Descent Theory

Descending quasi-coherent sheaves

Proposition 1. If $\phi: A\to B$ os a faithfully flat map, then

$$0\to A\xrightarrow{\phi}B\begin{array}{c} \xrightarrow{b\mapsto b\otimes 1} \\ \\[-4.3ex] \xrightarrow[b\mapsto 1\otimes b]{} \end{array}B\otimes_{A}B$$

is exact. More generally, if $M$ is an $A$-module, then

$$0\to M\xrightarrow{m\mapsto m\otimes 1} M\otimes_{A} B\begin{array}{c} \xrightarrow{m\otimes b\mapsto m\otimes b\otimes 1} \\ \\[-4.3ex] \xrightarrow[m\otimes b\mapsto m\otimes 1\otimes b]{} \end{array} M\otimes_{A}B\otimes_{A}B$$

is exact.

Proposition 2. Let $f:S'\to S$ be an fpqc morphism of schemes.

Let $\mathcal{F}$ amd $\mathcal{G}$ be quasi-coherent $\mathcal{O}_{S}$-modules. Let $p_{1}, p_{2}: S'\times_{S}S'$ be two projections and $q:S'\times_{S}S'\to S$ be the composition $f\circ p_{i}$/ Then the sequence

$$0\to \mathrm{Hom}_{\mathcal{O}_{s}}{\mathcal{F},\mathcal{G}}\xrightarrow{f^{*}}\mathrm{Hom}_{\mathcal{O}_{S}'}(f^{*}\mathcal{F},f^{*}\mathcal{G})\begin{array}{c} \xrightarrow{p_{1}^{*}} \\ \\[-4.3ex] \xrightarrow[p_{2}^{*}]{} \end{array}\mathrm{Hom}_{\mathcal{O}_{S'\times_{S}S'}}(q^{*}\mathcal{F},q^{*}\mathcal{G})$$

is exact.

Let $H$ be the quasi-coherent $\mathcal{O}_{S'}$-module and $\alpha:p_{1}^{*}H\to p_{2}^{*}$ be an isomorphism of $\mathcal{O}_{S'\times_{S}S'}$-module satisfying the cocycle condition $p_{23}^{*}\alpha\circ p_{12}^{*}\alpha =p_{13}^{*}\alpha$ on $S'\times_{S}S'\times_{S}S'$. Then there exists a quasi-coherent $\mathcal{O}_{S}$-module $\mathcal{G}$ and an isomorphism $\phi:H\to f^{*}\mathcal{G}$ such that $p_{1}^{*}\phi=p_{2}^{*}\phi\circ \alpha$ on $S'\times_{S}S'$. The data $(\mathcal{G},\phi)$ is unique up to isomorphism. The cocycle condition can be visualized as

Descending morphisms

Proposition 3. Let $Y$ be a scheme and $f:S'\to S$ be a fpqc morphism of schemes. If $g:S'\to Y$ is a morphism such that $g\circ p_{1}=g\circ p_{2}$ then there exists a unique morphism $h:S\to Y$ such that $g=h\circ f$.

Proposition 4. Let $f:S'\to S$ be an fpqc morphism of schemes.

If $S\to T$ os a morphism of schemes and $Y$ is a $T$-scheme then

$$\mathrm{Hom}(S,Y)\to \mathrm{Hom}(S',Y)\begin{array}{c} \to \\ \\[-4.6ex] \to \end{array}\mathrm{Hom}(S'\times_{S}S',Y)$$

is exact.

If $X$, $Y$ are schemes over $S$, then

$$\mathrm{Hom}_{S}(X,Y)\to \mathrm{Hom}_{S'}(X_{S'},Y_{S'})\begin{array}{c} \to \\ \\[-4.6ex] \to \end{array}\mathrm{Hom}_{S'\times_{S}S}(X_{S'\times_{S}S'},Y_{S'\times_{S}S'})$$

is exact.

Descending schemes

Proposition 5. Let $f:S'\to S$ be an fpqc morphism of schemes. If $Z'\subset S'$ is a closed (resp. open) subscheme such that $p_{1}^{-1}(Z')=p_{2}^{-1}(Z')$ as subschemes of $S'\times_{S}S'$, then there exists a closed (resp. open) subscheme $Z\subset S$ such that $Z'=f^{-1}(Z)$.

