
such that $s_i|_{T_i \times_T T_j} = s_j|_{T_i \times_T T_j}$.
We have to find the correspond section $s \in \Gamma(T, f^*\mathcal{F})$.
We can reinterpret the $s_i$ as a family of maps
$\varphi_i : f_i^*\mathcal{O}_T = \mathcal{O}_{T_i} \to f_i^*f^*\mathcal{F}$
compatible with the canonical descent data associated to the
quasi-coherent sheaves $\mathcal{O}_T$ and $f^*\mathcal{F}$ on $T$.
Hence by Proposition \ref{proposition-fpqc-descent-quasi-coherent}
we see that we may (uniquely) descend
these to a map $\mathcal{O}_T \to f^*\mathcal{F}$ which gives
us our section $s$.
\end{proof}
\noindent
We may in particular make the following definition.
\begin{definition}
\label{definition-structure-sheaf}
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0]
\etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
Let $S$ be a scheme.
Let $\Sch_\tau$ be a big site containing $S$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module.
\begin{enumerate}
\item The {\it structure sheaf of the big site $(\Sch/S)_\tau$}
is the sheaf of rings $T/S \mapsto \Gamma(T, \mathcal{O}_T)$ which is
denoted $\mathcal{O}$ or $\mathcal{O}_S$.
\item If $\tau = \etale$ the structure sheaf of the small site
$S_\etale$ is the sheaf of rings $T/S \mapsto \Gamma(T, \mathcal{O}_T)$
which is denoted $\mathcal{O}$ or $\mathcal{O}_S$.
\item The {\it sheaf of $\mathcal{O}$-modules associated to
$\mathcal{F}$} on the big site $(\Sch/S)_\tau$
is the sheaf of $\mathcal{O}$-modules
$(f : T \to S) \mapsto \Gamma(T, f^*\mathcal{F})$
which is denoted $\mathcal{F}^a$ (and often simply $\mathcal{F}$).
\item Let $\tau = \etale$ (resp.\ $\tau = Zariski$). The
{\it sheaf of $\mathcal{O}$-modules associated to $\mathcal{F}$}
on the small site $S_\etale$ (resp.\ $S_{Zar}$) is the sheaf of
$\mathcal{O}$-modules $(f : T \to S) \mapsto \Gamma(T, f^*\mathcal{F})$
which is denoted $\mathcal{F}^a$ (and often simply $\mathcal{F}$).
\end{enumerate}
\end{definition}
\noindent
Note how we use the same notation $\mathcal{F}^a$ in each case.
No confusion can really arise from this as by definition the rule
that defines the sheaf $\mathcal{F}^a$ is independent of the site
we choose to look at.
\begin{remark}
\label{remark-Zariski-site-space}
In Topologies, Lemma \ref{topologies-lemma-Zariski-usual}
we have seen that the small Zariski site of a scheme $S$ is
equivalent to $S$ as a topological space in the sense that the
category of sheaves are naturally equivalent. Now that $S_{Zar}$
is also endowed with a structure sheaf $\mathcal{O}$ we see
that sheaves of modules on the ringed site $(S_{Zar}, \mathcal{O})$
agree with sheaves of modules on the ringed space $(S, \mathcal{O}_S)$.
\end{remark}
\begin{remark}
\label{remark-change-topologies-ringed}
Let $f : T \to S$ be a morphism of schemes.
Each of the morphisms of sites $f_{sites}$ listed in
Topologies, Section \ref{topologies-section-change-topologies}
becomes a morphism of ringed sites. Namely, each of these morphisms of sites
$f_{sites} : (\Sch/T)_\tau \to (\Sch/S)_{\tau'}$, or
$f_{sites} : (\Sch/S)_\tau \to S_{\tau'}$ is given by the continuous
functor $S'/S \mapsto T \times_S S'/S$. Hence, given $S'/S$ we let
$$
f_{sites}^\sharp :
\mathcal{O}(S'/S)
\longrightarrow
f_{sites, *}\mathcal{O}(S'/S) =
\mathcal{O}(S \times_S S'/T)
$$
be the usual map
$\text{pr}_{S'}^\sharp : \mathcal{O}(S') \to \mathcal{O}(T \times_S S')$.
Similarly, the morphism
$i_f : \Sh(T_\tau) \to \Sh((\Sch/S)_\tau)$
for $\tau \in \{Zar, \etale\}$, see
Topologies, Lemmas \ref{topologies-lemma-put-in-T} and
\ref{topologies-lemma-put-in-T-etale},
becomes a morphism of ringed topoi because $i_f^{-1}\mathcal{O} = \mathcal{O}$.
Here are some special cases:
\begin{enumerate}
\item The morphism of big sites
$f_{big} : (\Sch/X)_{fppf} \to (\Sch/Y)_{fppf}$,
becomes a morphism of ringed sites
$$
(f_{big}, f_{big}^\sharp) :
((\Sch/X)_{fppf}, \mathcal{O}_X)
\longrightarrow
((\Sch/Y)_{fppf}, \mathcal{O}_Y)
$$
as in Modules on Sites, Definition \ref{sites-modules-definition-ringed-site}.
Similarly for the big syntomic, smooth, \'etale and Zariski sites.
\item The morphism of small sites
$f_{small} : X_\etale \to Y_\etale$
becomes a morphism of ringed sites
$$
(f_{small}, f_{small}^\sharp) :
(X_\etale, \mathcal{O}_X)
\longrightarrow
(Y_\etale, \mathcal{O}_Y)
$$
as in Modules on Sites, Definition \ref{sites-modules-definition-ringed-site}.
Similarly for the small Zariski site.
\end{enumerate}
\end{remark}
\noindent
Let $S$ be a scheme. It is clear that given an $\mathcal{O}$-module on (say)
$(\Sch/S)_{Zar}$ the pullback to (say) $(\Sch/S)_{fppf}$
is just the fppf-sheafification. To see what happens when comparing
big and small sites we have the following.
\begin{lemma}
\label{lemma-compare-sites}
Let $S$ be a scheme. Denote
$$
\begin{matrix}
\text{id}_{\tau, Zar} & : & (\Sch/S)_\tau \to S_{Zar}, &
\tau \in \{Zar, \etale, smooth, syntomic, fppf\} \\
\text{id}_{\tau, \etale} & : &
(\Sch/S)_\tau \to S_\etale, &
\tau \in \{\etale, smooth, syntomic, fppf\} \\
\text{id}_{small, \etale, Zar} & : & S_\etale \to S_{Zar},
\end{matrix}
$$
the morphisms of ringed sites of
Remark \ref{remark-change-topologies-ringed}.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_S$-modules
which we view a sheaf of $\mathcal{O}$-modules on $S_{Zar}$. Then
\begin{enumerate}
\item $(\text{id}_{\tau, Zar})^*\mathcal{F}$ is the $\tau$-sheafification
of the Zariski sheaf
$$
(f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F})
$$
on $(\Sch/S)_\tau$, and
\item $(\text{id}_{small, \etale, Zar})^*\mathcal{F}$ is the
\'etale sheafification of the Zariski sheaf
$$
(f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F})
$$
on $S_\etale$.
