## Resolution

Resolution is a widely used word with many different meanings. It can refer to resolution of equations, resolution of singularities (in Algebraic Geometry), resolution of modules or more sophisticated structures, etc. In a Block Design, a Partition of a BIBD's set of blocks into Parallel Classes, each of which in turn partitions the set , is called a resolution (Abel and Furino 1996).

A resolution of the Module over the Ring is a complex of -modules and morphisms and a Morphism such that

satisfying the following conditions:
1. The composition of any two consecutive morphisms is the zero map,

2. For all , ,

3. ,
where ker is the kernel and im is the image. Here, the quotient

is the th Homology Group.

If all modules are projective (free), then the resolution is called projective (free). There is a similar concept for resolutions to the right'' of , which are called injective resolutions.

Abel, R. J. R. and Furino, S. C. Resolvable and Near Resolvable Designs.'' §I.6 in The CRC Handbook of Combinatorial Designs (Ed. C. J. Colbourn and J. H. Dinitz). Boca Raton, FL: CRC Press, pp. 4 and 87-94, 1996.