Derived completion for comodules

Barthel, Tobias; Heard, Drew; Valenzuela, Gabriel

doi:10.1007/s00229-018-1094-0

Cited by 9 publications

(5 citation statements)

References 22 publications

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“…This observation clarified many things for the authors. In particular, Positselski has proved many results about the relation of derived completion with contramodules, see [15,16,17], but the relation with the work of Greenlees-May [8] and Barthel-Heard-Valenzuela [4,5] was unclear to the present authors. Definition 7.1.…”

Section: Contramodules and L I 0 -Completionmentioning

confidence: 68%

“…Positselski [15,16,17] has proved many results about the different forms of completions, and in particular, their relation to contramodules. Finally, we also note the work of Barthel-Heard-Valenzuela [5] who have investigated derived completion and L-completion for comodules over a flat Hopf algebroid. The relationship between these papers in the literature is not entirely clear -for instance, the use of L I 0 -completion seems restricted to authors coming from a topological standing, and contramodules from the algebraic side.…”

Section: Contribution Of This Paper and Related Workmentioning

confidence: 96%

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The homotopy theory of complete modules

Pol,

Williamson

2020

Preprint

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Given a commutative ring R and finitely generated ideal I, one can consider the classes of I-adically complete, L I 0 -complete and derived I-complete complexes. Under a mild assumption on the ideal I called weak pro-regularity, these three notions of completions interact well. We consider the classes of I-adically complete, L I 0 -complete and derived I-complete complexes and prove that they present the same homotopy theory. Given a ring homomorphism R → S, we then give necessary and sufficient conditions for the categories of complete R-complexes and the categories of complete S-complexes to have equivalent homotopy theories. This recovers and generalizes a result of Sather-Wagstaff and Wicklein on extended local (co)homology.

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Section: Contramodules and L I 0 -Completionmentioning

confidence: 68%

Section: Contribution Of This Paper and Related Workmentioning

confidence: 96%

The homotopy theory of complete modules

Pol,

Williamson

2020

Preprint

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“…Remark 12.13. The relationship between derived functors of the limit and derived completion in the context of comodules over a flat Hopf algebroid is studied further in [BHV20].…”

Section: Inverse Limits Of E * E-comodules and Continuous Cohomologymentioning

confidence: 99%

MORAVA K-THEORY AND FILTRATIONS BY POWERS

Barthel¹,

Pstrągowski

2023

J. Inst. Math. Jussieu

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We prove the convergence of the Adams spectral sequence based on Morava K-theory and relate it to the filtration by powers of the maximal ideal in the Lubin–Tate ring through a Miller square. We use the filtration by powers to construct a spectral sequence relating the homology of the K-local sphere to derived functors of completion and express the latter as cohomology of the Morava stabiliser group. As an application, we compute the zeroth limit at all primes and heights.

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“…Λ I the dual to the functor Γ I has been used widely to study local homology, see [7,9,18] among others. The functors Γ I and Λ I and their derived functors were key in [1,4,5,9,16,18,22,25] where notions such as Greenless-May (GM) Duality and Matlis-Greenless-May (MGM) Equivalence were studied in different settings.…”

Section: Introductionmentioning

confidence: 99%

Applications of reduced and coreduced modules I

Ssevviiri¹

2024

International Electronic Journal of Algebra

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This is the first in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We show that $I$-reduced $R$-modules and $I$-coreduced $R$-modules provide a setting in which the Matlis-Greenless-May (MGM) Equivalence and the Greenless-May (GM) Duality hold. These two notions have been hitherto only known to exist in the derived category setting. We realise the $I$-torsion and the $I$-adic completion functors as representable functors and under suitable conditions compute natural transformations between them and other functors.

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Derived completion for comodules

Cited by 9 publications

References 22 publications

The homotopy theory of complete modules

The homotopy theory of complete modules

MORAVA K-THEORY AND FILTRATIONS BY POWERS

Applications of reduced and coreduced modules I

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