Weakly Stable Torsion Classes

Vyas, Rishi

doi:10.1007/s10468-018-9817-1

Cited by 3 publications

(8 citation statements)

References 18 publications

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“…It can be shown that the torsion class T S is weakly stable if and only if it has dimension ≤ 1. This is [Vy,Theorem 4.4].…”

Section: Definition 34 Let T Be a Torsion Class In M(a)mentioning

confidence: 96%

“…Corollary 4.22 could be false for a torsion class T that is not defined by a finitely generated ideal. Indeed, in [Vy,Example 4.12] there is a weakly stable torsion class which is not quasi-compact (but is finite dimensional).…”

Section: Corollary 422mentioning

confidence: 99%

“…As already mentioned in Example 3.15, the torsion class T a ⊆ M(A) is weakly stable. It can be shown (see [Vy,Lemma 6.4]) that T a is also quasi-compact and finite dimensional. Therefore Theorem 8.4 applies.…”

Section: Proofmentioning

confidence: 99%

“…Compare to the commutative situation in Theorem 0.1, where Λ = Λ a is the a-adic completion functor. On the other hand, there are counterexamples where this fails (see [Vy,Example 6.2]).…”

Section: Proofmentioning

confidence: 99%

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Weak proregularity, weak stability, and the noncommutative MGM equivalence

Vyas

Yekutieli

2018

Journal of Algebra

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Let A be a commutative ring, and let a be a finitely generated Let A be a commutative ring, and let a be a finitely generated ideal in A. The category of A-modules is denoted by M(A). There are two functors on the category M(A) that the ideal a determines: the a-torsion functor Γ a , and the aadic completion functor Λ a . These are idempotent additive functors: Γ a • Γ a ∼ = Γ a and Λ a • Λ a ∼ = Λ a .The functors Γ a and Λ a seem as though they could be adjoint to each other. This is false however. What is true is that under suitable assumptions, the derived functors RΓ a , LΛ a :are adjoint to each other. Here D(A) is the (unbounded) derived category of Amodules. The most general condition under which this is known to hold is when the ideal a is weakly proregular. By definition, the ideal a is weakly proregular if it can be generated by a weakly proregular sequence. A sequence of elements a = (a 1 , . . . , a n ) in A is called weakly proregular if a rather complicated condition is satisfied by the Koszul complexes associated to powers of a; see Definition 4.1. This condition was first stated, without a name, by A. Grothendieck (see [LC] and [SGA2]); the name was only given by L. Alonso, A. Jeremias and J. Lipman in [AJL, Correction]. If A is noetherian, then every ideal in it is weakly proregular; but there are many non-noetherian examples (see Examples 4.10-4.12). The next theorem is the culmination of results by Grothendieck [LC], [SGA2]; E. Matlis [Ma]; J.P.C. Greenlees and J.P. May [GM]; Alonso, Jeremias and Lipman [AJL]; P. Schenzel [Sn]; M. Kashiwara and P. Schapira [KS3]; and M. Porta, L. Shaul and Yekutieli [PSY1]. There is parallel recent work on these matters by L. Positselski [Po]. Let a ⊆ A be a finitely generated ideal. A complex M ∈ D(A) is called derived a-torsion if the canonical morphism RΓ a (M ) → M is an isomorphism. Similarly, a complex M ∈ D(A) is called derived a-adically complete if the canonical morphism M → LΛ a (M ) is an isomorphism. (In [PSY1] these complexes were called cohomologically complete and cohomologically torsion, respectively; but now we realize that the adjective "derived" is better suited than "cohomologically". See Definition 3.7 below.) We denote by D(A) a-tor and D(A) a-com the full subcategory of D(A) on the derived a-torsion and the derived a-adically complete complexes, respectively. These are full triangulated subcategories.Theorem 0.1 (MGM Equivalence, [PSY1]). Let a be a weakly proregular ideal in a commutative ring A.(1) The functor LΛ a is right adjoint to the functor RΓ a .(2) The functors RΓ a and LΛ a are idempotent. (3) The categories D(A) a-tor and D(A) a-com are the essential images of the functors RΓ a and LΛ a , respectively. (4) The functor LΛ a : D(A) a-tor → D(A) a-comis an equivalence of triangulated categories, with quasi-inverse RΓ a .Actually, item (1) of this theorem is usually called GM Duality, and it is [PSY1, Erratum, Theorem 9]. Item (2) means that RΓ a •RΓ a ∼ = RΓ a and LΛ a •LΛ a ∼ = LΛ a .

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“…It can be shown that the torsion class T S is weakly stable if and only if it has dimension ≤ 1. This is [Vy,Theorem 4.4].…”

Section: Definition 34 Let T Be a Torsion Class In M(a)mentioning

confidence: 96%

Section: Corollary 422mentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

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Weak proregularity, weak stability, and the noncommutative MGM equivalence

Vyas

Yekutieli

2018

Journal of Algebra

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“…One version of the MGM Equivalence is Theorem 1.2 which appears as Theorem 7.11 in [17]. Recently, its version in the noncommutative setting was given, see [24].…”

Section: Introductionmentioning

confidence: 99%

Applications of reduced and coreduced modules I

Ssevviiri¹

2022

Preprint

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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 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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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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Weakly Stable Torsion Classes

Cited by 3 publications

References 18 publications

Weak proregularity, weak stability, and the noncommutative MGM equivalence

Weak proregularity, weak stability, and the noncommutative MGM equivalence

Applications of reduced and coreduced modules I

Applications of reduced and coreduced modules I

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