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(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.
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 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.
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Famous quotes containing the word resolution:
“A great many will find fault in the resolution that the negro shall be free and equal, because our equal not every human being can be; but free every human being has a right to be. He can only be equal in his rights.”
—Mrs. Chalkstone, U.S. suffragist. As quoted in History of Woman Suffrage, vol. 2, ch. 16, by Elizabeth Cady Stanton, Susan B. Anthony, and Matilda Joslyn Gage (1882)