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 balanced
incomplete block design, a set partition 
of the 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 chain complex of modules 
for 
, with morphisms 
for 
,
together with a morphism 
.
The augmented sequence
 |
(1)
|
satisfies the following conditions:
1. The composition of any two consecutive morphisms is the zero map,
2. For every 
, 
,
3. 
, and 
is surjective.
Here ker is the kernel and im is the image. Here, for 
, the quotient
 |
(2)
|
is the 
th
homology group.
If all modules 
are projective (free), then the resolution is called projective (free).
Dually, an injective resolution is a resolution
"to the right" of 
using injective modules.
In mathematical logic, the rule
 |
(3)
|
is known as resolution and is significant for automated theorem proving.
