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

(1)

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

(2)

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.

In mathematical logic, the rule

(3)

is known as resolution and is significant for automated theorem proving.

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.