Resolution (algebra)

In mathematics, particularly in abstract algebra and 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 in an abelian category), which is used to describe the structure of a specific module or object of this category. In particular, projective and injective resolutions induce a quasi-isomorphism between the exact sequence and the module, which may be regarded as a weak equivalence, with the resolution having nicer properties as a space.[2]

Generally, the objects in the sequence are restricted to have some property P (for example to be free). Thus one speaks of a P resolution: for example, a flat resolution, a free resolution, an injective resolution, a projective resolution. The sequence is supposed to be infinite to the left (to the right for a coresolution). However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero.

Resolutions of modules

Definitions

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

$\cdots\overset{d_{n+1}}{\longrightarrow}E_n\overset{d_n}{\longrightarrow}\cdots\overset{d_3}{\longrightarrow}E_2\overset{d_2}{\longrightarrow}E_1\overset{d_1}{\longrightarrow}E_0\overset{\epsilon}{\longrightarrow}M\longrightarrow0.$

The homomorphisms di are called boundary maps. The map ε is called an augmentation map. For succinctness, the resolution above can be written as

$E_\bullet\overset{\epsilon}{\longrightarrow}M\longrightarrow0.$

The dual notion is that of a right resolution (or coresolution, or simply resolution). Specifically, given a module M over a ring R, a right resolution is a possibly infinite exact sequence of R-modules

$0\longrightarrow M\overset{\epsilon}{\longrightarrow}C^0\overset{d^0}{\longrightarrow}C^1\overset{d^1}{\longrightarrow}C^2\overset{d^2}{\longrightarrow}\cdots\overset{d^{n-1}}{\longrightarrow}C^n\overset{d^n}{\longrightarrow}\cdots,$

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). For succinctness, the resolution above can be written as

$0\longrightarrow M\overset{\epsilon}{\longrightarrow}C^\bullet.$

A (co)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.[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 E0M etc. Dually, every R-module possesses an injective resolution. Flat 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 resolution P0M and P1M 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(M). 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(M) and flat dimension fd(M) 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 (or simplicial) 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 [π1(X)].

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 (resp. injective) resolution, then A is said to have enough projectives (resp. enough injectives). 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 MM' , together with injective resolutions

$0 \rightarrow M \rightarrow I_*, \ \ 0 \rightarrow M' \rightarrow I'_*,$

there is in general no functorial way of obtaining a map between $I_*$ and $I'_*$.

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: AB between two abelian categories, a resolution

$0 \rightarrow M \rightarrow E_0 \rightarrow E_1 \rightarrow E_2 \rightarrow \dots$

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.

For example, given a R module M, the tensor product   $\otimes_R M$ 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 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 $E_*$ of an object M, we have

$R_i F(M) = H_i F(E_*),$

where right hand side is the i-th homology object of the complex $F(E_*).$

This situation applies in many situations. For example, for the constant sheaf R on a differentiable manifold M can be resolved by the sheaves $\mathcal C^*(M)$ of smooth differential forms: $0 \rightarrow R \subset \mathcal C^0(M) \stackrel d \rightarrow \mathcal C^1(M) \stackrel d \rightarrow \dots \mathcal C^{dim M}(M) \rightarrow 0.$ The sheaves $\mathcal C^*(M)$ are fine sheaves, which are known to be acyclic with respect to the global section functor $\Gamma: \mathcal F \mapsto \mathcal F(M)$. Therefore, the sheaf cohomology, which is the derived functor of the global section functor Γ is computed as $\mathrm H^i(M, \mathbf R) = \mathrm H^i( \mathcal C^*(M)).$

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