nLab
class of adapted objects

Let A,BA,B be abelian categories, RObAR\subset \mathrm{Ob}A a class of objects in AA and F:ABF:A\to B an additive functor.

If FF is left exact functor we say that RR is a class of objects adapted to FF if FF sends bounded below cochain complexes to acyclic complexes (= having trivial cohomology), and every object in AA is a subobject of an object from RR.

If FF is right exact functor we say that RR is a class of objects adapted to FF if FF sends bounded above chain complexes to acyclic complexes (= having trivial homology), and every object in AA is a quotient object of an object from RR.

For example, if R=IR = I (resp. R=PR = P) is the class of all injective (resp. projective) objects in an abelian category AA with sufficiently many injectives (resp. projectives), then RR is adapted to any left (resp. right) exact functor whose domain is AA. If FF is left (resp. right) exact and a family RR of objects adapted to FF exists then the right derived functor D +(F):D +(A)D +(B)D^+(F):D^+(A)\to D^+(B) (resp. left derived functor D (F):D (A)D (B)D^-(F):D^-(A)\to D^-(B)) of FF exists by a standard construction using resolutions by objects in RR. Flexibility in choosing an adapted class is often useful.

References:

  • A. Grothendieck: Tohoku

  • S. I. Gel’fand, Yu. I. Manin, Methods of homological algebra, Moskva 1988 (Russian); Springer 1996 (English): chapter 3

  • (for generalizations in nonabelian setup) A. Rosenberg, Homological algebra of noncommutative ‘spaces’ I, preprint MPIM2008-91

Revised on June 24, 2009 18:10:44 by Toby Bartels (71.104.230.172)