In
mathematics, in the field of
homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology ...
, given an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
having enough injectives and an additive (covariant)
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
:
,
an acyclic object with respect to
, or simply an
-acyclic object, is an object
in
such that
:
for all
,
where
are the
right derived functors of
.
References
* {{cite book , last=Caenepeel , first=Stefaan , title=Brauer groups, Hopf algebras and Galois theory , zbl=0898.16001 , series=Monographs in Mathematics , volume=4 , location=Dordrecht , publisher=Kluwer Academic Publishers , year=1998 , isbn=1-4020-0346-3 , page=454
Homological algebra