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 \input{preamble} % OK, start here. % \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.