# 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

The homomorphisms *d _{i}* 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** (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

where each *C ^{i}* 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

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 *E*_{i} resolving the given module *M*. For example, a *free resolution* of a module *M* is a left resolution in which all the modules *E*_{i} are free *R*-modules. Likewise, *projective* and *flat* resolutions are left resolutions such that all the *E*_{i} are projective and flat *R*-modules, respectively. Injective resolutions are *right* resolutions whose *C*^{i} 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 *E*_{0} to be the free *R*-module generated by the elements of *M*, and then *E*_{1} to be the free *R*-module generated by the elements of the kernel of the natural map *E*_{0} → *M* 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 *P*_{0} → *M* and *P*_{1} → *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(*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 *E*_{i} may be graded in such a way that the *d*_{i} 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 *E*_{i} is minimal. The number of basis elements of each *E*_{i} 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 *E*_{i} 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** [*π*_{1}(*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 *E _{i}* and

*C*are objects in

^{i}*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 *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 *R*_{i}*F*(*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*_{i}*F* (of a left exact functor, and likewise *L*_{i}*F* 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.

## See also

## Notes

- ↑ Template:Harvnb uses
*coresolution*, though*right resolution*is more common, as in Template:Harvnb - ↑ Template:Nlab, Template:Nlab
- ↑ Template:Harvnb

## References

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