Faithfully flat descent

Given a faithfully flat ring homomorphism $A\; \backslash to\; B$, 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 $f\_1,\; \backslash dots,\; f\_r$ generating the unit ideal of A, $B\; =\; \backslash prod\_i\; A[f\_i^\{-1\}]$ is faithfully flat over $A$. Geometrically, $\backslash operatorname\{Spec\}(B)\; =\; \backslash bigcup\_\{i\; =\; 1\}^r\; \backslash operatorname\{Spec\}(A[f\_i^\{-1\}])$ is an open cover of $\backslash operatorname\{Spec\}(A)$ 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 $\backslash operatorname\{Spec\}(A[f\_i^\{-1\},\; f\_j^\{-1\}])$.

{{expand section|date=March 2023}}

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

:$\backslash varphi^1\; =\; \backslash varphi^0\; \backslash circ\; \backslash varphi^2$

where $\backslash varphi^i\; :\; N\; \backslash otimes\; B^\{\backslash otimes\; 2\}\; \backslash overset\{\backslash sim\}\backslash to\; N\; \backslash otimes\; B^\{\backslash otimes\; 2\}$ are given as:{{harvnb|Waterhouse|1979|loc=§ 17.1.}}

:$\backslash varphi^0(n\; \backslash otimes\; b\; \backslash otimes\; c)\; =\; \backslash rho^1(b)\; \backslash varphi(n\; \backslash otimes\; c)$

:$\backslash varphi^1(n\; \backslash otimes\; b\; \backslash otimes\; c)\; =\; \backslash rho^2(b)\; \backslash varphi(n\; \backslash otimes\; c)$

:$\backslash varphi^2(n\; \backslash otimes\; b\; \backslash otimes\; c)\; =\; \backslash varphi(n\; \backslash otimes\; b)\; \backslash otimes\; c$

with $\backslash rho^i(x)(y\_0\; \backslash otimes\; \backslash cdots\; \backslash otimes\; y\_r)\; =\; y\_0\; \backslash cdots\; y\_\{i-1\}\; \backslash otimes\; x\; \backslash otimes\; y\_i\; \backslash cdots\; y\_r$. Note the isomorphisms $\backslash varphi^i\; :\; N\; \backslash otimes\; B^\{\backslash otimes\; 2\}\; \backslash overset\{\backslash sim\}\backslash to\; N\; \backslash otimes\; B^\{\backslash otimes\; 2\}$ are determined only by $\backslash varphi$ 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 $B^\{\backslash otimes\; 2\}$-module isomorphism $\backslash varphi\; :\; N\; \backslash otimes\; B\; \backslash overset\{\backslash sim\}\backslash to\; N\; \backslash otimes\; B$ such that $\backslash varphi^1\; =\; \backslash varphi^0\; \backslash circ\; \backslash varphi^2$, an invariant submodule:

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

is such that $M\; \backslash otimes\; B\; =\; N$.{{harvnb|Waterhouse|1979|loc=§ 17.2.}}

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

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

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

{{math_theorem|name=Descent datum|Given a ring homomorphism $A\; \backslash to\; B$, a descent datum on a module N on $B$ is a $B^\{\backslash otimes\; 2\}$-module isomorphism

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

that satisfies the cocycle condition:{{harvnb|Vistoli|2008|loc=§ 4.2.1.}} NB: in the reference, the index starts with 1 instead of 0. $\backslash varphi\; \backslash otimes\_\{d^1\}\; B^\{\backslash otimes\; 3\}$ is the same as the composition $\backslash varphi\; \backslash otimes\_\{d^0\}\; B^\{\backslash otimes\; 3\}\; \backslash circ\; \backslash varphi\; \backslash otimes\_\{d^2\}\; B^\{\backslash otimes\; 3\}$.}}

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

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

Consider the natural map

:$M\; \backslash otimes\; B\; \backslash to\; N,\; \backslash ,\; x\; \backslash otimes\; a\; \backslash mapsto\; xa$.

The key point is that this map is an isomorphism if $A\; \backslash to\; B$ is faithfully flat.{{harvnb|SGA I|loc=Exposé VIII, Lemme 1.6.}} This is seen by considering the following:

:$\backslash begin\{array\}\{lccclcl\}$

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:

{{math_theorem|Given a faithfully flat ring homomorphism $A\; \backslash to\; B$, the functor

:$M\; \backslash mapsto\; (M\; \backslash otimes\_A\; B,\; \backslash varphi)$

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

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 $\backslash mathcal\{Q\}coh(X)$ 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 $U\_i\; \backslash subset\; X$ with $X\; =\; \backslash bigcup\; U\_i$ and isomorphisms $\backslash varphi\_\{ij\}\; :\; F\_i\; |\_\{U\_i\; \backslash cap\; U\_j\}\; \backslash overset\{\backslash sim\}\backslash to\; F\_j\; |\_\{U\_i\; \backslash cap\; U\_j\}$ such that (1) $\backslash varphi\_\{ii\}\; =\; \backslash operatorname\{id\}$ and (2) $\backslash varphi\_\{ik\}\; =\; \backslash varphi\_\{jk\}\; \backslash circ\; \backslash varphi\_\{ij\}$ on $U\_i\; \backslash cap\; U\_j\; \backslash cap\; U\_k$, then exists a unique quasi-coherent sheaf $F$ on X such that $F|\_\{U\_i\}\; \backslash simeq\; F\_i$ in a compatible way (i.e., $F|\_\{U\_j\}\; \backslash simeq\; F\_j$ restricts to $F|\_\{U\_i\; \backslash cap\; U\_j\}\; \backslash simeq\; F\_i|\_\{U\_i\; \backslash cap\; U\_j\}\; \backslash overset\{\backslash varphi\_\{ij\}\}\backslash underset\{\backslash sim\}\backslash to\; F\_j|\_\{U\_i\; \backslash cap\; U\_j\}$).{{harvnb|Hartshorne|1977|loc=Ch. II, Exercise 1.22.}}; NB: since "quasi-coherent" is a local property, gluing quasi-coherent sheaves results in a quasi-coherent one.

