Displaying similar documents to “A note on Frobenius divided modules in mixed characteristics”

Cominimaxness of local cohomology modules

Moharram Aghapournahr (2019)

Czechoslovak Mathematical Journal

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Let R be a commutative Noetherian ring, I an ideal of R . Let t 0 be an integer and M an R -module such that Ext R i ( R / I , M ) is minimax for all i t + 1 . We prove that if H I i ( M ) is FD 1 (or weakly Laskerian) for all i < t , then the R -modules H I i ( M ) are I -cominimax for all i < t and Ext R i ( R / I , H I t ( M ) ) is minimax for i = 0 , 1 . Let N be a finitely generated R -module. We prove that Ext R j ( N , H I i ( M ) ) and Tor j R ( N , H I i ( M ) ) are I -cominimax for all i and j whenever M is minimax and H I i ( M ) is FD 1 (or weakly Laskerian) for all i .

Some results on ( n , d ) -injective modules, ( n , d ) -flat modules and n -coherent rings

Zhanmin Zhu (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let n , d be two non-negative integers. A left R -module M is called ( n , d ) -injective, if Ext d + 1 ( N , M ) = 0 for every n -presented left R -module N . A right R -module V is called ( n , d ) -flat, if Tor d + 1 ( V , N ) = 0 for every n -presented left R -module N . A left R -module M is called weakly n - F P -injective, if Ext n ( N , M ) = 0 for every ( n + 1 ) -presented left R -module N . A right R -module V is called weakly n -flat, if Tor n ( V , N ) = 0 for every ( n + 1 ) -presented left R -module N . In this paper, we give some characterizations and properties of ( n , d ) -injective modules and ( n , d ) -flat modules in...

Recollements induced by good (co)silting dg-modules

Rongmin Zhu, Jiaqun Wei (2023)

Czechoslovak Mathematical Journal

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Let U be a dg- A -module, B the endomorphism dg-algebra of U . We know that if U is a good silting object, then there exist a dg-algebra C and a recollement among the derived categories 𝐃 ( C , d ) of C , 𝐃 ( B , d ) of B and 𝐃 ( A , d ) of A . We investigate the condition under which the induced dg-algebra C is weak nonpositive. In order to deal with both silting and cosilting dg-modules consistently, the notion of weak silting dg-modules is introduced. Thus, similar results for good cosilting dg-modules are obtained....

On τ -extending modules

Y. Talebi, R. Mohammadi (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we introduce the concept of τ -extending modules by τ -rational submodules and study some properties of such modules. It is shown that the set of all τ -rational left ideals of R R is a Gabriel filter. An R -module M is called τ -extending if every submodule of M is τ -rational in a direct summand of M . It is proved that M is τ -extending if and only if M = R e j M E ( R / τ ( R ) ) N , such that N is a τ -extending submodule of M . An example is given to show that the direct sum of τ -extending modules need not...

Local cohomology, cofiniteness and homological functors of modules

Kamal Bahmanpour (2022)

Czechoslovak Mathematical Journal

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Let I be an ideal of a commutative Noetherian ring R . It is shown that the R -modules H I j ( M ) are I -cofinite for all finitely generated R -modules M and all j 0 if and only if the R -modules Ext R i ( N , H I j ( M ) ) and Tor i R ( N , H I j ( M ) ) are I -cofinite for all finitely generated R -modules M , N and all integers i , j 0 .

S -depth on Z D -modules and local cohomology

Morteza Lotfi Parsa (2021)

Czechoslovak Mathematical Journal

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Let R be a Noetherian ring, and I and J be two ideals of R . Let S be a Serre subcategory of the category of R -modules satisfying the condition C I and M be a Z D -module. As a generalization of the S - depth ( I , M ) and depth ( I , J , M ) , the S - depth of ( I , J ) on M is defined as S - depth ( I , J , M ) = inf { S - depth ( 𝔞 , M ) : 𝔞 W ˜ ( I , J ) } , and some properties of this concept are investigated. The relations between S - depth ( I , J , M ) and H I , J i ( M ) are studied, and it is proved that S - depth ( I , J , M ) = inf { i : H I , J i ( M ) S } , where S is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology...

Enveloping algebras of Slodowy slices and the Joseph ideal

Alexander Premet (2007)

Journal of the European Mathematical Society

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Let G be a simple algebraic group over an algebraically closed field 𝕜 of characteristic 0, and 𝔤 = Lie G . Let ( e , h , f ) be an 𝔰 𝔩 2 -triple in 𝔤 with e being a long root vector in 𝔤 . Let ( · , · ) be the G -invariant bilinear form on 𝔤 with ( e , f ) = 1 and let χ 𝔤 * be such that χ ( x ) = ( e , x ) for all x 𝔤 . Let 𝒮 be the Slodowy slice at e through the adjoint orbit of e and let H be the enveloping algebra of 𝒮 ; see [31]. In this article we give an explicit presentation of H by generators and relations. As a consequence we deduce that H contains...

α -modules and generalized submodules

Rafiquddin Rafiquddin, Ayazul Hasan, Mohammad Fareed Ahmad (2019)

Communications in Mathematics

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A QTAG-module M is an α -module, where α is a limit ordinal, if M / H β ( M ) is totally projective for every ordinal β < α . In the present paper α -modules are studied with the help of α -pure submodules, α -basic submodules, and α -large submodules. It is found that an α -closed α -module is an α -injective. For any ordinal ω α ω 1 we prove that an α -large submodule L of an ω 1 -module M is summable if and only if M is summable.

Purity of level m stratifications

Marc-Hubert Nicole, Adrian Vasiu, Torsten Wedhorn (2010)

Annales scientifiques de l'École Normale Supérieure

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Let k be a field of characteristic p &gt; 0 . Let D m be a BT m over k (i.e., an m -truncated Barsotti–Tate group over k ). Let S be a k -scheme and let X be a BT m over S . Let S D m ( X ) be the subscheme of S which describes the locus where X is locally for the fppf topology isomorphic to D m . If p 5 , we show that S D m ( X ) is pure in S , i.e. the immersion S D m ( X ) S is affine. For p { 2 , 3 } , we prove purity if D m satisfies a certain technical property depending only on its p -torsion D m [ p ] . For p 5 , we apply the developed techniques to show that...

Some homological properties of amalgamated modules along an ideal

Hanieh Shoar, Maryam Salimi, Abolfazl Tehranian, Hamid Rasouli, Elham Tavasoli (2023)

Czechoslovak Mathematical Journal

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Let R and S be commutative rings with identity, J be an ideal of S , f : R S be a ring homomorphism, M be an R -module, N be an S -module, and let ϕ : M N be an R -homomorphism. The amalgamation of R with S along J with respect to f denoted by R f J was introduced by M. D’Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of ( R f J ) -module called the amalgamation of M and N along J with respect to ϕ , and denoted by M ϕ J N . We study some homological properties of the ( R f J ) -module M ϕ J N . Among...