Proposition 6. Let $f:S'\to S$ be an fpqc morphism of schemes. If $X'\to S'$ is affine (resp. quasi-affine) morphism and $\alpha:p_{1}^{*}(X')\xrightarrow{\sim} p_{2}^{*}(X')$ is an isomorphism over $S'\times_{S}S'$ satisfying $p_{23}^{*}\alpha\circ p_{12}^{*}\alpha=p_{13}^{*}\alpha$, then there exists an affine (resp. quasi-affine) morphism $X\to S$ of schemes, and an isomorphism $\phi:X'\to f^{*}(X)$ over $S'$ such that $p_{1}^{*}\phi=p_{2}^{*}\phi\circ \alpha$.

Theorem 1. Let $f:S'\to S$ be an fpqc morphism of schemes. If $X'\to S'$ is locally quasi-finite and separated morphism of schemes. $\alpha:p_{1}^{*}(X')\xrightarrow{\sim} p_{2}^{*}(X')$. Then there exists an locally quasi-finite and separated morphism $X\to S$ of schemes and an isomorphism $\phi:X'\to f^{*}(X)$ over $S'$ such that $p_{1}^{*}\phi=p_{2}^{*}\phi\circ \alpha$.

Proposition 7. Let $G\to T$ be an fppf affine group scheme and let $f:S'\to S$ be an fpqc morphism of schemes over $T$. If $P'\to S'$ is a principal $G$-bundle and $\alpha:p_{1}^{*}P'\xrightarrow{\sim}p_{2}^{*}P'$ is an isomorphism of principal $G$-bundle. Then there exists a principal $G$-bundle $P\to S$ and an isomorphism $\phi:P'\to f^{*}P$ of principal $G$-bundles such that $p_{1}\phi=p_{2}^{*}\phi\circ \alpha$.

Descending properties

Proposition 8. Let $f:S'\to S$ be an fpqc morphism of schemes.

A homomorphism $\mathcal{F}\to \mathcal{G}$ of quasi-coherent $\mathcal{O}_{S'}$-modules is an isomorphism (resp. injective, surjective) if and only if $f^{*}\mathcal{F}\to f^{*}\mathcal{G}$ is.

A quasi-coherent $\mathcal{O}_{S}$-module $\mathcal{F}$ is of finite type (resp. of finite presentation, flat, vector bundle, line bundle) if and only if $f^{*}\mathcal{G}$ is. If $S$ and $S'$ are noetherian, then the same holds for coherence.

A quasi-coherent $\mathcal{O}_{X}$-module $\mathcal{F}$ on a $S$-scheme $X$ is flat over $S$ if and only if the pullback of $\mathcal{F}$ to $X_{S'}$ is flat over $S'$.

Proposition 9. Let $X\to Y$ be an fpqc morphism of schemes. If $X$ is quasi-compact (resp. locally noetherian, noetherian, integral, reduced, normal, regular), then so is $Y$.

Proposition 10. Let $X'\to X$ be an fpqc morphism of schemes. If $X\to Y$ is a morphism such that $X'\to X\to Y$ is smooth (resp. etale), then $X\to Y$ is smooth (resp. etale).

Proposition 11. If $X\to Y$ is an fppf morphism of schemes, then $X$ is locally noetherian if and only if $Y$ is.

If $X\to Y$ is surjective smooth morphism of schemes, then $X$ is reduced (resp. normal, regular) if and only if $Y$ is.

Proposition 12. Let $S'\to S$ be an fpqc morphism of schemes and $\mathcal{P}$ be one of the following properties of a morphism of schemes: surjective, quasi-compact, quasi-separated, isomorphism, open immersion, closed immersion, monomorphism, affine, quasi-affine, quasi-compact locally closed immersion, locally of finite type, locally of finite presentation, separated, proper, universally closed, universally open, universally submersive, finite, locally quasi-finite, quasi-finite, flat, fppf, smooth, étale, unramified, or syntomic. Then $X\to S$ has $\mathcal{P}$ if and only if $X\times_{S} S'\to S'$ does.