Acyclic object
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In mathematics, in the field of homological algebra, given an abelian category [math]\displaystyle{ \mathcal{C} }[/math] having enough injectives and an additive (covariant) functor
- [math]\displaystyle{ F :\mathcal{C}\to\mathcal{D} }[/math],
an acyclic object with respect to [math]\displaystyle{ F }[/math], or simply an [math]\displaystyle{ F }[/math]-acyclic object, is an object [math]\displaystyle{ A }[/math] in [math]\displaystyle{ \mathcal{C} }[/math] such that
- [math]\displaystyle{ {\rm R}^i F (A) = 0 \,\! }[/math] for all [math]\displaystyle{ i\gt 0 \,\! }[/math],
where [math]\displaystyle{ {\rm R}^i F }[/math] are the right derived functors of [math]\displaystyle{ F }[/math].
References
- Caenepeel, Stefaan (1998). Brauer groups, Hopf algebras and Galois theory. Monographs in Mathematics. 4. Dordrecht: Kluwer Academic Publishers. p. 454. ISBN 1-4020-0346-3.
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