Cotilting sheaves on Noetherian schemes

Čoupek, Pavel; Šťovíček, Jan

doi:10.1007/s00209-019-02404-8

Cited by 16 publications

(12 citation statements)

References 49 publications

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“…We will end this section by introducing an application related to the tilting theory developed in recent years. Namely, Colpi and Fuller developed a theory of tilting objects of projective dimension ≤ 1 for abelian categories in [8], and Čoupek and Št'ovíček developed a theory of cotilting objects of injective dimension ≤ 1 for Grothendieck categories in [9]. A fundamental result needed in these theories is that…”

Section: Is An Isomorphism Of Abelian Groups For Every B ∈ Cmentioning

confidence: 99%

“…In this theory, the Baer sum can be extended to n-extensions, proving that the class Ext n C (C, A) of n-extensions of A by C is an abelian group. Recently, the generalization of homological techniques such as Gorenstein or tilting objects to abstract contexts [3,8,9,19], such as abelian categories that do not necessarily have enough projectives or injectives, claim for the introduction of an Ext functor that can be used without those constraints. The only problem is that it is not clear if the rich properties of the homological Ext are also valid for the Yoneda Ext.…”

Section: Introductionmentioning

confidence: 99%

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The Yoneda Ext and Arbitrary Coproducts in Abelian Categories

Alejandro¹

2021

Glasgow Math. J.

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There are well-known identities involving the Ext bifunctor, coproducts, and products in AB4 abelian categories with enough projectives. Namely, for every such category \[\mathcal{A}\] , given an object X and a set of objects \[{\{ {{\text{A}}_{\text{i}}}\} _{{\text{i}} \in {\text{I}}}}\] , an isomorphism \[Ext_\mathcal{A}^{\text{n}}({ \oplus _{{\text{i}} \in {\text{I}}}}{{\text{A}}_{\text{i}}},{\text{X}}) \cong \prod\nolimits_{{\text{i}} \in {\text{I}}} {Ext_\mathcal{A}^{\text{n}}({{\text{A}}_{\text{i}}},{\text{X}})} \] can be built, where \[Ex{t^{\text{n}}}\] is the nth derived functor of the Hom functor. The goal of this paper is to show a similar isomorphism for the nth Yoneda Ext, which is a functor equivalent to \[Ex{t^{\text{n}}}\] that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits in AB4 abelian categories with not necessarily enough projectives nor injectives, extending a result by Colpi and Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize AB4 categories. A dual result is also stated.

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Section: Is An Isomorphism Of Abelian Groups For Every B ∈ Cmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Yoneda Ext and Arbitrary Coproducts in Abelian Categories

Alejandro¹

2021

Glasgow Math. J.

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“…Next, following [3, Appendix B], let us briefly recall some relevant facts about the map of schemes

ι_{p} : Spec {scriptO}_{X, p} \to X

, where

p

is a point of

X

. This map arises as the composition of the natural maps

Spec {scriptO}_{X, p} ↪ Spec {scriptO}_{X} false(U false) ≅ U ↪ X,

where

U \subseteq X

is some open affine neighbourhood of

p

.…”

Section: Exactness Of the Internal Hommentioning

confidence: 99%

“…By [22, 0816],

ι_{p}

is a quasicompact quasiseparated map, hence the direct image functor

ι_{p, *}

preserves quasicoherence by [22, 01LC]. (Note that the assumption on

X

being locally Noetherian in [3] is superfluous.) To ease the notation, let us further compose this functor with the standard equivalence of categories

{scriptO}_{X, p} -Mod ≅ QCoh false(prefixSpec O_{X, p} false),

obtaining the functor

{\overset{ι}{true}}_{p, *} : {scriptO}_{X, p} -Mod \to QCoh false(X false) .

This functor can also be viewed as the right adjoint to the functor of taking stalks at

p

{(-)}_{p} : QCoh false(X false) \to {scriptO}_{X, p} -Mod

.…”

Section: Exactness Of the Internal Hommentioning

confidence: 99%

“…Next, following [3, Appendix B], let us briefly recall some relevant facts about the map of schemes ι p : Spec O X,p → X, where p is a point of X. This map arises as the composition of the natural maps (Note that the assumption on X being locally Noetherian in [3] is superfluous.) To ease the notation, let us further compose this functor with the standard equivalence of categories…”

Section: Exactness Of the Internal Hommentioning

confidence: 99%

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On flat generators and Matlis duality for quasicoherent sheaves

Slavik

Šťovíček

2020

Bull. London Math. Soc.

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t-Structures with Grothendieck hearts via functor categories

Saorín,

Št’ovíček

2023

Sel. Math. New Ser.

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We study when the heart of a t-structure in a triangulated category $$\mathcal {D}$$ D with coproducts is AB5 or a Grothendieck category. If $$\mathcal {D}$$ D satisfies Brown representability, a t-structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective t-cogenerating object. If $$\mathcal {D}$$ D is standard well generated, such a heart is automatically a Grothendieck category. For compactly generated t-structures (in any ambient triangulated category with coproducts), we prove that the heart is a locally finitely presented Grothendieck category. We use functor categories and the proofs rely on two main ingredients. Firstly, we express the heart of any t-structure in any triangulated category as a Serre quotient of the category of finitely presented additive functors for suitable choices of subcategories of the aisle or the co-aisle that we, respectively, call t-generating or t-cogenerating subcategories. Secondly, we study coproduct-preserving homological functors from $$\mathcal {D}$$ D to complete AB5 abelian categories with injective cogenerators and classify them, up to a so-called computational equivalence, in terms of pure-injective objects in $$\mathcal {D}$$ D . This allows us to show that any standard well generated triangulated category $$\mathcal {D}$$ D possesses a universal such coproduct-preserving homological functor, to develop a purity theory and to prove that pure-injective objects always cogenerate t-structures in such triangulated categories.

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Cotilting sheaves on Noetherian schemes

Cited by 16 publications

References 49 publications

The Yoneda Ext and Arbitrary Coproducts in Abelian Categories

The Yoneda Ext and Arbitrary Coproducts in Abelian Categories

On flat generators and Matlis duality for quasicoherent sheaves

t-Structures with Grothendieck hearts via functor categories

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