The dual of Brown representability for some derived categories

@article{Modoi2013TheDO,
  title={The dual of Brown representability for some derived categories},
  author={George Ciprian Modoi},
  journal={Arkiv f{\"o}r Matematik},
  year={2013},
  volume={54},
  pages={485-498}
}
  • G. Modoi
  • Published 26 May 2013
  • Mathematics
  • Arkiv för Matematik
Consider a complete abelian category which has an injective cogenerator. If its derived category is left-complete we show that the dual of this derived category satisfies Brown representability. In particular, this is true for the derived category of an abelian AB4∗$4^{*}$-n$n$ category and for the derived category of quasi-coherent sheaves over a nice enough scheme, including the projective finitely dimensional space. 

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References

SHOWING 1-10 OF 36 REFERENCES

Non-Left-Complete Derived Categories

We give some examples of abelian categories A for which the derived category D(A) is not left-complete. Perhaps the most natural of these is where A is the category of representations of the additive

Brown representability often fails for homotopy categories of complexes

We show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing

On Perfectly Generating Projective Classes in Triangulated Categories

We say that a projective class in a triangulated category with coproducts is perfect if the corresponding ideal is closed under coproducts of maps. We study perfect projective classes and the

Acyclicity Versus Total Acyclicity for Complexes over Noetherian Rings

It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective

The dual of the homotopy category of projective modules satisfies Brown representability

We show that the dual of the homotopy category of projective modules over an arbitrary ring satisfies Brown representability.

On the construction of triangle equivalences

We give a self-contained account of an alternative proof of J. Rickard's Morita-theorem for derived categories [135] and his theorem on the realization of derived equivalences as derived functors

The mock homotopy category of projectives and Grothendieck duality

The coherent sheaves defined on a separated noetherian scheme X reflect the underlying geometry, and they play a central role in modern algebraic geometry. Recent results have indicated that there

Derived Functors of Inverse Limits Revisited

We prove, correct and extend several results of an earlier paper of ours (using and recalling several of our later papers) about the derived functors of projective limit in abelian categories. In

An introduction to homological algebra

Preface 1. Generalities concerning modules 2. Tensor products and groups of homomorphisms 3. Categories and functors 4. Homology functors 5. Projective and injective modules 6. Derived functors 7.