The homotopy category of complexes of projective modules

By Peter Jørgensen. 2003

99 citations recorded by the source corpus.

Abstract The homotopy category of complexes of projective left-modules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite derived category of right-modules.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8f13862ef814006042c2050f75904b59a0b56a37

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Citations in the corpus

Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves (2013, methodology): “The exact model categories give in many respects a complementary approach to that of dg algebras, with different advantages and weaknesses, and it has a good potential for instance in Gorenstein homological algebra [39], homological algebra in the category of quasi-coherent sheaves [22, 23, 28], or ”
Coherent analogues of matrix factorizations and relative singularity categories (2011, background): “ring and injective quasi-coherent sheaves over a scheme were studied in the papers by Jørgensen [16] and Krause [19]; subsequently, Iyengar and Krause have constructed an equivalence between these two categories for rings with dualizing complexes [15].”
Approximations and adjoints in homotopy categories (2010, unknown): “The first result is an analogue of the flat cover conjecture for complexes of quasicoherent sheaves on a scheme; it has been established in the affine case by Neeman [33] and Enochs et al. [8], and for noetherian schemes by Murfet [29].”
Homotopy Equivalences induced by Balanced Pairs (2008, background): “Note that we have natural identifications K(R-Proj) c ≃ D b (R op -mod) op by Neeman [23, Proposition 6.12] (for this we need the right coherent property; cf. Jorgen sen [17, Theore m 3.2]), and K(R-Inj) c ≃ D b (R-mod) by Krause [20, Proposition 2.3(2)].”
On exact categories and applications to triangulated adjoints and model structures (2010, background): “As easy consequences, we recover some results by Jørgensen [32] and Neeman [50, §5].”
Homological algebra of semimodules and semicontramodules. Semi-infinite homological algebra of assoc (2007, background): “[29], where in the dual situation the role of the class F is played by flat modules).”
the stable derived category of a noetherian scheme (2004, background): “As before, A and B denote locally noetherian Grothendieck categories.”
Relative singularity categories and Gorenstein‐projective modules (2007, result): “The following result is essentially due to Jørgensen [17] and also Neeman [24].”
The homotopy category of flat modules, and Grothendieck duality (2008, background): “Jørgensen also computed for us the compact objects in K(R-Proj), in the special case where R is one of the rings for which he could prove K(R-Proj) compactly generated.”
Hereditary abelian model categories (2015, methodology): “P. Jørgensen, ‘The homotopy category of complexes of projective modules’, Adv.”
Acyclicity versus total acyclicity for complexes over noetherian rings (2005, unknown): “1 These classes fit into a commutative diagram of functors: Kc(PrjR) D⊗R− // Kc(InjR) Df (R) ≀P RHomR(−,D) // Df (R) ≀ I The functor P is induced by the composite K(PrjR) HomR(−,R) −−−−−−−→ K(R) can −−→ D(R) , and it is a theorem of Jørgensen [11] that P is an equivalence of categories.”