\end{enumerate}
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules
on $S_\etale$. Then
\begin{enumerate}
\item[(3)] $(\text{id}_{\tau, \etale})^*\mathcal{G}$ is the
$\tau$-sheafification of the \'etale sheaf
$$
(f : T \to S) \longmapsto \Gamma(T, f_{small}^*\mathcal{G})
$$
where $f_{small} : T_\etale \to S_\etale$
is the morphism of ringed small \'etale sites of
Remark \ref{remark-change-topologies-ringed}.
\end{enumerate}
\end{lemma}
\begin{proof}
Proof of (1). We first note that the result is true when $\tau = Zar$
because in that case we have the morphism of topoi
$i_f : \Sh(T_{Zar}) \to \Sh(\Sch/S)_{Zar})$
such that $\text{id}_{\tau, Zar} \circ i_f = f_{small}$ as morphisms
$T_{Zar} \to S_{Zar}$, see
Topologies, Lemmas \ref{topologies-lemma-put-in-T} and
\ref{topologies-lemma-morphism-big-small}.
Since pullback is transitive (see
Modules on Sites,
Lemma \ref{sites-modules-lemma-push-pull-composition-modules})
we see that
$i_f^*(\text{id}_{\tau, Zar})^*\mathcal{F} = f_{small}^*\mathcal{F}$
as desired. Hence, by the remark preceding this lemma we see that
$(\text{id}_{\tau, Zar})^*\mathcal{F}$ is the $\tau$-sheafification of
the presheaf $T \mapsto \Gamma(T, f^*\mathcal{F})$.
\medskip\noindent
The proof of (3) is exactly the same as the proof of (1), except that it
uses
Topologies, Lemmas \ref{topologies-lemma-put-in-T-etale} and
\ref{topologies-lemma-morphism-big-small-etale}.
We omit the proof of (2).
\end{proof}
\begin{remark}
\label{remark-change-topologies-ringed-sites}
Remark \ref{remark-change-topologies-ringed}
and
Lemma \ref{lemma-compare-sites}
have the following applications:
\begin{enumerate}
\item Let $S$ be a scheme.
The construction $\mathcal{F} \mapsto \mathcal{F}^a$ is
the pullback under the morphism of ringed sites
$\text{id}_{\tau, Zar} : ((\Sch/S)_\tau, \mathcal{O})
\to (S_{Zar}, \mathcal{O})$
or the morphism
$\text{id}_{small, \etale, Zar} :
(S_\etale, \mathcal{O}) \to (S_{Zar}, \mathcal{O})$.
\item Let $f : X \to Y$ be a morphism of schemes.
For any of the morphisms $f_{sites}$ of ringed sites of
Remark \ref{remark-change-topologies-ringed}
we have
$$
(f^*\mathcal{F})^a = f_{sites}^*\mathcal{F}^a.
$$
This follows from (1) and the fact that pullbacks are compatible with
compositions of morphisms of ringed sites, see
Modules on Sites,
Lemma \ref{sites-modules-lemma-push-pull-composition-modules}.
\end{enumerate}
\end{remark}
\begin{lemma}
\label{lemma-quasi-coherent-gives-quasi-coherent}
Let $S$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module.
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0]
\etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
\begin{enumerate}
\item The sheaf $\mathcal{F}^a$ is a quasi-coherent
$\mathcal{O}$-module on $(\Sch/S)_\tau$, as defined in
Modules on Sites, Definition \ref{sites-modules-definition-site-local}.
\item If $\tau = \etale$ (resp.\ $\tau = Zariski$), then the sheaf
$\mathcal{F}^a$ is a quasi-coherent $\mathcal{O}$-module on
$S_\etale$ (resp.\ $S_{Zar}$) as defined in
Modules on Sites, Definition \ref{sites-modules-definition-site-local}.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\{S_i \to S\}$ be a Zariski covering such that we have exact sequences
$$
\bigoplus\nolimits_{k \in K_i} \mathcal{O}_{S_i} \longrightarrow
\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{S_i} \longrightarrow
\mathcal{F} \longrightarrow 0
$$
for some index sets $K_i$ and $J_i$. This is possible by the definition
of a quasi-coherent sheaf on a ringed space
(See Modules, Definition \ref{modules-definition-quasi-coherent}).
\medskip\noindent
Proof of (1). Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0]
\etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
It is clear that $\mathcal{F}^a|_{(\Sch/S_i)_\tau}$ also
sits in an exact sequence
$$
\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau}
\longrightarrow
\bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau}
\longrightarrow
\mathcal{F}^a|_{(\Sch/S_i)_\tau} \longrightarrow 0
$$
Hence $\mathcal{F}^a$ is quasi-coherent by Modules on Sites,
Lemma \ref{sites-modules-lemma-local-final-object}.
\medskip\noindent
Proof of (2). Let $\tau = \etale$.
It is clear that $\mathcal{F}^a|_{(S_i)_\etale}$ also sits
in an exact sequence
$$
\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(S_i)_\etale}
\longrightarrow
\bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(S_i)_\etale}
\longrightarrow
\mathcal{F}^a|_{(S_i)_\etale} \longrightarrow 0
$$
Hence $\mathcal{F}^a$ is quasi-coherent by Modules on Sites,
Lemma \ref{sites-modules-lemma-local-final-object}.
The case $\tau = Zariski$ is similar (actually, it is really
tautological since the corresponding ringed topoi agree).
\end{proof}
\begin{lemma}
\label{lemma-standard-covering-Cech}
Let $S$ be a scheme. Let
\begin{enumerate}
\item[(a)] $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0]
\etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$
and $\mathcal{C} = (\Sch/S)_\tau$, or
\item[(b)] let $\tau = \etale$ and $\mathcal{C} = S_\etale$, or
\item[(c)] let $\tau = Zariski$ and $\mathcal{C} = S_{Zar}$.
\end{enumerate}
Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$.
Let $U \in \Ob(\mathcal{C})$ be affine.
