Resolution (algebra) - Acyclic Resolution

Acyclic Resolution

In many cases one is not really interested in the objects appearing in a resolution, but in the behavior of the resolution with respect to a given functor. Therefore, in many situations, the notion of acyclic resolutions is used: given a left exact functor F: A → B between two abelian categories, a resolution

of an object M of A is called F-acyclic, if the derived functors R_iF(E_n) vanish for all i>0 and n≥0. Dually, a left resolution is acyclic with respect to a right exact functor if its derived functors vanish on the objects of the resolution.

For example, given a R module M, the tensor product is a right exact functor Mod(R) → Mod(R). Every flat resolution is acyclic with respect to this functor. A flat resolution is acyclic for the tensor product by every M. Similarly, resolutions that are acyclic for all the functors Hom( ⋅, M) are the projective resolutions and those that are acyclic for the functors Hom(M, ⋅ ) are the injective resolutions.

Any injective (projective) resolution is F-acyclic for any left exact (right exact, respectively) functor.

The importance of acyclic resolutions lies in the fact that the derived functors R_iF (of a left exact functor, and likewise L_iF of a right exact functor) can be obtained from as the homology of F-acyclic resolutions: given an acyclic resolution of an object M, we have

where right hand side is the i-th homology object of the complex

This situation applies in many situations. For example, for the constant sheaf R on a differentiable manifold M can be resolved by the sheaves of smooth differential forms: The sheaves are fine sheaves, which are known to be acyclic with respect to the global section functor . Therefore, the sheaf cohomology, which is the derived functor of the global section functor Γ is computed as

Similarly Godement resolutions are acyclic with respect to the global sections functor.