Faithfully flat descent

Last updated January 15, 2024

Faithfully flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully flat morphism. Such morphisms, that are flat and surjective, are common, one example coming from an open cover.

Idea

Given a faithfully flat ring homomorphism $A\to B$ , the faithfully flat descent is, roughy, the statement that to give a module or an algebra over A is to give a module or an algebra over $B$ together with the so-called descent datum (or data). That is to say one can descend the objects (or even statements) on $B$ to $A$ provided some additional data.

For example, given some elements $f_{1},\dots ,f_{r}$ generating the unit ideal of A, $B=\prod _{i}A[f_{i}^{-1}]$ is faithfully flat over $A$ . Geometrically, $\operatorname {Spec} (B)=\bigcup _{i=1}^{r}\operatorname {Spec} (A[f_{i}^{-1}])$ is an open cover of $\operatorname {Spec} (A)$ and so descending a module from $B$ to $A$ would mean gluing modules $M_{i}$ on $A[f_{i}^{-1}]$ to get a module on A; the descend datum in this case amounts to the gluing data; i.e., how $M_{i},M_{j}$ are identified on overlaps $\operatorname {Spec} (A[f_{i}^{-1},f_{j}^{-1}])$ .

Affine case

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

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

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

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

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

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

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

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

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

is such that $M\otimes B=N$ .^[3]

Here is the precise definition of descent datum. Given a ring homomorphism $A\to B$ , we write:

d^{i}:B^{\otimes n}\to B^{\otimes {n+1}}

for the map given by inserting $A\to B$ in the i-th spot; i.e., $d^{0}$ is given as $B^{\otimes n}\simeq A\otimes _{A}B^{\otimes n}\to B\otimes _{A}B^{\otimes n}=B^{\otimes {n+1}}$ , $d^{1}$ as $B^{\otimes n}\simeq B\otimes A\otimes B^{\otimes n-1}\to B^{\otimes {n+1}}$ , etc. We also write $-\otimes _{d^{i}}B^{\otimes {n+1}}$ for tensoring over $B^{\otimes n}$ when $B^{\otimes {n+1}}$ is given the module structure by $d^{i}$ .

Descent datum — Given a ring homomorphism $A\to B$ , a descent datum on a module N on $B$ is a $B^{\otimes 2}$ -module isomorphism

\varphi :N\otimes _{d^{1}}B^{\otimes 2}{\overset {\sim }{\to }}N\otimes _{d^{0}}B^{\otimes 2}

that satisfies the cocycle condition:^[4] $\varphi \otimes _{d^{1}}B^{\otimes 3}$ is the same as the composition $\varphi \otimes _{d^{0}}B^{\otimes 3}\circ \varphi \otimes _{d^{2}}B^{\otimes 3}$ .

Now, given a $B$ -module $N$ with a descent datum $\varphi$ , define $M$ to be the kernel of

d^{0}-\varphi \circ d^{1}:N\to N\otimes _{d^{0}}B^{\otimes 2}

.

Consider the natural map

M\otimes B\to N,\,x\otimes a\mapsto xa

.

The key point is that this map is an isomorphism if $A\to B$ is faithfully flat.^[5] This is seen by considering the following:

{\begin{array}{lccclcl}0&\to &M\otimes _{A}B&\to &\quad N\otimes _{A}B&{\xrightarrow {d^{0}-\varphi \circ d^{1}}}&N\otimes _{d^{0}}B^{\otimes 2}\otimes _{A}B\\&&\downarrow &&\varphi \circ d^{1}\downarrow &&\quad \downarrow \varphi \otimes _{d^{0},d^{1}}B^{\otimes 3}\circ d^{2}\\0&\to &N&\to &\quad N\otimes _{d^{0}}B^{\otimes 2}&{\xrightarrow {d^{0}-d^{1}}}&N\otimes _{d^{0},d^{1}}B^{\otimes 3}\\\end{array}}

where the top row is exact by the flatness of B over A and the bottom row is the Amitsur complex, which is exact by a theorem of Grothendieck. The cocycle condition ensures that the above diagram is commutative. Since the second and the third vertical maps are isomorphisms, so is the first one.

The forgoing can be summarized simply as follows:

Theorem — Given a faithfully flat ring homomorphism $A\to B$ , the functor

M\mapsto (M\otimes _{A}B,\varphi )

from the category of A-modules to the category of pairs $(N,\varphi )$ consisting of a B-module N and a descent datum $\varphi$ on it is an equivalence.