Let $\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}$ be a standard affine
$\tau$-covering in $\mathcal{C}$. Then
\begin{enumerate}
\item $\mathcal{V} = \{\coprod_{i = 1, \ldots, n} U_i \to U\}$ is a
$\tau$-covering of $U$,
\item $\mathcal{U}$ is a refinement of $\mathcal{V}$, and
\item the induced map on {\v C}ech complexes
(Cohomology on Sites,
Equation (\ref{sites-cohomology-equation-map-cech-complexes}))
$$
\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F})
\longrightarrow
\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})
$$
is an isomorphism of complexes.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows because
$$
(\coprod\nolimits_{i_0 = 1, \ldots, n} U_{i_0}) \times_U
\ldots \times_U
(\coprod\nolimits_{i_p = 1, \ldots, n} U_{i_p})
=
\coprod\nolimits_{i_0, \ldots, i_p \in \{1, \ldots, n\}}
U_{i_0} \times_U \ldots \times_U U_{i_p}
$$
and the fact that $\mathcal{F}(\coprod_a V_a) = \prod_a \mathcal{F}(V_a)$
since disjoint unions are $\tau$-coverings.
\end{proof}
\begin{lemma}
\label{lemma-standard-covering-Cech-quasi-coherent}
Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$.
Let $\tau$, $\mathcal{C}$, $U$, $\mathcal{U}$ be as in
Lemma \ref{lemma-standard-covering-Cech}. Then there is an isomorphism
of complexes
$$
\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^a)
\cong
s((A/R)_\bullet \otimes_R M)
$$
(see Section \ref{section-descent-modules})
where $R = \Gamma(U, \mathcal{O}_U)$, $M = \Gamma(U, \mathcal{F}^a)$
and $R \to A$ is a faithfully flat ring map. In particular
$$
\check{H}^p(\mathcal{U}, \mathcal{F}^a) = 0
$$
for all $p \geq 1$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-standard-covering-Cech} we see that
$\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^a)$
is isomorphic to $\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}^a)$
where $\mathcal{V} = \{V \to U\}$ with $V = \coprod_{i = 1, \ldots n} U_i$
affine also. Set $A = \Gamma(V, \mathcal{O}_V)$. Since $\{V \to U\}$
is a $\tau$-covering we see that $R \to A$ is faithfully flat.
On the other hand, by definition of $\mathcal{F}^a$ we have
that the degree $p$ term $\check{\mathcal{C}}^p(\mathcal{V}, \mathcal{F}^a)$
is
$$
\Gamma(V \times_U \ldots \times_U V, \mathcal{F}^a)
=
\Gamma(\Spec(A \otimes_R \ldots \otimes_R A), \mathcal{F}^a)
=
A \otimes_R \ldots \otimes_R A \otimes_R M
$$
We omit the verification that the maps of the {\v C}ech complex agree with
the maps in the complex $s((A/R)_\bullet \otimes_R M)$. The vanishing
of cohomology is Lemma \ref{lemma-ff-exact}.
\end{proof}
\begin{proposition}
\label{proposition-same-cohomology-quasi-coherent}
\begin{slogan}
Cohomology of quasi-coherent sheaves is the same no matter which
topology you use.
\end{slogan}
Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$.
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0]
\etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
\begin{enumerate}
\item There is a canonical isomorphism
$$
H^q(S, \mathcal{F}) = H^q((\Sch/S)_\tau, \mathcal{F}^a).
$$
\item There are canonical isomorphisms
$$
H^q(S, \mathcal{F}) =
H^q(S_{Zar}, \mathcal{F}^a) =
H^q(S_\etale, \mathcal{F}^a).
$$
\end{enumerate}
\end{proposition}
\begin{proof}
The result for $q = 0$ is clear from the definition of $\mathcal{F}^a$.
Let $\mathcal{C} = (\Sch/S)_\tau$, or $\mathcal{C} = S_\etale$,
or $\mathcal{C} = S_{Zar}$.
\medskip\noindent
We are going to apply
Cohomology on Sites,
Lemma \ref{sites-cohomology-lemma-cech-vanish-collection}
with $\mathcal{F} = \mathcal{F}^a$,
$\mathcal{B} \subset \Ob(\mathcal{C})$ the set of affine schemes
in $\mathcal{C}$, and $\text{Cov} \subset \text{Cov}_\mathcal{C}$ the
set of standard affine $\tau$-coverings. Assumption (3) of
the lemma is satisfied by
Lemma \ref{lemma-standard-covering-Cech-quasi-coherent}.
Hence we conclude that $H^p(U, \mathcal{F}^a) = 0$ for every
affine object $U$ of $\mathcal{C}$.
\medskip\noindent
Next, let $U \in \Ob(\mathcal{C})$ be any separated object.
Denote $f : U \to S$ the structure morphism.
Let $U = \bigcup U_i$ be an affine open covering.
We may also think of this as a $\tau$-covering
$\mathcal{U} = \{U_i \to U\}$ of $U$ in $\mathcal{C}$.
Note that
$U_{i_0} \times_U \ldots \times_U U_{i_p} =
U_{i_0} \cap \ldots \cap U_{i_p}$ is affine as we assumed $U$ separated.
By
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-cech-spectral-sequence-application}
and the result above we see that
$$
H^p(U, \mathcal{F}^a) = \check{H}^p(\mathcal{U}, \mathcal{F}^a)
= H^p(U, f^*\mathcal{F})
$$
the last equality by
Cohomology of Schemes, Lemma
\ref{coherent-lemma-cech-cohomology-quasi-coherent}.
In particular, if $S$ is separated we can take $U = S$ and
$f = \text{id}_S$ and the proposition is proved.
We suggest the reader skip the rest of the proof (or rewrite it
to give a clearer exposition).
\medskip\noindent
Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ on $S$.
Choose an injective resolution $\mathcal{F}^a \to \mathcal{J}^\bullet$
on $\mathcal{C}$. Denote $\mathcal{J}^n|_S$ the restriction of $\mathcal{J}^n$
to opens of $S$; this is a sheaf on the topological space $S$ as open
coverings are $\tau$-coverings. We get a complex
$$
0 \to \mathcal{F} \to \mathcal{J}^0|_S \to \mathcal{J}^1|_S \to \ldots
$$
which is exact since its sections over any affine open $U \subset S$
is exact (by the vanishing of $H^p(U, \mathcal{F}^a)$, $p > 0$ seen
above). Hence by
Derived Categories, Lemma \ref{derived-lemma-morphisms-lift}
there exists map of complexes
$\mathcal{J}^\bullet|_S \to \mathcal{I}^\bullet$ which in particular
induces a map
$$
R\Gamma(\mathcal{C}, \mathcal{F}^a)
=
\Gamma(S, \mathcal{J}^\bullet)
\longrightarrow
\Gamma(S, \mathcal{I}^\bullet)
=
R\Gamma(S, \mathcal{F}).
$$
Taking cohomology gives the map
$H^n(\mathcal{C}, \mathcal{F}^a) \to H^n(S, \mathcal{F})$ which
we have to prove is an isomorphism.
