(complex)
In an abelian category \(\mca\), a complex \(A^\bullet\) is a family of objects \(A^i\), \(i \in \ZZ\), with maps \(\delta^i: A^i \to A^{i+1}\) such that \(\delta^{i+1}\circ \delta^i = 0\) for all \(i\); equivalently \(\im \delta^i \subseteq \ker \delta^{i+1}\). Its cohomology is \(h^i(A^\bullet) \da \ker \delta^i / \im \delta^{i-1}\).
A morphism of complexes \(f: A^\bullet \to B^\bullet\) is a family \(f^i: A^i \to B^i\) commuting with the differentials, \(\delta^i \circ f^i = f^{i+1}\circ \delta^i\).
(homotopic)
Morphisms of complexes \(f, g: A^\bullet \to B^\bullet\) are homotopic, written \(f \sim g\), if there are maps \(h^i: A^i \to B^{i-1}\) with \[ f^i - g^i = \delta^{i-1}h^i + h^{i+1}\delta^i \] for all \(i\).
Homotopic morphisms of complexes induce the same map \(h^i(A^\bullet) \to h^i(B^\bullet)\) on cohomology.
Any two injective resolutions \(I^\bullet\), \(J^\bullet\) of an object \(A\) are homotopy equivalent through morphisms lifting \(\id_A\). An additive functor \(F\) carries this homotopy equivalence to a homotopy equivalence \(F(I^\bullet)\simeq F(J^\bullet)\), so \(h^i(F(I^\bullet))\cong h^i(F(J^\bullet))\) and \(R^iF(A)\) is independent of the injective resolution up to canonical isomorphism.
Dependencies
D-DEFABCATAbelian categoriesuses
Backlinks
T-DEFFHHFThe FHHF theorem, on functors and cohomologyuses