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.
