Resolution (algebra)


In mathematics, and more specifically in homological algebra, a resolution is an exact sequence of modules, which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for resolutions, and to the right for right resolutions. 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 or the rightmost object is the zero-object.
Generally, the objects in the sequence are restricted to have some property P. Thus one speaks of a P resolution. In particular, every module has free resolutions, projective resolutions and flat resolutions, which are left resolutions consisting, respectively of free modules, projective modules or flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules.

Resolutions of modules

Definitions

Given a module M over a ring R, a left resolution of M is an exact sequence of R-modules
The homomorphisms di are called boundary maps. The map ε is called an augmentation map. For succinctness, the resolution above can be written as
The dual notion is that of a right resolution. 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. For succinctness, the resolution above can be written as
A resolution is said to be finite if only finitely many of the modules involved are non-zero. The length of a finite resolution is the maximum index n labeling a nonzero module in the finite resolution.

Free, projective, injective, and flat resolutions

In many circumstances conditions are imposed on the modules Ei resolving the given module M. For example, a free resolution of a module M is a left resolution in which all the modules Ei are free R-modules. Likewise, projective and flat resolutions are left resolutions such that all the Ei are projective and flat R-modules, respectively. Injective resolutions are right resolutions whose Ci are all injective modules.
Every R-module possesses a free left resolution. 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 resolutions can be used to compute Tor functors.
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.
Resolutions are used to define homological dimensions. The minimal length of a finite projective resolution of a module M is called its projective dimension and denoted pd. For example, a module has projective dimension zero if and only if it is a projective module. If M does not admit a finite projective resolution then the projective dimension is infinite. For example, for a commutative local ring R, the projective dimension is finite if and only if R is regular and in this case it coincides with the Krull dimension of R. Analogously, the injective dimension id and flat dimension fd are defined for modules also.
The injective and projective dimensions are used on the category of right R modules to define a homological dimension for R called the right global dimension of R. Similarly, flat dimension is used to define weak global dimension. The behavior of these dimensions reflects characteristics of the ring. For example, a ring has right global dimension 0 if and only if it is a semisimple ring, and a ring has weak global dimension 0 if and only if it is a von Neumann regular ring.

Graded modules and algebras

Let M be a graded module over a graded algebra, which is generated over a field by its elements of positive degree. Then M has a free resolution in which the free modules Ei may be graded in such a way that the di and ε are graded linear maps. 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.

Examples

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. Then every singular chain complex of E is a free resolution of the module Z not only over the ring Z but also over the group ring Z .

Resolutions in abelian categories

The definition of resolutions of an object M in an abelian category A is the same as above, but the Ei and Ci are objects in A, and all maps involved are morphisms in A.
The analogous notion of projective and injective modules are projective and injective objects, and, accordingly, projective and injective resolutions. However, such resolutions need not exist in a general abelian category A. If every object of A has a projective resolution, then A is said to have enough projectives. Even if they do exist, such resolutions are often difficult to work with. For example, as pointed out above, every R-module has an injective resolution, but this resolution is not functorial, i.e., given a homomorphism M → M' , together with injective resolutions
there is in general no functorial way of obtaining a map between and.

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 RiF 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 → Mod. 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 are the projective resolutions and those that are acyclic for the functors Hom are the injective resolutions.
Any injective resolution is F-acyclic for any left exact functor.
The importance of acyclic resolutions lies in the fact that the derived functors RiF 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.