Regular Differential Forms and Relative Duality

Hübl, Reinhold; Sastry, Pramathanath

doi:10.2307/2375012

Cited by 18 publications

(10 citation statements)

References 9 publications

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“…) and res reg Z (= X/Y,Z ) can be found in [HS,Theorem]. In brief, here are the main points of this result:…”

Section: Regular Differential Formsmentioning

confidence: 84%

“…commutes [HS,p.752,(iii) (The Residue Theorem)]. (iv) If Z is a closed subscheme of X such that Z lies in the smooth locus of f and Z → Y is an isomorphism, then res reg Z = res Z (see [HK1,p.62,Cor.…”

Section: Regular Differential Formsmentioning

confidence: 99%

“…1.13] and [HK1,p.78,4.4] as well as the formulae in Remark 5.3.3). (v) The map ϕ is compatible with flat base change to excellent schemes without embedded associated points [KW,3.13] and [HS,(ii) and (iv)]. In particular ϕ is compatible with étale base change.…”

Section: Regular Differential Formsmentioning

confidence: 99%

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Grothendieck Duality and Transitivity II: Traces and Residues via Verdier's isomorphism

Nayak¹,

Sastry²

2019

Preprint

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For a smooth map between noetherian schemes, Verdier relates the top relative differentials of the map with the twisted inverse image functor "upper shriek" [V]. We show that the associated traces for smooth proper maps can be rendered concrete by showing that the resulting theory of residues satisfy the residue formulas (R1)-(R10) in Hartshorne's Residues and Duality [RD]. We show that the resulting abstract transitivity map relating the twisted image functors for the composite of two smooth maps satisfies an explicit formula involving differential forms. We also give explicit formulas for traces of differential forms for finite flat maps (arising from Verdier's isomorphism) between schemes which are smooth over a common base, and use this to relate Verdier's isomorphism to Kunz and Waldi's regular differentials. These results also give concrete realisations of traces and residues for Lipman's fundamental class map via the results of Lipman and Neeman [LN2] relating the fundamental class to Verdier's isomorphism.All schemes formal or ordinary are assumed to be noetherian. The category of O X -modules for a formal scheme X is denoted A(X ) and its derived category D(X ) as in [NS1]. In general we use the notations of ibid. Thus the torsion functor Γ ′ X on O X -modules is defined by the formulawhere I is any ideal of definition of the formal scheme X . A torsion module F is an object in A(X ) such that Γ ′ X F = F . The reader is advised to look at [NS1, § 2] for further definitions and notations, especially the definitions of A c (X ), A c (X ), A qc (X ), A qct (X ), and the triangulated full subcategories of D(X), D c (X ), D c (X ), D qc (X ), D qct (X ), and their various bounded versions (e.g., D + c (X ) . . . ).

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“…) and res reg Z (= X/Y,Z ) can be found in [HS,Theorem]. In brief, here are the main points of this result:…”

Section: Regular Differential Formsmentioning

confidence: 84%

Section: Regular Differential Formsmentioning

confidence: 99%

See 1 more Smart Citation

Grothendieck Duality and Transitivity II: Traces and Residues via Verdier's isomorphism

Nayak¹,

Sastry²

2019

Preprint

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“…397,Theorem 3], [HK2,p. 84,Duality theorem] for projective maps with the base which is free of embedded points, [HS,p. 750,Duality Theorem] for maps where the base is free of embedded points).…”

Section: Introductionmentioning

confidence: 99%

Grothendieck Duality and Transitivity I: Formal Schemes

Nayak,

Sastry

2019

Preprint

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For a proper map f : X → Y of noetherian ordinary schemes, one has a well-known natural transformation, Lf * (−)obtained via the projection formula, which extends, using Nagata's compactification, to the case where f is separated and of finite type. In this paper we extend this transformation to the situation where f is a pseudo-finite-type map of noetherian formal schemes which is a composite of compactifiable maps, and show it is compatible with the pseudofunctorial structures involved. This natural transformation has implications for the abstract theory of residues and traces, giving Fubini type results for iterated maps. These abstractions are rendered concrete in a sequel to this paper.

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“…The principal result ("Residue Theorem") reifies c f as a globalization of the local residue maps at the points of X, leading to explicit versions of local and global duality and the relation between them. This is generalized to certain maps of noetherian schemes in [HS93].…”

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confidence: 99%

On the fundamental class of an essentially smooth scheme-map

Lipman¹,

Neeman²

2018

Alg. Geom.

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Let f : X → Z be a separated, essentially finite-type, flat map of noetherian schemes and δ : X → X × Z X the diagonal map. The fundamental class C f (globalizing residues) is a map from the relative Hochschild functor Lδ * δ * f * to the relative dualizing functor f ! . We show a compatibility between C f and the derived tensor product. The main result is that, in a suitable sense, C f generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f ! to relative d-forms.

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Regular Differential Forms and Relative Duality

Cited by 18 publications

References 9 publications

Grothendieck Duality and Transitivity II: Traces and Residues via Verdier's isomorphism

Grothendieck Duality and Transitivity II: Traces and Residues via Verdier's isomorphism

Grothendieck Duality and Transitivity I: Formal Schemes

On the fundamental class of an essentially smooth scheme-map

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