# Resolution

In homological algebra a right resolution of a module $A$ is a complex (in homological algebra) $C$: $C _ {0} \rightarrow C _ {1} \rightarrow \dots$, defined for positive degrees and provided with a supplementary homomorphism $A \rightarrow C ^ {0}$ such that the sequence $0 \rightarrow A \rightarrow C ^ {0} \rightarrow C ^ {1} \rightarrow \dots$ is exact (cf. Exact sequence).

The supplementary homomorphism $A \rightarrow C ^ {0}$ can also be seen as a homomorphism of complexes $A \rightarrow C$, where $A$ is viewed as a complex concentrated in degree zero. The right resolution $0 \rightarrow A \rightarrow C ^ {0} \rightarrow \dots$ is called injective if the modules $C ^ {i}$ are all injective (cf. Injective module). Dually, a left resolution is an exact sequence $\dots \rightarrow P _ {1} \rightarrow P _ {0} \rightarrow A \rightarrow 0$. Such a left resolution is called projective if all the modules $P _ {i}$ are projective, free if all the $P _ {i}$ are free, and flat if all the $P _ {i}$ are flat (cf. Projective module; Flat module).
More generally, the notion of a resolution of an object can be defined in any Abelian category in a completely similar way, [a1]. E.g., in the category of sheaves of Abelian groups on a topological space an injective resolution of a sheaf $A$ is an exact sequence $0 \rightarrow A \rightarrow C ^ {0} \rightarrow \dots$ of sheaves of Abelian groups with each $C ^ {i}$ an injective sheaf. In sheaf theory one often uses resolutions by flabby or soft sheaves (cf. Flabby sheaf; Soft sheaf). For the case of sheaves over a topos see [a5], [a6].