Editing Flat module (section)

== Flat resolutions ==
A '''flat resolution''' of a module <math>M</math> is a [[resolution of a module|resolution]] of the form
: <math>\cdots \to F_2 \to F_1 \to F_0 \to M \to 0,</math>
where the <math>F_i</math> are all flat modules. Any free or projective resolution is necessarily a flat resolution. Flat resolutions can be used to compute the [[Tor functor]].

The ''length'' of a finite flat resolution is the first subscript ''n'' such that <math>F_n</math> is nonzero and <math>F_i=0</math> for <math>i>n</math>. If a module <math>M</math> admits a finite flat resolution, the minimal length among all finite flat resolutions of <math>M</math> is called its [[flat dimension]]{{sfn|Lam|1999|p=183|ps=none}} and denoted <math>\operatorname{fd}(M)</math>. If <math>M</math> does not admit a finite flat resolution, then by convention the flat dimension is said to be infinite. As an example, consider a module <math>M</math> such that <math>\operatorname{fd}(M)=0</math>. In this situation, the exactness of the sequence <math>0 \to F_0 \to M \to 0</math> indicates that the arrow in the center is an isomorphism, and hence <math>M</math> itself is flat.{{efn|A module isomorphic to a flat module is of course flat.}}

In some areas of module theory, a flat resolution must satisfy the additional requirement that each map is a flat pre-cover of the kernel of the map to the right. For projective resolutions, this condition is almost invisible: a projective pre-cover is simply an [[epimorphism]] from a projective module. These ideas are inspired from Auslander's work in approximations. These ideas are also familiar from the more common notion of minimal projective resolutions, where each map is required to be a [[projective cover]] of the kernel of the map to the right. However, projective covers need not exist in general, so minimal projective resolutions are only of limited use over rings like the integers.