Let $\mathcal{U} : S = \bigcup U_i$ be an affine open covering
which we may think of as a $\tau$-covering also.
By the above we get a map of double complexes
$$
\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{J})
=
\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{J}|_S)
\longrightarrow
\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}).
$$
This map induces a map of spectral sequences
$$
{}^\tau\! E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}^a))
\longrightarrow
E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))
$$
The first spectral sequence converges to
$H^{p + q}(\mathcal{C}, \mathcal{F})$ and the second to
$H^{p + q}(S, \mathcal{F})$. On the other hand, we have seen
that the induced maps ${}^\tau\! E_2^{p, q} \to E_2^{p, q}$ are
bijections (as all the intersections are separated being opens in affines).
Whence also the maps $H^n(\mathcal{C}, \mathcal{F}^a) \to H^n(S, \mathcal{F})$
are isomorphisms, and we win.
\end{proof}
\begin{proposition}
\label{proposition-equivalence-quasi-coherent}
Let $S$ be a scheme.
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0]
\etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
\begin{enumerate}
\item The functor $\mathcal{F} \mapsto \mathcal{F}^a$
defines an equivalence of categories
$$
\QCoh(\mathcal{O}_S)
\longrightarrow
\QCoh((\Sch/S)_\tau, \mathcal{O})
$$
between the category of quasi-coherent sheaves on $S$ and the category
of quasi-coherent $\mathcal{O}$-modules on the big $\tau$ site of $S$.
\item Let $\tau = \etale$, or $\tau = Zariski$.
The functor $\mathcal{F} \mapsto \mathcal{F}^a$
defines an equivalence of categories
$$
\QCoh(\mathcal{O}_S)
\longrightarrow
\QCoh(S_\tau, \mathcal{O})
$$
between the category of quasi-coherent sheaves on $S$ and the category
of quasi-coherent $\mathcal{O}$-modules on the small $\tau$ site of $S$.
\end{enumerate}
\end{proposition}
\begin{proof}
We have seen in Lemma \ref{lemma-quasi-coherent-gives-quasi-coherent}
that the functor is well defined.
It is straightforward to show that the functor is fully faithful (we omit
the verification). To finish the proof we will show that a
quasi-coherent $\mathcal{O}$-module on $(\Sch/S)_\tau$ gives
rise to a descent datum for quasi-coherent sheaves relative to a
$\tau$-covering of $S$. Having produced this descent datum we will appeal
to Proposition \ref{proposition-fpqc-descent-quasi-coherent} to get the
corresponding quasi-coherent sheaf on $S$.
\medskip\noindent
Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}$-modules on
the big $\tau$ site of $S$. By
Modules on Sites, Definition \ref{sites-modules-definition-site-local}
there exists a $\tau$-covering $\{S_i \to S\}_{i \in I}$ of $S$
such that each of the restrictions
$\mathcal{G}|_{(\Sch/S_i)_\tau}$ has a global presentation
$$
\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau}
\longrightarrow
\bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau}
\longrightarrow
\mathcal{G}|_{(\Sch/S_i)_\tau} \longrightarrow 0
$$
for some index sets $J_i$ and $K_i$. We claim that this implies
that $\mathcal{G}|_{(\Sch/S_i)_\tau}$ is $\mathcal{F}_i^a$
for some quasi-coherent sheaf $\mathcal{F}_i$ on $S_i$. Namely,
this is clear for the direct sums
$\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau}$
and
$\bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau}$.
Hence we see that $\mathcal{G}|_{(\Sch/S_i)_\tau}$ is a
cokernel of a map $\varphi : \mathcal{K}_i^a \to \mathcal{L}_i^a$
for some quasi-coherent sheaves $\mathcal{K}_i$, $\mathcal{L}_i$
on $S_i$. By the fully faithfulness of $(\ )^a$ we see that
$\varphi = \phi^a$ for some map of quasi-coherent sheaves
$\phi : \mathcal{K}_i \to \mathcal{L}_i$ on $S_i$. Then it is
clear that
$\mathcal{G}|_{(\Sch/S_i)_\tau} \cong \Coker(\phi)^a$
as claimed.
\medskip\noindent
Since $\mathcal{G}$ lives on all of the category
$(\Sch/S_i)_\tau$ we see that
$$
(\text{pr}_0^*\mathcal{F}_i)^a
\cong
\mathcal{G}|_{(\Sch/(S_i \times_S S_j))_\tau}
\cong
(\text{pr}_1^*\mathcal{F})^a
$$
as $\mathcal{O}$-modules on $(\Sch/(S_i \times_S S_j))_\tau$.
Hence, using fully faithfulness again we get canonical isomorphisms
$$
\phi_{ij} :
\text{pr}_0^*\mathcal{F}_i
\longrightarrow
\text{pr}_1^*\mathcal{F}_j
$$
of quasi-coherent modules over $S_i \times_S S_j$. We omit the verification
that these satisfy the cocycle condition. Since they do we see by
effectivity of descent for quasi-coherent sheaves and the covering
$\{S_i \to S\}$ (Proposition \ref{proposition-fpqc-descent-quasi-coherent})
that there exists a quasi-coherent sheaf $\mathcal{F}$ on $S$
with $\mathcal{F}|_{S_i} \cong \mathcal{F}_i$ compatible
with the given descent data. In other words we are given
$\mathcal{O}$-module isomorphisms
$$
\phi_i :
\mathcal{F}^a|_{(\Sch/S_i)_\tau}
\longrightarrow
\mathcal{G}|_{(\Sch/S_i)_\tau}
$$
which agree over $S_i \times_S S_j$. Hence, since
$\SheafHom_\mathcal{O}(\mathcal{F}^a, \mathcal{G})$ is
a sheaf (Modules on Sites, Lemma \ref{sites-modules-lemma-internal-hom}),
we conclude that
there is a morphism of $\mathcal{O}$-modules $\mathcal{F}^a \to \mathcal{G}$
recovering the isomorphisms $\phi_i$ above. Hence this is an isomorphism
and we win.
\medskip\noindent
The case of the sites $S_\etale$ and $S_{Zar}$ is proved in the
exact same manner.
\end{proof}
\begin{lemma}
\label{lemma-equivalence-quasi-coherent-properties}
Let $S$ be a scheme.
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0]
\etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
Let $\mathcal{P}$ be one of the properties of modules\footnote{The list is:
free, finite free, generated by global sections,
generated by $r$ global sections, generated by finitely many global sections,
having a global presentation, having a global finite presentation,
locally free, finite locally free, locally generated by sections,
locally generated by $r$ sections, finite type, of finite presentation,
coherent, or flat.} defined in
Modules on Sites, Definitions \ref{sites-modules-definition-global},
\ref{sites-modules-definition-site-local}, and
\ref{sites-modules-definition-flat}.
