Resolution

In homological algebra a right resolution of a module $ A $ is a 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).

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). 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; 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 \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; Soft sheaf). For the case of sheaves over a topos see [a5], [a6].

Resolutions are the main tool in the calculation of derived functors (cf. 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 [a4].

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

References

[a1]	A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohôku Math. J. , 9 (1957) pp. 119–221 MR0102537
[a2]	S. Lang, "Algebra" , Addison-Wesley (1984) MR0783636 Zbl 0712.00001
[a3]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381 MR0463157 Zbl 0367.14001
[a4]	M. André, "Méthode simpliciale en algèbre homologique et algèbre commutative" , Lect. notes in math. , 32 , Springer (1967) MR0214644 Zbl 0154.01402
[a5]	P. Berthelot, A. Ogus, "Notes on crystalline cohomology" , Princeton Univ. Press (1978) MR0491705 Zbl 0383.14010
[a6]	J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980) MR0559531 Zbl 0433.14012
[a7]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) MR0077480 Zbl 0075.24305
[a8]	S. MacLane, "Homology" , Springer (1963) pp. 16, 260 Zbl 0818.18001 Zbl 0328.18009
[a9]	R. Godement, "Théorie des faisceaux" , Hermann (1964) MR0345092 MR0130684 MR0102797 Zbl 0275.55010 Zbl 0101.15701 Zbl 0080.16201