On flat generators and Matlis duality for quasicoherent sheaves

Slavik, Alexander; Šťovíček, Jan

doi:10.1112/blms.12398

Cited by 4 publications

(2 citation statements)

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“…A straightforward proof of this fact, suggested by Neeman, is given in [EP15, Appendix A, Lemma 1]. It is very notable too that any quasi-compact and quasi-separated scheme possessing a generating set of flat sheaves is necessarily semiseparated, by [SS21]. So with our approach of using flat generators this is the best result we can expect.…”

Section: The Stable Derived Category and Recollementmentioning

confidence: 85%

“…To do this, we first construct, see Theorem 4.1, an injective model structure for Krause's stable derived category. Another point of interest here is that any quasi-compact and quasi-separated scheme possessing a generating set of flat (quasi-coherent) sheaves is necessarily semiseparated, by [SS21]. So with our approach of using flat generators, this is the best result we can expect.…”

Section: Introductionmentioning

confidence: 99%

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Quillen equivalences inducing Grothendieck duality for unbounded chain complexes of sheaves

Estrada¹,

Gillespie²

2021

Preprint

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Let X be a semiseparated Noetherian scheme with a dualizing complex D. We lift some well-known triangulated equivalences associated with Grothendieck duality to Quillen equivalences of model categories. In the process we are able to show that the Gorenstein flat model structure, on the category of quasi-coherent sheaves on X, is Quillen equivalent to the Gorenstein injective model structure. Also noteworthy is that we extend the recollement of Krause to hold without the Noetherian condition. Using a set of flat generators, it holds for any quasi-compact semiseparated scheme X. With this we also show that the Gorenstein injective quasi-coherent sheaves are the fibrant objects of a cofibrantly generated abelian model structure for any semiseparated Noetherian scheme X. Finally, we consider both the injective and (mock) projective approach to Tate cohomology of quasi-coherent sheaves. They agree whenever X is a semiseparated Gorenstein scheme of finite Krull dimension.

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Section: The Stable Derived Category and Recollementmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%