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.