Zariski descent

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 ${\mathcal {Q}}coh(X)$ denote the category of quasi-coherent sheaves on a scheme X. Then Zariski descent states that, given quasi-coherent sheaves $F_{i}$ on open subsets $U_{i}\subset X$ with $X=\bigcup U_{i}$ and isomorphisms $\varphi _{ij}:F_{i}|_{U_{i}\cap U_{j}}{\overset {\sim }{\to }}F_{j}|_{U_{i}\cap U_{j}}$ such that (1) $\varphi _{ii}=\operatorname {id}$ and (2) $\varphi _{ik}=\varphi _{jk}\circ \varphi _{ij}$ on $U_{i}\cap U_{j}\cap U_{k}$ , then exists a unique quasi-coherent sheaf $F$ on X such that $F|_{U_{i}}\simeq F_{i}$ in a compatible way (i.e., $F|_{U_{j}}\simeq F_{j}$ restricts to $F|_{U_{i}\cap U_{j}}\simeq F_{i}|_{U_{i}\cap U_{j}}{\overset {\varphi _{ij}}{\underset {\sim }{\to }}}F_{j}|_{U_{i}\cap U_{j}}$ ).^[6]

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

Descent for quasi-coherent sheaves

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.)

Theorem — The prestack of quasi-coherent sheaves over a base scheme S is a stack with respect to the fpqc topology.^[7]

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

Here "quasi-compact" cannot be eliminated.^{[ citation needed ]}

Example: a vector space

Let F be a finite Galois field extension of a field k. Then, for each vector space V over F,

V\otimes _{k}F\simeq \prod _{\sigma }V,\,v\otimes a\mapsto \sigma (a)v

where the product runs over the elements in the Galois group of $F/k$ .

Specific descents

fpqc descent

Étale descent

An étale descent is a consequence of a faithfully descent.

Galois descent

Notes

↑ Deligne, Pierre (1990), Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Progress in Math., vol. 87, Birkhäuser, pp. 111–195
↑ Waterhouse 1979 , § 17.1.
↑ Waterhouse 1979 , § 17.2.
↑ Vistoli 2008, § 4.2.1. NB: in the reference, the index starts with 1 instead of 0.
↑ SGA I , Exposé VIII, Lemme 1.6.
↑ Hartshorne 1977 , Ch. II, Exercise 1.22.; NB: since "quasi-coherent" is a local property, gluing quasi-coherent sheaves results in a quasi-coherent one.
↑ Fantechi, Barbara (2005). Fundamental Algebraic Geometry: Grothendieck's FGA Explained. American Mathematical Soc. p. 82. ISBN 9780821842454 . Retrieved 3 March 2018.

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References

SGA 1, Exposé VIII – this is the main reference (but it depends on a result from Giraud (1964), which replaced (in much more general form) the unpublished Exposé VII of SGA1)
Deligne, P. (2007), "Catégories tannakiennes", The Grothendieck Festschrift, Volume II, Modern Birkhäuser Classics, pp. 111–195, doi:10.1007/978-0-8176-4575-5_3, ISBN 978-0-8176-4567-0
Giraud, Jean (1964), "Méthode de la descent", Mémoires de la Société Mathématique de France, 2: 1–150, doi: 10.24033/msmf.2 , MR 0190142
Hartshorne, Robin (1977), Algebraic Geometry , Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Street, Ross (2004), "Categorical and Combinatorial Aspects of Descent Theory", Applied Categorical Structures, 12 (5–6): 537–576, arXiv: math/0303175 , doi:10.1023/B:APCS.0000049317.24861.36 (a detailed discussion of a 2-category)
Vistoli, Angelo (September 2, 2008). "Notes on Grothendieck topologies, fibered categories and descent theory" (PDF).
Waterhouse, William (1979), Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-6217-6, ISBN 978-0-387-90421-4, MR 0547117

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[1] Deligne, Pierre (1990), Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Progress in Math., vol. 87, Birkhäuser, pp. 111–195

[2] Waterhouse 1979 , § 17.1.

[3] Waterhouse 1979 , § 17.2.

[4] Vistoli 2008, § 4.2.1. NB: in the reference, the index starts with 1 instead of 0.

[5] SGA I , Exposé VIII, Lemme 1.6.

[6] Hartshorne 1977 , Ch. II, Exercise 1.22.; NB: since "quasi-coherent" is a local property, gluing quasi-coherent sheaves results in a quasi-coherent one.

[Fantechi2005-7] Fantechi, Barbara (2005). Fundamental Algebraic Geometry: Grothendieck's FGA Explained. American Mathematical Soc. p. 82. ISBN 9780821842454 . Retrieved 3 March 2018.

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