Resolution (algebra)
In
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 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 (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 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
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.
Projective resolution of a module M is unique up to a
Resolutions are used to define
The injective and projective dimensions are used on the category of right R-modules to define a homological dimension for R called the right
Graded modules and algebras
Let M be a
If I is a
Examples
A classic example of a free resolution is given by the
Let X be an
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
there is in general no functorial way of obtaining a map between and .
Abelian categories without projective resolutions in general
One class of examples of Abelian categories without projective resolutions are the categories of coherent sheaves on a scheme . For example, if is projective space, any coherent sheaf on has a presentation given by an exact sequence
The first two terms are not in general projective since for . But, both terms are locally free, and locally flat. Both classes of sheaves can be used in place for certain computations, replacing projective resolutions for computing some derived functors.
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
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 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 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
Similarly Godement resolutions are acyclic with respect to the global sections functor.
See also
- Standard resolution
- Hilbert–Burch theorem
- Hilbert's syzygy theorem
- Free presentation
- Matrix factorizations (algebra)
Notes
- ^ Jacobson 2009, §6.5 uses coresolution, though right resolution is more common, as in Weibel 1994, Chap. 2
- ^ projective resolution at the nLab, resolution at the 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
- Zbl 0819.13001
- ISBN 978-0-486-47187-7
- Zbl 0848.13001
- OCLC 36131259.