nLab
derived limit functor

If AA is a category and II a small category then we can consider the category of functors A IA^I, whose objects are the functors from II to AA. If AA admits limits of shape II, then the limit lim=lim I:A IAlim = lim_I:A^I\to A is a functor which is right adjoint to the constant diagram functor hence it commutes with limits, and in particular it is left exact.

If AA has some notion of homotopy theory then usually A IA^I has the notion as well (e.g. if we work with model category structures or say Abelian categories) hence we can then form right derived functors of limlim which are usually denoted lim nlim^n and called the derived limit functors. As usual we can also assemble them into the total right derived functor lim\mathbb{R}lim.

An important example is where AA is a category of modules over a commutative ring, then one has Ch(A)Ch(A) and any M:IAM:I\to A can be thought of as M:ICh(A)M:I\to Ch(A) by thinking of each M(i)M(i) as being a chain complex concentrated in dimension 0.

These include homotopy limit, lim^1 and Milnor sequences and cohomology of small categories?, this latter in the case of coefficients in a category of modules. This is a special case of the more general Baues-Wirsching cohomology.

References

A classic text with links to the theory of modules is

  • C. U. Jensen, 1972, Les foncteurs dérivés de Lim et leurs applications en théorie de modules, volume 254 of Springer Lecture Notes in Maths.

A proof that derived limit functors give invariants of a corresponding pro-obect can be found in

  • John Duskin, Pro-objects (after Verdier), Sém. Heidelberg- Strasbourg1966 -67, Exposé 6, I.R.M.A.Strasbourg.

Some results on the vanishing of ‘derived limits’ are in

  • Barbara Osofsky?, The subscript of n\aleph_n projective dimension, and the vanishing of lim (n)lim^{(n)}, Bull. Amer. Math. Soc. Volume 80, Number 1 (1974), 8 - 26.

Last revised on September 18, 2017 at 12:00:55. See the history of this page for a list of all contributions to it.