Faithfully flat descent

Let ${\displaystyle A\to B}$ be a faithfully flat ring homomorphism. Given an ${\displaystyle A}$-module ${\displaystyle M}$, we get the ${\displaystyle B}$-module ${\displaystyle N=M\otimes _{A}B}$ and because ${\displaystyle A\to B}$ is faithfully flat, we have the inclusion ${\displaystyle M\hookrightarrow M\otimes _{A}B}$. Moreover, we have the isomorphism ${\displaystyle \varphi :N\otimes B{\overset {\sim }{\to }}N\otimes B}$ of ${\displaystyle B^{\otimes 2}}$-modules that is induced by the isomorphism ${\displaystyle B^{\otimes 2}\simeq B^{\otimes 2},x\otimes y\mapsto y\otimes x}$ and that satisfies the cocycle condition:

${\displaystyle \varphi ^{1}=\varphi ^{0}\circ \varphi ^{2}}$

where ${\displaystyle \varphi ^{i}:N\otimes B^{\otimes 2}{\overset {\sim }{\to }}N\otimes B^{\otimes 2}}$ are given as:[2]

${\displaystyle \varphi ^{0}(n\otimes b\otimes c)=\rho ^{1}(b)\varphi (n\otimes c)}$

${\displaystyle \varphi ^{1}(n\otimes b\otimes c)=\rho ^{2}(b)\varphi (n\otimes c)}$

${\displaystyle \varphi ^{2}(n\otimes b\otimes c)=\varphi (n\otimes b)\otimes c}$

with ${\displaystyle \rho ^{i}(x)(y_{0}\otimes \cdots \otimes y_{r})=y_{0}\cdots y_{i-1}\otimes x\otimes y_{i}\cdots y_{r}}$. Note the isomorphisms ${\displaystyle \varphi ^{i}:N\otimes B^{\otimes 2}{\overset {\sim }{\to }}N\otimes B^{\otimes 2}}$ are determined only by ${\displaystyle \varphi }$ and do not involve ${\displaystyle M.}$

Now, the most basic form of faithfully flat descent says that the above construction can be reversed; i.e., given a ${\displaystyle B}$-module ${\displaystyle N}$ and a ${\displaystyle B^{\otimes 2}}$-module isomorphism ${\displaystyle \varphi :N\otimes B{\overset {\sim }{\to }}N\otimes B}$ such that ${\displaystyle \varphi ^{1}=\varphi ^{0}\circ \varphi ^{2}}$, an invariant submodule:

${\displaystyle M=\{n\in N|\varphi (n\otimes 1)=n\otimes 1\}\subset N}$

is such that ${\displaystyle M\otimes B=N}$.[3]

The Zariski descent refers simply to the fact that a quasi-coherent sheaf can be obtained by gluing those on a (Zariski-)open cover. It is a special case of a faithfully flat descent but is frequently used to reduce the descent problem to the affine case.

In details, let ${\displaystyle {\mathcal {Q}}coh(X)}$ denote the category of quasi-coherent sheaves on a scheme X. Then Zariski descent states that, given quasi-coherent sheaves ${\displaystyle F_{i}}$ on open subsets ${\displaystyle U_{i}\subset X}$ with ${\displaystyle X=\bigcup U_{i}}$ and isomorphisms ${\displaystyle \varphi _{ij}:F_{i}|_{U_{i}\cap U_{j}}{\overset {\sim }{\to }}F_{j}|_{U_{i}\cap U_{j}}}$ such that (1) ${\displaystyle \varphi _{ii}=\operatorname {id} }$ and (2) ${\displaystyle \varphi _{ik}=\varphi _{jk}\circ \varphi _{ij}}$ on ${\displaystyle U_{i}\cap U_{j}\cap U_{k}}$, then exists a unique quasi-coherent sheaf ${\displaystyle F}$ on X such that ${\displaystyle F|_{U_{i}}\simeq F_{i}}$ in a compatible way (i.e., ${\displaystyle F|_{U_{j}}\simeq F_{j}}$ restricts to ${\displaystyle F|_{U_{i}\cap U_{j}}\simeq F_{i}|_{U_{i}\cap U_{j}}{\overset {\varphi _{ij}}{\underset {\sim }{\to }}}F_{j}|_{U_{i}\cap U_{j}}}$).[4]

In a fancy language, the Zariski descent states that, with respect to the Zariski topology, ${\displaystyle {\mathcal {Q}}coh}$ is a stack; i.e., a category ${\displaystyle {\mathcal {C}}}$ equipped with the functor ${\displaystyle p:{\mathcal {C}}\to }$ the category of (relative) schemes that has an effective descent theory. Here, let ${\displaystyle {\mathcal {Q}}coh}$ denote the category consisting of pairs ${\displaystyle (U,F)}$ consisting of a (Zariski)-open subset U and a quasi-coherent sheaf on it and ${\displaystyle p}$ the forgetful functor ${\displaystyle (U,F)\mapsto U}$.

There is a succinct statement for the major result in this area: (the prestack of quasi-coherent sheaves over a scheme S means that, for any S-scheme X, each X-point of the prestack is a quasi-coherent sheaf on X.)

The proof uses Zariski descent and the faithfully flat descent in the affine case.

Here "quasi-compact" cannot be eliminated; see https://mathoverflow.net/questions/127362/counter-example-to-faithfully-flat-descent/

• Amitsur complex
• Hilbert scheme
• Quot scheme

• SGA 1, Ch VIII – this is the main reference
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Street, Ross (20 Mar 2003). "Categorical and combinatorial aspects of descent theory". arXiv:math/0303175. (a detailed discussion of a 2-category)
• Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (Updated September 2, 2008)
• Waterhouse, William (1979), Introduction to affine group schemes, Graduate Texts in Mathematics, 66, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-6217-6, ISBN 978-0-387-90421-4, MR 0547117