# Resolution (algebra)

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), 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 (left) 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 (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). 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** (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. Projective resolutions (and, more generally, 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 resolutions *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.

## Notes

- Jacobson 2009, §6.5 uses
*coresolution*, though*right resolution*is more common, as in Weibel 1994, Chap. 2 - projective resolution in
*nLab*, resolution in*nLab* - Jacobson 2009, §6.5

## References

- Iain T. Adamson (1972),
*Elementary rings and modules*, University Mathematical Texts, Oliver and Boyd, ISBN 0-05-002192-3 - Eisenbud, David (1995),
*Commutative algebra. With a view toward algebraic geometry*, Graduate Texts in Mathematics,**150**, Berlin, New York: Springer-Verlag, ISBN 3-540-94268-8, MR 1322960, Zbl 0819.13001 - Jacobson, Nathan (2009) [1985],
*Basic algebra II*(Second ed.), Dover Publications, ISBN 978-0-486-47187-7 - Lang, Serge (1993),
*Algebra*(Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001 - Weibel, Charles A. (1994).
*An introduction to homological algebra*. Cambridge Studies in Advanced Mathematics.**38**. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.