The Stacks project

Proof. The pushforward of a quasi-coherent module along a quasi-compact and quasi-separated morphism is quasi-coherent, see Schemes, Lemma 26.24.1. Hence $s_*t^*\mathcal{G}$ is quasi-coherent. With notation as in Lemma 39.13.4 we have

The middle equality because $t \circ c = t \circ \text{pr}_0$ as morphisms $R \times _{s, U, t} R \to U$, and the first and the last equality because we know that base change and pushforward commute in these steps by Cohomology of Schemes, Lemma 30.5.2.

To verify the cocycle condition of Definition 39.14.1 for $\alpha $ and the adjointness property we describe the construction $\mathcal{G} \mapsto (s_*t^*\mathcal{G}, \alpha )$ in another way. Consider the groupoid scheme $(R, R \times _{t, U, t} R, \text{pr}_0, \text{pr}_1, \text{pr}_{02})$ associated to the equivalence relation $R \times _{t, U, t} R$ on $R$, see Lemma 39.13.3. There is a morphism

of groupoid schemes given by $s : R \to U$ and $R \times _{t, U, t} R \to R$ given by $(r_0, r_1) \mapsto r_0^{-1} \circ r_1$; we omit the verification of the commutativity of the required diagrams. Since $t, s : R \to U$ are quasi-compact, quasi-separated, and flat, and since we have a cartesian square

by Lemma 39.13.5 it follows that Lemma 39.14.4 applies to $f$. Thus pushforward and pullback of quasi-coherent modules along $f$ are adjoint functors. To finish the proof we will identify these functors with the functors described above. To do this, note that

is an equivalence by the theory of descent of quasi-coherent sheaves as $\{ t : R \to U\} $ is an fpqc covering, see Descent, Proposition 35.5.2.

Pushforward along $f$ precomposed with the equivalence $t^*$ sends $\mathcal{G}$ to $(s_*t^*\mathcal{G}, \alpha )$; we omit the verification that the isomorphism $\alpha $ obtained in this fashion is the same as the one constructed above.

Pullback along $f$ postcomposed with the inverse of the equivalence $t^*$ sends $(\mathcal{F}, \beta )$ to the descent relative to $\{ t : R \to U\} $ of the module $s^*\mathcal{F}$ endowed with the descent datum $\gamma $ on $R \times _{t, U, t} R$ which is the pullback of $\beta $ by $(r_0, r_1) \mapsto r_0^{-1} \circ r_1$. Consider the isomorphism $\beta : t^*\mathcal{F} \to s^*\mathcal{F}$. The canonical descent datum (Descent, Definition 35.2.3) on $t^*\mathcal{F}$ relative to $\{ t : R \to U\} $ translates via $\beta $ into the map

Since $\beta $ satisfies the cocycle condition, this is equal to the pullback of $\beta $ by $(r_0, r_1) \mapsto r_0^{-1} \circ r_1$. To see this take the actual cocycle relation in Definition 39.14.1 and pull it back by the morphism $(\text{pr}_0, c \circ (i, 1)) : R \times _{t, U, t} R \to R \times _{s, U, t} R$ which also plays a role in the commutative diagram of Lemma 39.13.5. It follows that $(s^*\mathcal{F}, \gamma )$ is isomorphic to $(t^*\mathcal{F}, can)$. All in all, we conclude that pullback by $f$ postcomposed with the inverse of the equivalence $t^*$ is isomorphic to the forgetful functor $(\mathcal{F}, \beta ) \mapsto \mathcal{F}$. $\square$

Proof. We will freely use the characterization of coherent modules of Cohomology of Schemes, Lemma 30.9.1 as well as the fact that coherent modules form a Serre subcategory of $\mathit{QCoh}(\mathcal{O}_ X)$, see Cohomology of Schemes, Lemma 30.9.3. If $f$ has a section $\sigma $, then we see that $\mathcal{F} \cap f_*\mathcal{G}$ is contained in the image of $\sigma ^*\mathcal{G} \to \sigma ^*f^*\mathcal{F} = \mathcal{F}$, hence coherent. In general, to show that $\mathcal{F} \cap f_*\mathcal{G}$ is coherent, it suffices the show that $f^*(\mathcal{F} \cap f_*\mathcal{G})$ is coherent (see Descent, Lemma 35.7.1). Since $f$ is flat this is equal to $f^*\mathcal{F} \cap f^*f_*\mathcal{G}$. Since $f$ is flat, quasi-compact, and quasi-separated we see $f^*f_*\mathcal{G} = p_*q^*\mathcal{G}$ where $p, q : Y \times _ X Y \to Y$ are the projections, see Cohomology of Schemes, Lemma 30.5.2. Since $p$ has a section we win. $\square$

