Faithfully flat descent

Faithfully flat descent or 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.

In practice, from an affine point of view, this technique allows one to prove some statement about a ring or scheme after faithfully flat base change.

In the language of stacks, flat descent is exactly the statement that the prestack of quasi-coherent sheaves is a stack with respect to étale (or fpqc) topology.

"Vanilla" faithfully flat descent is generally false; instead, faithfully flat descent is valid under some finiteness conditions (e.g., quasi-compact or locally of finite presentation).

A faithfully flat descent is a special case of Beck's monadicity theorem.

Idea

Given a faithfully flat ring homomorphism

A

→

B

{\displaystyle A\to B}

, the faithfully flat descent is, roughly, the statement that to give a module or an algebra over A is to give a module or an algebra over

B

{\displaystyle B}

together with the so-called descent datum (or data). That is to say one can descend the objects (or even statements) on

B

{\displaystyle B}

to

A

{\displaystyle A}

provided some additional data.

For example, given some elements

f

1

,

…

,

f

r

{\displaystyle f_{1},\dots ,f_{r}}

generating the unit ideal of A,

B

=

∏

i

A

[

f

i

−

1

]

{\displaystyle B=\prod _{i}A[f_{i}^{-1}]}

is faithfully flat over

A

{\displaystyle A}

. Geometrically,

Spec

⁡

(

B

)

=

⋃

i

=

1

r

Spec

⁡

(

A

[

f

i

−

1

]

)

{\displaystyle \operatorname {Spec} (B)=\bigcup _{i=1}^{r}\operatorname {Spec} (A[f_{i}^{-1}])}

is an open cover of

Spec

⁡

(

A

)

{\displaystyle \operatorname {Spec} (A)}

and so descending a module from

B

{\displaystyle B}

to

A

{\displaystyle A}

would mean gluing modules

M

i

{\displaystyle M_{i}}

on

A

[

f

i

−

1

]

{\displaystyle 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

{\displaystyle M_{i},M_{j}}

are identified on overlaps

Spec

⁡

(

A

[

f

i

−

1

,

f

j

−

1

]

)

{\displaystyle \operatorname {Spec} (A[f_{i}^{-1},f_{j}^{-1}])}

.