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