The equivalences of categories
$$
\QCoh(\mathcal{O}_S)
\longrightarrow
\QCoh((\Sch/S)_\tau, \mathcal{O})
\quad\text{and}\quad
\QCoh(\mathcal{O}_S)
\longrightarrow
\QCoh(S_\tau, \mathcal{O})
$$
defined by the rule $\mathcal{F} \mapsto \mathcal{F}^a$ seen in
Proposition \ref{proposition-equivalence-quasi-coherent}
have the property
$$
\mathcal{F}\text{ has }\mathcal{P}
\Leftrightarrow
\mathcal{F}^a\text{ has }\mathcal{P}\text{ as an }\mathcal{O}\text{-module}
$$
except (possibly) when $\mathcal{P}$ is ``locally free'' or ``coherent''.
If $\mathcal{P}=$``coherent'' the equivalence
holds for $\QCoh(\mathcal{O}_S) \to \QCoh(S_\tau, \mathcal{O})$
when $S$ is locally Noetherian and $\tau$ is Zariski or \'etale.
\end{lemma}
\begin{proof}
This is immediate for the global properties, i.e., those defined in
Modules on Sites, Definition \ref{sites-modules-definition-global}.
For the local properties we can use
Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}
to translate ``$\mathcal{F}^a$ has $\mathcal{P}$'' into a property
on the members of a covering of $X$. Hence the result follows from
Lemmas \ref{lemma-finite-type-descends},
\ref{lemma-finite-presentation-descends},
\ref{lemma-locally-generated-by-r-sections-descends},
\ref{lemma-flat-descends}, and
\ref{lemma-finite-locally-free-descends}.
Being coherent for a quasi-coherent module is the same as being
of finite type over a locally Noetherian scheme (see
Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian})
hence this reduces
to the case of finite type modules (details omitted).
\end{proof}
\begin{lemma}
\label{lemma-equivalence-quasi-coherent-limits}
Let $S$ be a scheme.
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0]
\etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
The functors
$$
\QCoh(\mathcal{O}_S)
\longrightarrow
\textit{Mod}((\Sch/S)_\tau, \mathcal{O})
\quad\text{and}\quad
\QCoh(\mathcal{O}_S)
\longrightarrow
\textit{Mod}(S_\tau, \mathcal{O})
$$
defined by the rule $\mathcal{F} \mapsto \mathcal{F}^a$ seen in
Proposition \ref{proposition-equivalence-quasi-coherent}
are
\begin{enumerate}
\item fully faithful,
\item compatible with direct sums,
\item compatible with colimits,
\item right exact,
\item exact as a functor
$\QCoh(\mathcal{O}_S) \to \textit{Mod}(S_\etale, \mathcal{O})$,
\item {\bf not} exact as a functor
$\QCoh(\mathcal{O}_S) \to
\textit{Mod}((\Sch/S)_\tau, \mathcal{O})$
in general,
\item given two quasi-coherent $\mathcal{O}_S$-modules
$\mathcal{F}$, $\mathcal{G}$ we have
$(\mathcal{F} \otimes_{\mathcal{O}_S} \mathcal{G})^a =
\mathcal{F}^a \otimes_\mathcal{O} \mathcal{G}^a$,
\item given two quasi-coherent $\mathcal{O}_S$-modules
$\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$
is of finite presentation we have
$(\SheafHom_{\mathcal{O}_S}(\mathcal{F}, \mathcal{G}))^a =
\SheafHom_\mathcal{O}(\mathcal{F}^a, \mathcal{G}^a)$, and
\item given a short exact sequence
$0 \to \mathcal{F}_1^a \to \mathcal{E} \to \mathcal{F}_2^a \to 0$
of $\mathcal{O}$-modules then $\mathcal{E}$ is
quasi-coherent\footnote{Warning: This is misleading. See part (6).}, i.e.,
$\mathcal{E}$ is in the essential image of the functor.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (1) we saw in
Proposition \ref{proposition-equivalence-quasi-coherent}.
\medskip\noindent
We have seen in
Schemes, Section \ref{schemes-section-quasi-coherent}
that a colimit of quasi-coherent sheaves on a scheme is a quasi-coherent
sheaf. Moreover, in
Remark \ref{remark-change-topologies-ringed-sites}
we saw that $\mathcal{F} \mapsto \mathcal{F}^a$ is the pullback functor
for a morphism of ringed sites, hence commutes with all colimits, see
Modules on Sites, Lemma
\ref{sites-modules-lemma-exactness-pushforward-pullback}.
Thus (3) and its special case (3) hold.
\medskip\noindent
This also shows that the functor is right exact (i.e., commutes with
finite colimits), hence (4).
\medskip\noindent
The functor $\QCoh(\mathcal{O}_S) \to
\QCoh(S_\etale, \mathcal{O})$,
$\mathcal{F} \mapsto \mathcal{F}^a$
is left exact because an \'etale morphism is flat, see
Morphisms, Lemma \ref{morphisms-lemma-etale-flat}.
This proves (5).
\medskip\noindent
To see (6), suppose that $S = \Spec(\mathbf{Z})$.
Then $2 : \mathcal{O}_S \to \mathcal{O}_S$ is injective but the associated
map of $\mathcal{O}$-modules on $(\Sch/S)_\tau$ isn't
injective because $2 : \mathbf{F}_2 \to \mathbf{F}_2$ isn't injective
and $\Spec(\mathbf{F}_2)$ is an object of $(\Sch/S)_\tau$.
\medskip\noindent
We omit the proofs of (7) and (8).
\medskip\noindent
Let $0 \to \mathcal{F}_1^a \to \mathcal{E} \to \mathcal{F}_2^a \to 0$
be a short exact sequence of $\mathcal{O}$-modules with $\mathcal{F}_1$
and $\mathcal{F}_2$ quasi-coherent on $S$. Consider the restriction
$$
0 \to \mathcal{F}_1 \to \mathcal{E}|_{S_{Zar}} \to \mathcal{F}_2
$$
to $S_{Zar}$. By
Proposition \ref{proposition-same-cohomology-quasi-coherent}
we see that on any affine $U \subset S$ we have
$H^1(U, \mathcal{F}_1^a) = H^1(U, \mathcal{F}_1) = 0$.
