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. ,

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.

**References**

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.

Jacobson, N. *Basic Algebra II, 2nd ed.* New York: W. H. Freeman, p. 339, 1989.

© 1996-9

1999-05-25