Proof. We will use the characterization of Cohomology of Schemes, Lemma 30.9.1 of coherent modules on locally Noetherian scheme without further mention. We can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i$ as the filtered colimit of coherent submodules $\mathcal{H}_ i \subset \mathcal{F}$, see Cohomology of Schemes, Lemma 30.10.4. Given a quasi-coherent sheaf $\mathcal{H}$ on $U$ we denote $(s_*t^*\mathcal{H}, \alpha )$ the quasi-coherent sheaf on $(U, R, s, t, c)$ of Lemma 39.15.1. Consider the adjunction map $(\mathcal{F}, \beta ) \to (s_*t^*\mathcal{F}, \alpha )$ in $\mathit{QCoh}(U, R, s, t, c)$, see Remark 39.15.2. Set

in $\mathit{QCoh}(U, R, s, t, c)$. Since restriction to $U$ is an exact functor on $\mathit{QCoh}(U, R, s, t, c)$ by the proof of Lemma 39.14.6 we obtain a pullback diagram

in other words $\mathcal{F}_ i = \mathcal{F} \cap s_*t^*\mathcal{H}_ i$. By the description of the adjunction map in Remark 39.15.2 this diagram is isomorphic to the diagram

where the right vertical arrow is the result of appplying $s_*$ to the map

This arrow is injective as $t$ is a flat morphism. It follows that $\mathcal{F}_ i$ is coherent by Lemma 39.15.3. Finally, because $s$ is quasi-compact and quasi-separated we see that $s_*$ commutes with colimits (see Cohomology of Schemes, Lemma 30.6.1). Hence $s_*t^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits s_*t^*\mathcal{H}_ i$ and hence $(\mathcal{F}, \beta ) = \mathop{\mathrm{colim}}\nolimits (\mathcal{F}_ i, \beta _ i)$ as desired. $\square$

Proof. The assumption means that $\mathcal{O}_ R(R)$ is a free $\mathcal{O}_ U(U)$-module via $s$ with basis $e_ i$. Hence for any quasi-coherent $\mathcal{O}_ U$-module $\mathcal{G}$ we see that $s^*\mathcal{G}(R) = \bigoplus _ i \mathcal{G}(U)e_ i$. We will write $s(-)$ to indicate pullback of sections by $s$ and similarly for other morphisms. Let $(\mathcal{F}, \alpha )$ be a quasi-coherent module on $(U, R, s, t, c)$. Let $\sigma \in \mathcal{F}(U)$. By the above we can write

for some unique $\sigma _ i \in \mathcal{F}(U)$ (all but finitely many are zero of course). We can also write

as functions on $R \times _{s, U, t}R$. Then the commutativity of the diagram in Definition 39.14.1 means that

(calculation omitted). Picking off the coefficients of $\text{pr}_0(e_ l)$ we see that $\alpha (t(\sigma _ l)) = \sum s(\sigma _ i)f_{il}$. Hence the submodule $\mathcal{G} \subset \mathcal{F}$ generated by the elements $\sigma _ i$ defines a finite type quasi-coherent module preserved by $\alpha $. Hence it is a subobject of $\mathcal{F}$ in $\mathit{QCoh}(U, R, s, t, c)$. This submodule contains $\sigma $ (as one sees by pulling back the first relation by $e$). Hence we win. $\square$

Proof. For each quasi-coherent module $\mathcal{F}$ on $U$ there is a (possibly empty) set of maps $\alpha : t^*\mathcal{F} \to s^*\mathcal{F}$ such that $(\mathcal{F}, \alpha )$ is a quasi-coherent modules on $(U, R, s, t, c)$. By Properties, Lemma 28.23.2 there exists a set of isomorphism classes of $\kappa $-generated quasi-coherent $\mathcal{O}_ U$-modules. $\square$

Proof. In the statement of the lemma and in this proof a submodule of a quasi-coherent module $(\mathcal{F}, \alpha )$ is a quasi-coherent submodule $\mathcal{G} \subset \mathcal{F}$ such that $\alpha (t^*\mathcal{G}) = s^*\mathcal{G}$ as subsheaves of $s^*\mathcal{F}$. This makes sense because since $s, t$ are flat the pullbacks $s^*$ and $t^*$ are exact, i.e., preserve subsheaves. The proof will be a repeat of the proof of Properties, Lemma 28.23.3. We urge the reader to read that proof first.

