Descent Theory for Schemes

Mesablishvili, Bachuki

doi:10.1023/b:apcs.0000049314.33172.0d

Cited by 6 publications

(16 citation statements)

References 15 publications

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“…and then the result is a consequence of [41,Theorem 5.12 (xi) ⇔ (xii)]. The rest is [42,Theorem 6.5].…”

Section: Algebramentioning

confidence: 88%

“…We observe that a composite of quasicompact pure morphisms is pure by Proposition 49 (3). If a composite ∘ of quasicompact morphisms of schemes is pure, so is [42,Corollary 6.2,Theorem 6.5]. A quasicompact faithfully flat morphism of schemes is pure [41,Remark 3.13].…”

Section: Algebramentioning

confidence: 95%

“…We are thus led to consider a class of scheme morphisms introduced by Mesablishvili, that is, a generalization to schemes of the class of pure ring morphisms [41,42]. We refer the reader to [41,42] for a definition of arbitrary pure morphisms of schemes. We will only consider quasicompact morphisms.…”

Section: Pure Morphismsmentioning

confidence: 99%

“…If a composite ∘ of quasicompact morphisms of schemes is pure, so is [42,Corollary 6.2,Theorem 6.5]. A quasicompact faithfully flat morphism of schemes is pure [41,Remark 3.13]. A quasicompact pure morphism of schemes : → is surjective.…”

Section: Algebramentioning

confidence: 99%

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Recent Progress on Submersions: A Survey and New Properties

Picavet

2013

Algebra

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This paper is a survey about recent progress on submersive morphisms of schemes combined with new results that we prove. They concern the class of quasicompact universally subtrusive morphisms that we introduced about 30 years ago. They are revisited in a recent paper by Rydh, with substantial complements and key results. We use them to show Artin-Tate-like results about the 14th problem of Hilbert, for a base scheme either Noetherian or the spectrum of a valuation domain. We look at faithfully flat morphisms and get “almost” Artin-Tate-like results by considering the Goldman (finite type) points of a scheme. Bjorn Poonen recently proved that universally closed morphisms are quasicompact. By introducing incomparable morphisms of schemes, we are able to characterize universally closed surjective morphisms that are either integral or finite. Next we consider pure morphisms of schemes introduced by Mesablishvili. In the quasicompact case, they are universally schematically dominant morphisms. This leads us to a characterization of universally subtrusive morphisms by purity. Some results on the schematically dominant property are given. The paper ends with properties of monomorphisms and topological immersions, a dual notion of submersions.

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“…and then the result is a consequence of [41,Theorem 5.12 (xi) ⇔ (xii)]. The rest is [42,Theorem 6.5].…”

Section: Algebramentioning

confidence: 88%

Section: Algebramentioning

confidence: 95%

Section: Pure Morphismsmentioning

confidence: 99%

Section: Algebramentioning

confidence: 99%

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Recent Progress on Submersions: A Survey and New Properties

Picavet

2013

Algebra

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In this paper we continue the investigation of some aspects of descent theory for schemes that was begun in [11]. Let SCH be a category of schemes.

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confidence: 99%

More on Descent Theory for Schemes

Mesablishvili

2004

GMJ

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In this paper we continue the investigation of some aspects of descent theory for schemes that was begun in [Mesablishvili, Appl. Categ. Structures]. Let 𝐒𝐂𝐇 be a category of schemes. We show that quasi-compact pure morphisms of schemes are effective descent morphisms with respect to 𝐒𝐂𝐇-indexed categories given by (i) quasi-coherent modules of finite type, (ii) flat quasi-coherent modules, (iii) flat quasi-coherent modules of finite type, (iv) locally projective quasicoherent modules of finite type. Moreover, we prove that a quasi-compact morphism of schemes is pure precisely when it is a stable regular epimorphism in 𝐒𝐂𝐇. Finally, we present an alternative characterization of pure morphisms of schemes.

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Descent in Locally Presentable Categories

Mesablishvili

2013

Appl Categor Struct

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Descent Theory for Schemes

Cited by 6 publications

References 15 publications

Recent Progress on Submersions: A Survey and New Properties

Recent Progress on Submersions: A Survey and New Properties

More on Descent Theory for Schemes

Descent in Locally Presentable Categories

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