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The homotopy category of injectives

Neeman, Amnon

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Krause studied the homotopy category K.(Inj A) of complexes of injectives in a locally noetherian Grothendieck abelian category A. Because A is assumed locally noetherian, we know that arbitrary direct sums of injectives are injective, and hence, the category K.InjA/ has coproducts. It turns out that K. (Inj A) is compactly generated, and Krause studies the relation between the compact objects in K. (Inj A)/, the derived category D.A/, and the category Kac. (Inj A) of acyclic objects in K. (Inj...[Show more]

dc.contributor.authorNeeman, Amnon
dc.date.accessioned2015-12-13T22:27:08Z
dc.identifier.issn1937-0652
dc.identifier.urihttp://hdl.handle.net/1885/73810
dc.description.abstractKrause studied the homotopy category K.(Inj A) of complexes of injectives in a locally noetherian Grothendieck abelian category A. Because A is assumed locally noetherian, we know that arbitrary direct sums of injectives are injective, and hence, the category K.InjA/ has coproducts. It turns out that K. (Inj A) is compactly generated, and Krause studies the relation between the compact objects in K. (Inj A)/, the derived category D.A/, and the category Kac. (Inj A) of acyclic objects in K. (Inj A). We wish to understand what happens in the nonnoetherian case, and this paper begins the study. We prove that, for an arbitrary Grothendieck abelian category A, the category K. (Inj A) has coproducts and is μ-compactly generated for some sufficiently large μ. The existence of coproducts follows easily from a result of Krause: the point is that the natural inclusion of K. (Inj A) into K.A has a left adjoint and the existence of coproducts is a formal corollary. But in order to prove anything about these coproducts, for example the μ-compact generation, we need to have a handle on this adjoint. Also interesting is the counterexample at the end of the article: we produce a locally noetherian Grothendieck abelian category in which products of acyclic complexes need not be acyclic. It follows that D.A is not compactly generated. I believe this is the first known example of such a thing.
dc.publisherMathematical Sciences Publishers
dc.rightsAuthor/s retain copyright
dc.sourceAlgebra & Number Theory
dc.titleThe homotopy category of injectives
dc.typeJournal article
local.description.notesImported from ARIES
local.identifier.citationvolume8
dc.date.issued2014
local.identifier.absfor010101 - Algebra and Number Theory
local.identifier.ariespublicationU3488905xPUB3841
local.type.statusPublished Version
local.contributor.affiliationNeeman, Amnon, College of Physical and Mathematical Sciences, ANU
local.bibliographicCitation.issue2
local.bibliographicCitation.startpage429
local.bibliographicCitation.lastpage456
local.identifier.doi10.2140/ant.2014.8.429
dc.date.updated2015-12-11T08:28:47Z
local.identifier.scopusID2-s2.0-84902954290
local.identifier.thomsonID000338424200006
dcterms.accessRightsOpen Access
CollectionsANU Research Publications

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