Flat Module - Flat Resolutions

Flat Resolutions

A flat resolution of a module M is a resolution of the form

... → F₂ → F₁ → F₀ → M → 0

where the F_i 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 F_n is nonzero and F_i=0 for i greater than n. If a module M admits a finite flat resolution, the minimal length among all finite flat resolutions of M is called its flat dimension and denoted fd(M). If M does not admit a finite flat resolution, then by convention the flat dimension is said to be infinite. As an example, consider a module M such that fd(M) = 0. In this situation, the exactness of the sequence 0 → F₀ → M → 0 indicates that the arrow in the center is an isomorphism, and hence M itself is 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.

While projective covers for modules do not always exist, it was speculated that for general rings, every module would have a flat cover, that is, every module would be the epimorphic image of a flat module under a homomorphism with superfluous kernel. This flat cover conjecture was explicitly first stated in (Enochs 1981, p 196). The conjecture turned out to be true, resolved positively and proved simultaneously by L. Bican, R. El Bashir and E. Enochs. The was preceded by important contributions by P. Eklof, J. Trlifaj and J. Xu.

Since flat covers exist for all modules over all rings, minimal flat resolutions can take the place of minimal projective resolutions in many circumstances. The measurement of the departure of flat resolutions from projective resolutions is called relative homological algebra, and is covered in classics such as (MacLane 1963) and in more recent works focussing on flat resolutions such as (Enochs & Jenda 2000).