Pure exact structures and the pure derived category of a scheme

Estrada, Sergio; Gillespie, James; Odabasi, Sinem

doi:10.1017/s0305004116000980

Cited by 12 publications

(6 citation statements)

References 15 publications

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“…In the literature, this kind of purity is often called geometrically purity (as opposed to categorically purity, mentioned above). The study of geometrically purity was initiated in [21] and was recently continued in [17] and [20]. Below we establish the geometrically pure exact structure, E ⊗ , on V 0 , and show that the exact category (V 0 , E ⊗ ) has enough relative injectives (Propositions 3.7 and 3.12).…”

Section: Exact Categories ([39]mentioning

confidence: 88%

“…Note that if V happens to be locally λ-presentable (which will often be the case), then it also makes sense to consider the categorically pure exact structure, E λ , from ( * ). As mentioned in [20,Rem. 2.8], one always has E λ ⊆ E ⊗ , but in general these two exact structures are different!…”

Section: Introductionmentioning

confidence: 89%

“…As V is closed symmetric monoidal, it can be equipped with the so-called geometrically pure exact structure, E ⊗ , in which the admissible monomorphisms are geometrically pure monomorphisms introduced by Fox [21] (see Definition 3.4). The exact category (V 0 , E ⊗ ) has recently been studied in works of Enochs, Estrada, Gillespie, and Odabas ¸ı [17,20], and we continue to investigate it in this paper. Note that if V happens to be locally λ-presentable (which will often be the case), then it also makes sense to consider the categorically pure exact structure, E λ , from ( * ).…”

Section: Introductionmentioning

confidence: 98%

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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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Section: Exact Categories ([39]mentioning

confidence: 88%

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

The tensor embedding for a grothendieck cosmos

Holm¹,

Odabasi²

2019

Preprint

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“…The two theorems show that

{}_{\mathcal {C}}\mathcal {W}

is the left side of a cotorsion pair that is so nice that it represents a model structure on

\textnormal {Ch}(R)

with

{}_{\mathcal {C}}\mathcal {W}

the class of trivial objects. To prove these theorems, the author builds on some techniques he learned from his co‐authors in [9].…”

Section: Introductionmentioning

confidence: 99%

K‐flat complexes and derived categories

Gillespie

2022

Bulletin of London Math Soc

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Let 𝑅 be a ring with identity. Inspired by recent work in Emmanouil, Preprint, 2021, we show that the derived category of 𝑅 is equivalent to the chain homotopy category of all K-flat complexes with pure-injective components. This is implicitly related to a recollement we exhibit. It expresses  𝑝𝑢𝑟 (𝑅), the pure derived category of 𝑅, as an attachment of the usual derived category (𝑅) with Emmanouil's quotient category  K-flat (𝑅) ∶= 𝐾(𝑅)∕𝐾-Flat, which we call here the K-flat derived category. It follows that this Verdier quotient is a compactly generated triangulated category. We obtain our results by using methods of cotorsion pairs to construct (cofibrantly generated) monoidal abelian model structures on the exact category of chain complexes along with the degreewise pure exact structure. In fact, most of our model structures are obtained as corollaries of a general method which associates an abelian model structure to any class of so-called -acyclic complexes, where  is any given class of chain complexes. Finally, we also give a new characterization of K-flat complexes in terms of the pure derived category of 𝑅.

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“…The pure derived category with respect to this exact structure is defined by [Ne90] and denoted by D pur (X). This category was first appeared in [EGO16]. These pure derived categories encouraged us to ask the following question.…”

Section: Introductionmentioning

confidence: 99%

The pure derived category of quasi-coherent sheaves

Hosseini

2019

Communications in Algebra

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Let X be a quasi-compact and quasi-separated (not necessarily semiseparated) scheme. The category QcoX of all quasi-coherent sheaves of OX -modules has several different pure derived categories. Recently, categorical pure derived categories of X have been studied in more details. In this work, we focus on the geometrical purity and find a replacement for the geometrical pure derived category of X.

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Pure exact structures and the pure derived category of a scheme

Cited by 12 publications

References 15 publications

The tensor embedding for a grothendieck cosmos

The tensor embedding for a grothendieck cosmos

K‐flat complexes and derived categories

The pure derived category of quasi-coherent sheaves

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