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 $R$ of a BIBD's set of blocks $B$ into Parallel Classes, each of which in turn partitions the set $V$, is called a resolution (Abel and Furino 1996).

A resolution of the Module $M$ over the Ring $R$ is a complex of $R$-modules $C_i$ and morphisms $d_i$ and a Morphism $\epsilon$ such that

$$\cdots \rightarrow C_i\rightarrow^{d_i}C_{i-1}\rightarrow\cdots\rightarrow C_0\rightarrow^{\epsilon} M\rightarrow 0$$

satisfying the following conditions:

1. The composition of any two consecutive morphisms is the zero map,

2. For all $i$, $(\mathop{\rm ker} d_i)/(\mathop{\rm im} d_{i+1}) = 0$,

3. $C_0/(\mathop{\rm ker}\epsilon) \simeq M$,

where ker is the kernel and im is the image. Here, the quotient

$${(\mathop{\rm ker} d_i)\over (\mathop{\rm im} d_{i+1})}$$

is the $i$th Homology Group.

If all modules $C_i$ are projective (free), then the resolution is called projective (free). There is a similar concept for resolutions ``to the right'' of $M$, which are called injective resolutions.

See also Homology Group, Module, Morphism, Ring

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.