Choose an affine open covering $U = \bigcup _{i \in I} U_ i$. For each pair $i, j$ choose affine open coverings

Write $U_ i = \mathop{\mathrm{Spec}}(A_ i)$, $U_{ijk} = \mathop{\mathrm{Spec}}(A_{ijk})$, $W_{ijk} = \mathop{\mathrm{Spec}}(B_{ijk})$. Let $\kappa $ be any infinite cardinal $\geq $ than the cardinality of any of the sets $I$, $I_{ij}$, $J_{ij}$.

Let $(\mathcal{F}, \alpha )$ be a quasi-coherent module on $(U, R, s, t, c)$. Set $M_ i = \mathcal{F}(U_ i)$, $M_{ijk} = \mathcal{F}(U_{ijk})$. Note that

and that $\alpha $ gives isomorphisms

see Schemes, Lemma 26.7.3. Using the axiom of choice we choose a map

which associates to every $i, j \in I$, $k \in I_{ij}$ or $k \in J_{ij}$ and $m \in M_ i$ a finite subset $S(i, j, k, m) \subset M_ j$ such that we have

in $M_{ijk}$ for some $a_{m'} \in A_{ijk}$ or $b_{m'} \in B_{ijk}$. Moreover, let's agree that $S(i, i, k, m) = \{ m\} $ for all $i, j = i, k, m$ when $k \in I_{ij}$. Fix such a collection $S(i, j, k, m)$

Given a family $\mathcal{S} = (S_ i)_{i \in I}$ of subsets $S_ i \subset M_ i$ of cardinality at most $\kappa $ we set $\mathcal{S}' = (S'_ i)$ where

Note that $S_ i \subset S'_ i$. Note that $S'_ i$ has cardinality at most $\kappa $ because it is a union over a set of cardinality at most $\kappa $ of finite sets. Set $\mathcal{S}^{(0)} = \mathcal{S}$, $\mathcal{S}^{(1)} = \mathcal{S}'$ and by induction $\mathcal{S}^{(n + 1)} = (\mathcal{S}^{(n)})'$. Then set $\mathcal{S}^{(\infty )} = \bigcup _{n \geq 0} \mathcal{S}^{(n)}$. Writing $\mathcal{S}^{(\infty )} = (S^{(\infty )}_ i)$ we see that for any element $m \in S^{(\infty )}_ i$ the image of $m$ in $M_{ijk}$ can be written as a finite sum $\sum m' \otimes a_{m'}$ with $m' \in S_ j^{(\infty )}$. In this way we see that setting

we have

as submodules of $M_{ijk}$ or $M_ j \otimes _{A_ j, s} B_{ijk}$. Thus there exists a quasi-coherent submodule $\mathcal{G} \subset \mathcal{F}$ with $\mathcal{G}(U_ i) = N_ i$ such that $\alpha (t^*\mathcal{G}) = s^*\mathcal{G}$ as submodules of $s^*\mathcal{F}$. In other words, $(\mathcal{G}, \alpha |_{t^*\mathcal{G}})$ is a submodule of $(\mathcal{F}, \alpha )$. Moreover, by construction $\mathcal{G}$ is $\kappa $-generated.

Let $\{ (\mathcal{G}_ t, \alpha _ t)\} _{t \in T}$ be the set of $\kappa $-generated quasi-coherent submodules of $(\mathcal{F}, \alpha )$. If $t, t' \in T$ then $\mathcal{G}_ t + \mathcal{G}_{t'}$ is also a $\kappa $-generated quasi-coherent submodule as it is the image of the map $\mathcal{G}_ t \oplus \mathcal{G}_{t'} \to \mathcal{F}$. Hence the system (ordered by inclusion) is directed. The arguments above show that every section of $\mathcal{F}$ over $U_ i$ is in one of the $\mathcal{G}_ t$ (because we can start with $\mathcal{S}$ such that the given section is an element of $S_ i$). Hence $\mathop{\mathrm{colim}}\nolimits _ t \mathcal{G}_ t \to \mathcal{F}$ is both injective and surjective as desired. $\square$