Hence the sequence above is also exact on the right. By
Schemes, Section \ref{schemes-section-quasi-coherent}
we conclude that $\mathcal{F} = \mathcal{E}|_{S_{Zar}}$ is
quasi-coherent. Thus we obtain a commutative diagram
$$
\xymatrix{
& \mathcal{F}_1^a \ar[r] \ar[d] &
\mathcal{F}^a \ar[r] \ar[d] &
\mathcal{F}_2^a \ar[r] \ar[d] & 0 \\
0 \ar[r] &
\mathcal{F}_1^a \ar[r] &
\mathcal{E} \ar[r] &
\mathcal{F}_2^a \ar[r] & 0
}
$$
To finish the proof it suffices to show that the top row is also
right exact. To do this, denote once more $U = \Spec(A) \subset S$
an affine open of $S$. We have seen above that
$0 \to \mathcal{F}_1(U) \to \mathcal{E}(U) \to \mathcal{F}_2(U) \to 0$
is exact. For any affine scheme $V/U$,
$V = \Spec(B)$ the map $\mathcal{F}_1^a(V) \to \mathcal{E}(V)$
is injective. We have $\mathcal{F}_1^a(V) = \mathcal{F}_1(U) \otimes_A B$
by definition. The injection
$\mathcal{F}_1^a(V) \to \mathcal{E}(V)$ factors as
$$
\mathcal{F}_1(U) \otimes_A B \to
\mathcal{E}(U) \otimes_A B \to \mathcal{E}(U)
$$
Considering $A$-algebras $B$ of the form $B = A \oplus M$
we see that $\mathcal{F}_1(U) \to \mathcal{E}(U)$ is
universally injective (see
Algebra, Definition \ref{algebra-definition-universally-injective}).
Since $\mathcal{E}(U) = \mathcal{F}(U)$ we conclude that
$\mathcal{F}_1 \to \mathcal{F}$ remains injective after any base change,
or equivalently that $\mathcal{F}_1^a \to \mathcal{F}^a$ is injective.
\end{proof}
\begin{proposition}
\label{proposition-equivalence-quasi-coherent-functorial}
Let $f : T \to S$ be a morphism of schemes.
\begin{enumerate}
\item The equivalences of categories of
Proposition \ref{proposition-equivalence-quasi-coherent}
are compatible with pullback.
More precisely, we have $f^*(\mathcal{G}^a) = (f^*\mathcal{G})^a$
for any quasi-coherent sheaf $\mathcal{G}$ on $S$.
\item The equivalences of categories of
Proposition \ref{proposition-equivalence-quasi-coherent} part (1)
are {\bf not} compatible with pushforward in general.
\item If $f$ is quasi-compact and quasi-separated, and
$\tau \in \{Zariski, \etale\}$ then $f_*$ and $f_{small, *}$
preserve quasi-coherent sheaves and the diagram
$$
\xymatrix{
\QCoh(\mathcal{O}_T)
\ar[rr]_{f_*} \ar[d]_{\mathcal{F} \mapsto \mathcal{F}^a} & &
\QCoh(\mathcal{O}_S)
\ar[d]^{\mathcal{G} \mapsto \mathcal{G}^a} \\
\QCoh(T_\tau, \mathcal{O}) \ar[rr]^{f_{small, *}} & &
\QCoh(S_\tau, \mathcal{O})
}
$$
is commutative, i.e., $f_{small, *}(\mathcal{F}^a) = (f_*\mathcal{F})^a$.
\end{enumerate}
\end{proposition}
\begin{proof}
Part (1) follows from the discussion in
Remark \ref{remark-change-topologies-ringed-sites}.
Part (2) is just a warning, and can be explained in the following way:
First the statement cannot be made precise since $f_*$ does not
transform quasi-coherent sheaves into quasi-coherent sheaves in general.
Even if this is the case for $f$ (and any base change of $f$), then the
compatibility over the big sites would mean that formation of $f_*\mathcal{F}$
commutes with any base change, which does not hold in general.
An explicit example is the quasi-compact open immersion
$j : X = \mathbf{A}^2_k \setminus \{0\} \to \mathbf{A}^2_k = Y$
where $k$ is a field. We have $j_*\mathcal{O}_X = \mathcal{O}_Y$
but after base change to $\Spec(k)$ by the $0$ map
we see that the pushforward is zero.
\medskip\noindent
Let us prove (3) in case $\tau = \etale$. Note that $f$, and any
base change of $f$, transforms quasi-coherent sheaves
into quasi-coherent sheaves, see
Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}.
The equality $f_{small, *}(\mathcal{F}^a) = (f_*\mathcal{F})^a$
means that for any \'etale morphism $g : U \to S$ we have
$\Gamma(U, g^*f_*\mathcal{F}) = \Gamma(U \times_S T, (g')^*\mathcal{F})$
where $g' : U \times_S T \to T$ is the projection. This is true by
Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}.
\end{proof}
\begin{lemma}
\label{lemma-higher-direct-images-small-etale}
Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $T$. For either the \'etale
or Zariski topology, there are canonical isomorphisms
$R^if_{small, *}(\mathcal{F}^a) = (R^if_*\mathcal{F})^a$.
\end{lemma}
\begin{proof}
We prove this for the \'etale topology; we omit the proof in the case
of the Zariski topology. By Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images}
the sheaves $R^if_*\mathcal{F}$ are quasi-coherent so that the assertion
makes sense. The sheaf $R^if_{small, *}\mathcal{F}^a$ is the sheaf associated
to the presheaf
$$
U \longmapsto H^i(U \times_S T, \mathcal{F}^a)
$$
where $g : U \to S$ is an object of $S_\etale$, see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}.
By our conventions the right hand side is the \'etale
cohomology of the restriction of $\mathcal{F}^a$ to the localization
$T_\etale/U \times_S T$ which equals
$(U \times_S T)_\etale$. By
Proposition \ref{proposition-same-cohomology-quasi-coherent}
this is presheaf the same as the presheaf
$$
U \longmapsto
H^i(U \times_S T, (g')^*\mathcal{F}),
$$
where $g' : U \times_S T \to T$ is the projection. If $U$ is affine
then this is the same as $H^0(U, R^if'_*(g')^*\mathcal{F})$, see
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images-application}.
By
Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}
this is equal to $H^0(U, g^*R^if_*\mathcal{F})$ which is the value
of $(R^if_*\mathcal{F})^a$ on $U$.
Thus the values of the sheaves of modules
$R^if_{small, *}(\mathcal{F}^a)$ and $(R^if_*\mathcal{F})^a$
on every affine object of $S_\etale$ are canonically isomorphic
which implies they are canonically isomorphic.
\end{proof}
\noindent
The results in this section say there is virtually no difference between
quasi-coherent sheaves on $S$ and quasi-coherent sheaves on any of the
sites associated to $S$ in the chapter on topologies. Hence one often
sees statements on quasi-coherent sheaves formulated in either language,
without restatements in the other.
