On the Homology of Completion and Torsion

Porta, Marco; Shaul, Liran; Yekutieli, Amnon

doi:10.1007/s10468-012-9385-8

Cited by 92 publications

(168 citation statements)

References 21 publications

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“…In the language of [25,Section 7], this says that we have RΓ 0/m (R) ≃ R p . The proof of [25,Lemma 7.2] exhibits an exact triangle of the following form.…”

Section: Computing Derived Functorsmentioning

confidence: 99%

Extended Local Cohomology and Local Homology

Sather-Wagstaff

Wicklein

2016

Algebr Represent Theor

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Abstract. We present an in-depth exploration of the module structures of local (co)homology modules (moreover, for complexes) over the completion R a of a commutative noetherian ring R with respect to a proper ideal a. In particular, we extend Greenlees-May Duality and MGM Equivalence to track behavior over R a , not just over R. We apply this to the study of two recent versions of homological finiteness for complexes, and to certain isomorphisms, with a view toward further applications. We also discuss subtleties and simplifications in the computations of these functors.

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“…In the language of [25,Section 7], this says that we have RΓ 0/m (R) ≃ R p . The proof of [25,Lemma 7.2] exhibits an exact triangle of the following form.…”

Section: Computing Derived Functorsmentioning

confidence: 99%

Extended Local Cohomology and Local Homology

Sather-Wagstaff

Wicklein

2016

Algebr Represent Theor

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“…The mistake in the proof of [2,Theorem 7.12] was as follows. We had claimed that for any K-projective complex P , the homomorphism of complexes…”

Section: (T M)mentioning

confidence: 99%

“…There is no implication of this error on the rest of the paper, since WPR ideals are by definition finitely generated. We thank R. Vyas for mentioning this error to us.In the rest of the note we recall the definitions and results from [2], that are needed to prove Theorem 9.Let A be a commutative ring, and let a be a weakly proregular ideal in it (see [2, Definition 4.21]). We choose a finite sequence a = (a 1 , .…”

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Erratum to: On the Homology of Completion and Torsion

Porta

Shaul

Yekutieli

2015

Algebr Represent Theor

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In this note we correct two errors in our paper [2].(1) There is an error in the proof of Theorem 7.12 (GM Duality) of [2]. The statement itself is correct, and it is repeated as Theorem 9 below, with a correct proof. (2) Just before formula (3.12) in [2], we said that "Mod a-tor A is a thick abelian subcategory of Mod A". This is true if the ideal a is finitely generated, but might be false otherwise. There is no implication of this error on the rest of the paper, since WPR ideals are by definition finitely generated. We thank R. Vyas for mentioning this error to us.In the rest of the note we recall the definitions and results from [2], that are needed to prove Theorem 9.Let A be a commutative ring, and let a be a weakly proregular ideal in it (see [2, Definition 4.21]). We choose a finite sequence a = (a 1 , . . . , a n ) that generThe online version of the original article can be found at http://dx

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Remarks on derived complete modules and complexes

Positselski

2022

Mathematische Nachrichten

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Let R be a commutative ring and I⊂R$I\subset R$ a finitely generated ideal. We discuss two definitions of derived I‐adically complete (also derived I‐torsion) complexes of R‐modules, which appear in the literature: the idealistic and the sequential ones. The two definitions are known to be equivalent for a weakly proregular ideal I; we show that they are different otherwise. We argue that the sequential approach works well, but the idealistic one needs to be reinterpreted or properly understood. We also consider I‐adically flat R‐modules.

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On the Homology of Completion and Torsion

Cited by 92 publications

References 21 publications

Extended Local Cohomology and Local Homology

Extended Local Cohomology and Local Homology

Erratum to: On the Homology of Completion and Torsion

Remarks on derived complete modules and complexes

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