In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution[1]) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to define invariants characterizing the structure of a specific module or object of this category.
However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the zero-object.
[2] Generally, the objects in the sequence are restricted to have some property P (for example to be free).
Given a module M over a ring R, a left resolution (or simply resolution) of M is an exact sequence (possibly infinite) of R-modules The homomorphisms di are called boundary maps.
Specifically, given a module M over a ring R, a right resolution is a possibly infinite exact sequence of R-modules where each Ci is an R-module (it is common to use superscripts on the objects in the resolution and the maps between them to indicate the dual nature of such a resolution).
Likewise, projective and flat resolutions are left resolutions such that all the Ei are projective and flat R-modules, respectively.
Every R-module possesses a free left resolution.
[3] A fortiori, every module also admits projective and flat resolutions.
The proof idea is to define E0 to be the free R-module generated by the elements of M, and then E1 to be the free R-module generated by the elements of the kernel of the natural map E0 → M etc.
Dually, every R-module possesses an injective resolution.
Projective resolution of a module M is unique up to a chain homotopy, i.e., given two projective resolutions P0 → M and P1 → M of M there exists a chain homotopy between them.
The behavior of these dimensions reflects characteristics of the ring.
Among these graded free resolutions, the minimal free resolutions are those for which the number of basis elements of each Ei is minimal.
The number of basis elements of each Ei and their degrees are the same for all the minimal free resolutions of a graded module.
If I is a homogeneous ideal in a polynomial ring over a field, the Castelnuovo–Mumford regularity of the projective algebraic set defined by I is the minimal integer r such that the degrees of the basis elements of the Ei in a minimal free resolution of I are all lower than r-i.
A classic example of a free resolution is given by the Koszul complex of a regular sequence in a local ring or of a homogeneous regular sequence in a graded algebra finitely generated over a field.
Let X be an aspherical space, i.e., its universal cover E is contractible.
However, such resolutions need not exist in a general abelian category A.
has a presentation given by an exact sequence The first two terms are not in general projective since
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 RiF(En) 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.
Every flat resolution is acyclic with respect to this functor.
The importance of acyclic resolutions lies in the fact that the derived functors RiF (of a left exact functor, and likewise LiF of a right exact functor) can be obtained from as the homology of F-acyclic resolutions: given an acyclic resolution
For example, for the constant sheaf R on a differentiable manifold M can be resolved by the sheaves
are fine sheaves, which are known to be acyclic with respect to the global section functor
Similarly Godement resolutions are acyclic with respect to the global sections functor.