\section{Parasitic modules}
\label{section-parasitic}
\noindent
Parasitic modules are those which are zero when restricted
to schemes flat over the base scheme. Here is the formal definition.
\begin{definition}
\label{definition-parasitic}
Let $S$ be a scheme. Let $\tau \in \{Zar, \etale,
smooth, syntomic, fppf\}$. Let $\mathcal{F}$ be a presheaf
of $\mathcal{O}$-modules on $(\Sch/S)_\tau$.
\begin{enumerate}
\item $\mathcal{F}$ is called
{\it parasitic}\footnote{This may be nonstandard notation.}
if for every flat morphism $U \to S$ we have $\mathcal{F}(U) = 0$.
\item $\mathcal{F}$ is called {\it parasitic for the $\tau$-topology}
if for every $\tau$-covering $\{U_i \to S\}_{i \in I}$ we have
$\mathcal{F}(U_i) = 0$ for all $i$.
\end{enumerate}
\end{definition}
\noindent
If $\tau = fppf$ this means that $\mathcal{F}|_{U_{Zar}} = 0$ whenever
$U \to S$ is flat and locally of finite presentation; similar for
the other cases.
\begin{lemma}
\label{lemma-cohomology-parasitic}
Let $S$ be a scheme. Let $\tau \in \{Zar, \etale, smooth,
syntomic, fppf\}$. Let $\mathcal{G}$ be a presheaf of
$\mathcal{O}$-modules on $(\Sch/S)_\tau$.
\begin{enumerate}
\item If $\mathcal{G}$ is parasitic for the $\tau$-topology, then
$H^p_\tau(U, \mathcal{G}) = 0$ for every $U$ open in $S$,
resp.\ \'etale over $S$,
resp.\ smooth over $S$,
resp.\ syntomic over $S$,
resp.\ flat and locally of finite presentation over $S$.
\item If $\mathcal{G}$ is parasitic then $H^p_\tau(U, \mathcal{G}) = 0$
for every $U$ flat over $S$.
\end{enumerate}
\end{lemma}
\begin{proof}
Proof in case $\tau = fppf$; the other cases are proved in the
exact same way. The assumption means that $\mathcal{G}(U) = 0$ for any
$U \to S$ flat and locally of finite presentation. Apply
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection}
to the subset $\mathcal{B} \subset \Ob((\Sch/S)_{fppf})$ consisting
of $U \to S$ flat and locally of finite presentation and the collection
$\text{Cov}$ of all fppf coverings of elements of $\mathcal{B}$.
\end{proof}
\begin{lemma}
\label{lemma-direct-image-parasitic}
Let $f : T \to S$ be a morphism of schemes. For any parasitic
$\mathcal{O}$-module on $(\Sch/T)_\tau$ the pushforward
$f_*\mathcal{F}$ and the higher direct images $R^if_*\mathcal{F}$
are parasitic $\mathcal{O}$-modules on $(\Sch/S)_\tau$.
\end{lemma}
\begin{proof}
Recall that $R^if_*\mathcal{F}$ is the sheaf associated to the
presheaf
$$
U \mapsto H^i((\Sch/U \times_S T)_\tau, \mathcal{F})
$$
see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}.
If $U \to S$ is flat, then $U \times_S T \to T$ is flat as a base change.
Hence the displayed group is zero by
Lemma \ref{lemma-cohomology-parasitic}.
If $\{U_i \to U\}$ is a $\tau$-covering then
$U_i \times_S T \to T$ is also flat.
Hence it is clear that the sheafification of the displayed
presheaf is zero on schemes $U$ flat over $S$.
\end{proof}
\begin{lemma}
\label{lemma-quasi-coherent-and-flat-base-change}
Let $S$ be a scheme. Let $\tau \in \{Zar, \etale\}$.
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules on
$(\Sch/S)_{fppf}$ such that
\begin{enumerate}
\item $\mathcal{G}|_{S_\tau}$ is quasi-coherent, and
\item for every flat, locally finitely presented morphism
$g : U \to S$ the canonical map
$g_{\tau, small}^*(\mathcal{G}|_{S_\tau}) \to \mathcal{G}|_{U_\tau}$
is an isomorphism.
\end{enumerate}
Then $H^p(U, \mathcal{G}) = H^p(U, \mathcal{G}|_{U_\tau})$
for every $U$ flat and locally of finite presentation over $S$.
\end{lemma}
\begin{proof}
Let $\mathcal{F}$ be the pullback of $\mathcal{G}|_{S_\tau}$
to the big fppf site $(\Sch/S)_{fppf}$. Note that $\mathcal{F}$
is quasi-coherent. There is a canonical
comparison map $\varphi : \mathcal{F} \to \mathcal{G}$ which by
assumptions (1) and (2) induces an isomorphism
$\mathcal{F}|_{U_\tau} \to \mathcal{G}|_{U_\tau}$
for all $g : U \to S$ flat and locally of finite presentation.
Hence in the short exact sequences
$$
0 \to \Ker(\varphi) \to \mathcal{F} \to \Im(\varphi) \to 0
$$
and
$$
0 \to \Im(\varphi) \to \mathcal{G} \to \Coker(\varphi) \to 0
$$
the sheaves $\Ker(\varphi)$ and $\Coker(\varphi)$ are
parasitic for the fppf topology. By
Lemma \ref{lemma-cohomology-parasitic}
we conclude that $H^p(U, \mathcal{F}) \to H^p(U, \mathcal{G})$
is an isomorphism for $g : U \to S$ flat and locally of finite presentation.
Since the result holds for $\mathcal{F}$ by
Proposition \ref{proposition-same-cohomology-quasi-coherent}
we win.
\end{proof}
\section{Fpqc coverings are universal effective epimorphisms}
\label{section-fpqc-universal-effective-epimorphisms}
\noindent
We apply the material above to prove an interesting result, namely
Lemma \ref{lemma-fpqc-universal-effective-epimorphisms}.
By Sites, Section \ref{sites-section-representable-sheaves}
this lemma implies that the representable
presheaves on any of the sites $(\Sch/S)_\tau$ are sheaves for
$\tau \in \{Zariski, fppf, \etale, smooth, syntomic\}$. First
we prove a helper lemma.
\begin{lemma}
\label{lemma-equiv-fibre-product}
For a scheme $X$ denote $|X|$ the underlying set.
Let $f : X \to S$ be a morphism of schemes.
Then
$$
|X \times_S X| \to |X| \times_{|S|} |X|
$$
is surjective.
\end{lemma}
\begin{proof}
Follows immediately from the description of points on the
fibre product in Schemes, Lemma \ref{schemes-lemma-points-fibre-product}.
\end{proof}
\begin{lemma}
\label{lemma-open-fpqc-covering}
Let $\{f_i : T_i \to T\}_{i \in I}$ be a fpqc covering.
