Faithfully flat descent is a technique from
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, allowing one to draw conclusions about objects on the target of a
faithfully flat morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,
:f_P\colon \mathcal_ \t ...
. 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.
"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
In category theory, a branch of mathematics, Beck's monadicity theorem gives a criterion that characterises monadic functors, introduced by in about 1964. It is often stated in dual form for comonads. It is sometimes called the Beck tripleabi ...
.
Idea
Given a
faithfully flat ring homomorphism
In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact seq ...
, the faithfully flat descent is, roughy, the statement that to give a module or an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
over ''A'' is to give a module or an algebra over
together with the so-called descent datum (or data). That is to say one can ''descend'' the objects (or even statements) on
to
provided some additional data.
For example, given some elements
generating the unit ideal of ''A'',