In a fancy language, the Zariski descent states that, with respect to the Zariski topology, $\backslash mathcal\{Q\}coh$ is a stack; i.e., a category $\backslash mathcal\{C\}$ equipped with the functor $p\; :\; \backslash mathcal\{C\}\; \backslash to$ the category of (relative) schemes that has an effective descent theory. Here, let $\backslash mathcal\{Q\}coh$ 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 $(U,\; F)\; \backslash mapsto\; U$.

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.)

{{math_theorem|The prestack of quasi-coherent sheaves over a base scheme S is a stack with respect to the fpqc topology.{{cite book|last=Fantechi|first=Barbara|title=Fundamental Algebraic Geometry: Grothendieck's FGA Explained|url=https://books.google.com/books?id=KxH0BwAAQBAJ&pg=PA82|accessdate=3 March 2018|year=2005|publisher=American Mathematical Soc.|isbn=9780821842454|page=82}}}}

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

Here "quasi-compact" cannot be eliminated.{{Cite web |last=Benoist |first=Olivier |title=Counter-example to faithfully flat descent |url=https://mathoverflow.net/q/127373 }}

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

:$V\; \backslash otimes\_k\; F\; \backslash simeq\; \backslash prod\_\{\backslash sigma\}\; V,\; \backslash ,\; v\; \backslash otimes\; a\; \backslash mapsto\; \backslash sigma(a)v$

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

{{expand section|date=March 2023}}

An étale descent is a consequence of a faithfully descent.

Let $f\; :\; X\; \backslash to\; Y$ be a morphism of schemes and $f\_*,\; f^*$ denote the pushforward as well the pullback for quasi-coherent sheaves (here, for simplicity, assume $f\_*\; :\; \backslash textrm\{QCoh\}(X)\; \backslash to\; \backslash textrm\{QCoh\}(Y)$ is well-defined.See https://math.stackexchange.com/questions/1109747/when-is-the-pushforward-of-a-quasi-coherent-sheaf-quasi-coherent-hartshorne-pro/3665838#3665838 for this kind of matter.) Since $f^*$ is a left adjoint of $f\_*$, the composition $T\; =\; f^*\; f\_*$ together with the counit and the comultiplication induced by the adjunction is a comonad. Then Beck's monadicity theorem, if applicable, says that the functor

:$f^*\; :\; \backslash textrm\{QCoh\}(Y)\; \backslash to\; T-\backslash textrm\{Coalg\}$

is an equivalence, where $T-\backslash textrm\{Coalg\}$ is the Eilenberg–Moore category of $T$-coalgebras; i.e., roughly, the category consists of objects in $\backslash textrm\{QCoh\}(X)$ with $T$-coactions $a\; :\; F\; \backslash to\; T(F)$ (despite the name, they are more like comodules than coalgebras). Then the key point here is that a $T$-action amounts to a descent data and thus $T-\backslash textrm\{Alg\}$ can be identified as the category of quasi-coherent sheaves on $X$ together with descent data. Hence, the above exactly states the flat descent.

For example,{{harvnb|Deligne|2007|loc=§ 4.2.}} if $f$ is a faithfully flat morphism between affine schemes, then the monadicity theorem applies and the above recovers the flat descent in the affine case. More generally, the theorem applies if $f$ is faithfully flat and has some finiteness property; e.g., a fpqc morphism.{{fact|date=April 2025}}

• Amitsur complex
• Hilbert scheme
• Quot scheme

{{reflist}}

• {{citation |ref={{harvid|SGA I}} |title=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)}}
• {{Citation |doi=10.1007/978-0-8176-4575-5_3 |chapter=Catégories tannakiennes |title=The Grothendieck Festschrift, Volume II |series=Modern Birkhäuser Classics |date=2007 |last1=Deligne |first1=P. |pages=111–195 |isbn=978-0-8176-4567-0 }}
• {{Citation | last1=Giraud | first1=Jean | author1-link= Jean Giraud (mathematician) | title=Méthode de la descent | journal=Mémoires de la Société Mathématique de France | year=1964 | volume=2 | pages=1–150 | doi=10.24033/msmf.2 | mr=190142| doi-access=free }}
• {{Hartshorne AG}}
• {{Citation |doi=10.1023/B:APCS.0000049317.24861.36 |title=Categorical and Combinatorial Aspects of Descent Theory |date=2004 |last1=Street |first1=Ross |journal=Applied Categorical Structures |volume=12 |issue=5–6 |pages=537–576 |arxiv=math/0303175 }} (a detailed discussion of a 2-category)
• {{cite web |first=Angelo |last=Vistoli |url=http://homepage.sns.it/vistoli/descent.pdf |title=Notes on Grothendieck topologies, fibered categories and descent theory |date=September 2, 2008}}
• {{Citation | last1=Waterhouse | first1=William | author1-link=William C. Waterhouse | title=Introduction to affine group schemes | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90421-4 | year=1979 | volume=66 | doi=10.1007/978-1-4612-6217-6 | mr=0547117}}

• https://mathoverflow.net/questions/444034/using-comonadicity-to-prove-faithfully-flat-descent

Category:Algebraic geometry