Suppose that for each $i$ we have an open subset $W_i \subset T_i$
such that for all $i, j \in I$ we have
$\text{pr}_0^{-1}(W_i) = \text{pr}_1^{-1}(W_j)$ as open
subsets of $T_i \times_T T_j$. Then there exists a unique open subset
$W \subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$.
\end{lemma}
\begin{proof}
Apply
Lemma \ref{lemma-equiv-fibre-product}
to the map $\coprod_{i \in I} T_i \to T$.
It implies there exists a subset $W \subset T$ such that
$W_i = f_i^{-1}(W)$ for each $i$, namely $W = \bigcup f_i(W_i)$.
To see that $W$ is open we may work Zariski locally on $T$.
Hence we may assume that $T$ is affine. Using the definition
of a fpqc covering, this reduces us to the case where
$\{f_i : T_i \to T\}$ is a standard fpqc covering. In this case we
may apply
Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology}
to the morphism
$\coprod T_i \to T$ to conclude that $W$ is open.
\end{proof}
\begin{lemma}
\label{lemma-fpqc-universal-effective-epimorphisms}
Let $\{T_i \to T\}$ be an fpqc covering, see
Topologies, Definition \ref{topologies-definition-fpqc-covering}.
Then $\{T_i \to T\}$ is a universal effective epimorphism
in the category of schemes, see
Sites, Definition \ref{sites-definition-universal-effective-epimorphisms}.
In other words, every representable functor on the category of schemes
satisfies the sheaf condition for the fpqc topology, see
Topologies, Definition \ref{topologies-definition-sheaf-property-fpqc}.
\end{lemma}
\begin{proof}
Let $S$ be a scheme. We have to show the following:
Given morphisms $\varphi_i : T_i \to S$
such that $\varphi_i|_{T_i \times_T T_j} = \varphi_j|_{T_i \times_T T_j}$
there exists a unique morphism $T \to S$ which restricts
to $\varphi_i$ on each $T_i$.
In other words, we have to show that the functor
$h_S = \Mor_{\Sch}( - , S)$ satisfies
the sheaf property for the fpqc topology.
\medskip\noindent
Thus Topologies, Lemma \ref{topologies-lemma-sheaf-property-fpqc}
reduces us to the case of a Zariski covering and a covering
$\{\Spec(A) \to \Spec(R)\}$ with $R \to A$ faithfully flat.
The case of a Zariski covering follows from
Schemes, Lemma \ref{schemes-lemma-glue}.
\medskip\noindent
Suppose that $R \to A$ is a faithfully flat ring map.
Denote $\pi : \Spec(A) \to \Spec(R)$ the
corresponding morphism of schemes. It is surjective and flat.
Let $f : \Spec(A) \to S$ be a morphism
such that $f \circ \text{pr}_1 = f \circ \text{pr}_2$
as maps $\Spec(A \otimes_R A) \to S$.
By Lemma \ref{lemma-equiv-fibre-product} we see that
as a map on the underlying
sets $f$ is of the form $f = g \circ \pi$ for some
(set theoretic) map $g : \Spec(R) \to S$.
By Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology}
and the fact that $f$ is continuous we see that $g$
is continuous.
\medskip\noindent
Pick $x \in \Spec(R)$.
Choose $U \subset S$ affine open containing $g(x)$.
Say $U = \Spec(B)$.
By the above we may choose an $r \in R$ such that
$x \in D(r) \subset g^{-1}(U)$.
The restriction of $f$ to $\pi^{-1}(D(r))$ into $U$
corresponds to a ring map $B \to A_r$. The two induced
ring maps $B \to A_r \otimes_{R_r} A_r = (A \otimes_R A)_r$ are equal
by assumption on $f$.
Note that $R_r \to A_r$ is faithfully flat.
By Lemma \ref{lemma-ff-exact} the equalizer of
the two arrows $A_r \to A_r \otimes_{R_r} A_r$ is $R_r$.
We conclude that $B \to A_r$ factors uniquely through a map $B \to R_r$.
This map in turn gives a morphism of schemes $D(r) \to U \to S$,
see Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}.
\medskip\noindent
What have we proved so far? We have shown that for any prime
$\mathfrak p \subset R$, there exists a standard affine open
$D(r) \subset \Spec(R)$ such that the morphism
$f|_{\pi^{-1}(D(r))} : \pi^{-1}(D(r)) \to S$ factors uniquely
though some morphism of schemes $D(r) \to S$. We omit the
verification that these morphisms glue to the desired
morphism $\Spec(R) \to S$.
\end{proof}
\begin{lemma}
\label{lemma-coequalizer-fpqc-local}
Consider schemes $X, Y, Z$ and morphisms $a, b : X \to Y$ and
a morphism $c : Y \to Z$ with $c \circ a = c \circ b$. Set
$d = c \circ a = c \circ b$. If there exists an
fpqc covering $\{Z_i \to Z\}$ such that
\begin{enumerate}
\item for all $i$ the morphism $Y \times_{c, Z} Z_i \to Z_i$
is the coequalizer of $(a, 1) : X \times_{d, Z} Z_i \to Y \times_{c, Z} Z_i$
and $(b, 1) : X \times_{d, Z} Z_i \to Y \times_{c, Z} Z_i$, and
\item for all $i$ and $i'$ the morphism
$Y \times_{c, Z} (Z_i \times_Z Z_{i'}) \to (Z_i \times_Z Z_{i'})$
is the coequalizer of
$(a, 1) : X \times_{d, Z} (Z_i \times_Z Z_{i'}) \to
Y \times_{c, Z} (Z_i \times_Z Z_{i'})$ and
$(b, 1) : X \times_{d, Z} (Z_i \times_Z Z_{i'}) \to
Y \times_{c, Z} (Z_i \times_Z Z_{i'})$
\end{enumerate}
then $c$ is the coequalizer of $a$ and $b$.
\end{lemma}
\begin{proof}
Namely, for a scheme $T$ a morphism $Z \to T$ is the same thing as
a collection of morphism $Z_i \to T$ which agree on overlaps by
Lemma \ref{lemma-fpqc-universal-effective-epimorphisms}.
\end{proof}
\section{Descent of finiteness properties of morphisms}
\label{section-descent-finiteness-morphisms}
\noindent
Another application of flat descent for modules is the following
amusing and useful result. There is an algebraic version and a
scheme theoretic version. (The ``Noetherian'' reader should
consult Lemma \ref{lemma-finite-type-local-source-fppf-algebra}
instead of the next lemma.)
\begin{lemma}
\label{lemma-flat-finitely-presented-permanence-algebra}
Let $R \to A \to B$ be ring maps.