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In [[Homological algebra|homological algebra]] a right resolution of a module  $  A $
is a [[Complex (in homological algebra)|complex (in homological algebra)]]  $  C $: 
$  C _ {0} \rightarrow C _ {1} \rightarrow \cdots $, 
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 \cdots $
is exact (cf. [[Exact sequence|Exact sequence]]).

====Comments====
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 \cdots $
is called injective if the modules  $  C  ^ {i} $
are all injective (cf. [[Injective module|Injective module]]). Dually, a left resolution is an exact sequence  $  \cdots \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|Projective module]]; [[Flat module|Flat module]]).

More generally, the notion of a resolution of an object can be defined in any [[Abelian category|Abelian category]] in a completely similar way, [[#References|[a1]]]. E.g., in the category of sheaves of Abelian groups on a topological space an injective resolution of a [[Sheaf|sheaf]]  $  A $
is an exact sequence  $  0 \rightarrow A \rightarrow C  ^ {0} \rightarrow \cdots $
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|Flabby sheaf]]; [[Soft sheaf|Soft sheaf]]). For the case of sheaves over a topos see [[#References|[a5]]], [[#References|[a6]]].

Resolutions are the main tool in the calculation of derived functors (cf. [[Derived functor|Derived functor]]) and in the approach to homology and cohomology as derived functors. In order to construct derived functors in a non-additive category, the technique of simplicial resolutions is used [[#References|[a4]]].

In many cases it is useful to use resolutions of very special forms. One such is the resolution afforded by the [[Koszul complex|Koszul complex]], which is something like an [[Exterior algebra|exterior algebra]] pulled apart.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Grothendieck, "Sur quelques points d'algèbre homologique" ''Tohôku Math. J.'' , '''9''' (1957) pp. 119–221 {{MR|0102537}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1984) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. André, "Méthode simpliciale en algèbre homologique et algèbre commutative" , ''Lect. notes in math.'' , '''32''' , Springer (1967) {{MR|0214644}} {{ZBL|0154.01402}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Berthelot, A. Ogus, "Notes on crystalline cohomology" , Princeton Univ. Press (1978) {{MR|0491705}} {{ZBL|0383.14010}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980) {{MR|0559531}} {{ZBL|0433.14012}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) {{MR|0077480}} {{ZBL|0075.24305}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> S. MacLane, "Homology" , Springer (1963) pp. 16, 260 {{MR|}} {{ZBL|0818.18001}} {{ZBL|0328.18009}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> R. Godement, "Théorie des faisceaux" , Hermann (1964) {{MR|0345092}} {{MR|0130684}} {{MR|0102797}} {{ZBL|0275.55010}} {{ZBL|0101.15701}} {{ZBL|0080.16201}} </TD></TR></table>