The derived category of quasi-coherent sheaves and axiomatic stable homotopy

Tarrío, Leovigildo Alonso; López, Ana Jeremías; Rodríguez, Marta Pérez; Gonsalves, María Jesús Vale

doi:10.1016/j.aim.2008.03.011

Cited by 20 publications

(6 citation statements)

References 15 publications

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“…Recall that a perfect complex is a complex locally isomorphic to a bounded complex of vector bundles. In the case of D qc (X), a perfect complex is strongly dualizable [ATJLPRVG08,Prop. 4.4].…”

Section: Other Duality Contextsmentioning

confidence: 99%

“…Moreover D qc (X) is compactly generated by a single perfect complex G [BvdB03, Thm. 3.1.1(2)] and is a stable category [ATJLPRVG08]. In particular, it has a closed symmetric monoidal structure, where we write Hom Dqc(X) (E, F ) for the internal mapping object.…”

Section: Other Duality Contextsmentioning

confidence: 99%

“…The relationship between the two is the following: There is a (nonderived) functor, called the coherator, Q : Mod OX → QCoh(X), right adjoint to the exact inclusion functor ι, which gives rise to a functor RQ : D OX (X) → D qc (X). We then have an equivalence Hom Dqc(X) (E, F ) ≃ RQHom D O X (X) (E, F ), see [ATJLPRVG08,Sec. 3.7].…”

Section: Other Duality Contextsmentioning

confidence: 99%

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Local duality in algebra and topology

Barthel

Heard

Valenzuela

2018

Advances in Mathematics

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The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable ∞-category C together with a collection of compact objects K ⊂ C we construct local cohomology and local homology functors satisfying an abstract version of local duality. When specialized to the derived category of a commutative ring A and a suitable ideal in A, we recover the classical local duality due to Grothendieck as well as generalizations by Greenlees and May. More generally, applying our result to the derived category of quasi-coherent sheaves on a quasi-compact and separated scheme X implies the local duality theorem of Alonso Tarrío, Jeremías López, and Lipman.As a second objective, we establish local duality for quasi-coherent sheaves over many algebraic stacks, in particular those arising naturally in stable homotopy theory. After constructing an appropriate model of the derived category in terms of comodules over a Hopf algebroid, we show that, in familiar cases, the resulting local cohomology and local homology theories coincide with functors previously studied by Hovey and Strickland. Furthermore, our framework applies to global and local stable homotopy theory, in a way which is compatible with the algebraic avatars of these theories. In order to aid computability, we provide spectral sequences relating the algebraic and topological local duality contexts.

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Section: Other Duality Contextsmentioning

confidence: 99%

Section: Other Duality Contextsmentioning

confidence: 99%

Section: Other Duality Contextsmentioning

confidence: 99%

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Local duality in algebra and topology

Barthel

Heard

Valenzuela

2018

Advances in Mathematics

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“…It is well-known, and completely formal, that the functor H om qc X := Q X H om X yields a closed structure on (Qcoh(X), ⊗ X , O X ). The category Qcoh(X) is Grothendieck by [4,Lem. 1.3].…”

Section: Monoidal Categories ([30]mentioning

confidence: 99%

“…In [3,Thm. 4.2] it is shown that if K is a small V-category, then the ordinary category [K, V] 0 of V-functors K → V is abelian too, and even Grothendieck if V is 4 . Moreover, (co)limits, in particular, (co)kernels, in the category [K, V] 0 are formed objectwise.…”

Section: Observationmentioning

confidence: 99%

The tensor embedding for a grothendieck cosmos

Holm¹,

Odabasi²

2019

Preprint

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While the Yoneda embedding and its generalizations have been studied extensively in the literature, the so-called tensor embedding has only received little attention. In this paper, we study the tensor embedding for closed symmetric monoidal categories and show how it is connected to the notion of geometrically purity, which has recently been investigated in works of Enochs, Estrada, Gillespie, and Odabas ¸ı. More precisely, for a Grothendieck cosmos-that is, a bicomplete Grothendick category V with a closed symmetric monoidal structure-we prove that the geometrically pure exact category (V, E ⊗ ) has enough relative injectives; in fact, every object has a geometrically pure injective envelope. We also show that for some regular cardinal λ, the tensor embedding yields an exact equivalence between (V, E ⊗ ) and the category of λ-cocontinuous V-functors from Pres λ (V) to V, where the former is the full V-subcategory of λ-presentable objects in V. In many cases of interest, λ can be chosen to be ℵ 0 and the tensor embedding identifies the geometrically pure injective objects in V with the (categorically) injective objects in the abelian category of V-functors from fp(V) to V. As we explain, the developed theory applies e.g. to the category Ch(R) of chain complexes of modules over a commutative ring R and to the category Qcoh(X) of quasi-coherent sheaves over a (suitably nice) scheme X. ( * ) Any locally finitely presentable (= locally ℵ 0 -presentable) abelian 1 category C can be equipped with the categorically pure exact structure, E ℵ 0 , consisting of exact sequences 0 → X → Y → Z → 0 in C for which the sequence 0 −→ Hom C (C, X) −→ Hom C (C, Y) −→ Hom C (C, Z) −→ 0

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Dualizable and semi-flat objects in abstract module categories

Bak

2020

Math. Z.

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The derived category of quasi-coherent sheaves and axiomatic stable homotopy

Cited by 20 publications

References 15 publications

Local duality in algebra and topology

Local duality in algebra and topology

The tensor embedding for a grothendieck cosmos

Dualizable and semi-flat objects in abstract module categories

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