Displaying similar documents to “Homological dimensions for endomorphism algebras of Gorenstein projective modules”

Rings whose nonsingular right modules are R -projective

Yusuf Alagöz, Sinem Benli, Engin Büyükaşık (2021)

Commentationes Mathematicae Universitatis Carolinae

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A right R -module M is called R -projective provided that it is projective relative to the right R -module R R . This paper deals with the rings whose all nonsingular right modules are R -projective. For a right nonsingular ring R , we prove that R R is of finite Goldie rank and all nonsingular right R -modules are R -projective if and only if R is right finitely Σ - C S and flat right R -modules are R -projective. Then, R -projectivity of the class of nonsingular injective right modules is also considered....

Strongly ( 𝒯 , n ) -coherent rings, ( 𝒯 , n ) -semihereditary rings and ( 𝒯 , n ) -regular rings

Zhanmin Zhu (2020)

Czechoslovak Mathematical Journal

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Let 𝒯 be a weak torsion class of left R -modules and n a positive integer. A left R -module M is called ( 𝒯 , n ) -injective if Ext R n ( C , M ) = 0 for each ( 𝒯 , n + 1 ) -presented left R -module C ; a right R -module M is called ( 𝒯 , n ) -flat if Tor n R ( M , C ) = 0 for each ( 𝒯 , n + 1 ) -presented left R -module C ; a left R -module M is called ( 𝒯 , n ) -projective if Ext R n ( M , N ) = 0 for each ( 𝒯 , n ) -injective left R -module N ; the ring R is called strongly ( 𝒯 , n ) -coherent if whenever 0 K P C 0 is exact, where C is ( 𝒯 , n + 1 ) -presented and P is finitely generated projective, then K is ( 𝒯 , n ) -projective; the ring R is called...

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...

Ding projective and Ding injective modules over trivial ring extensions

Lixin Mao (2023)

Czechoslovak Mathematical Journal

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Let R M be a trivial extension of a ring R by an R - R -bimodule M such that M R , R M , ( R , 0 ) R M and R M ( R , 0 ) have finite flat dimensions. We prove that ( X , α ) is a Ding projective left R M -module if and only if the sequence M R M R X M α M R X α X is exact and coker ( α ) is a Ding projective left R -module. Analogously, we explicitly describe Ding injective R M -modules. As applications, we characterize Ding projective and Ding injective modules over Morita context rings with zero bimodule homomorphisms.

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 .

One-sided Gorenstein subcategories

Weiling Song, Tiwei Zhao, Zhaoyong Huang (2020)

Czechoslovak Mathematical Journal

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We introduce the right (left) Gorenstein subcategory relative to an additive subcategory 𝒞 of an abelian category 𝒜 , and prove that the right Gorenstein subcategory r 𝒢 ( 𝒞 ) is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When 𝒞 is self-orthogonal, we give a characterization for objects in r 𝒢 ( 𝒞 ) , and prove that any object in 𝒜 with finite r 𝒢 ( 𝒞 ) -projective dimension is isomorphic to a kernel (or a cokernel) of a morphism from an object in 𝒜 with finite 𝒞 -projective...

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.

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...

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...

Matlis dual of local cohomology modules

Batoul Naal, Kazem Khashyarmanesh (2020)

Czechoslovak Mathematical Journal

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Let ( R , 𝔪 ) be a commutative Noetherian local ring, 𝔞 be an ideal of R and M a finitely generated R -module such that 𝔞 M M and cd ( 𝔞 , M ) - grade ( 𝔞 , M ) 1 , where cd ( 𝔞 , M ) is the cohomological dimension of M with respect to 𝔞 and grade ( 𝔞 , M ) is the M -grade of 𝔞 . Let D ( - ) : = Hom R ( - , E ) be the Matlis dual functor, where E : = E ( R / 𝔪 ) is the injective hull of the residue field R / 𝔪 . We show that there exists the following long exact sequence 0 H 𝔞 n - 2 ( D ( H 𝔞 n - 1 ( M ) ) ) H 𝔞 n ( D ( H 𝔞 n ( M ) ) ) D ( M ) H 𝔞 n - 1 ( D ( H 𝔞 n - 1 ( M ) ) ) H 𝔞 n + 1 ( D ( H 𝔞 n ( M ) ) ) H 𝔞 n ( D ( H ( x 1 , ... , x n - 1 ) n - 1 ( M ) ) ) H 𝔞 n ( D ( H ( n - 1 M ) ) ) ... , where n : = cd ( 𝔞 , M ) is a non-negative integer, x 1 , ... , x n - 1 is a regular sequence in 𝔞 on M and, for an R -module L , H 𝔞 i ( L ) is the i th local cohomology module...

On the Configuration Spaces of Grassmannian Manifolds

Sandro Manfredini, Simona Settepanella (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let h i ( k , n ) be the i -th ordered configuration space of all distinct points H 1 , ... , H h in the Grassmannian G r ( k , n ) of k -dimensional subspaces of n , whose sum is a subspace of dimension i . We prove that h i ( k , n ) is (when non empty) a complex submanifold of G r ( k , n ) h of dimension i ( n - i ) + h k ( i - k ) and its fundamental group is trivial if i = m i n ( n , h k ) , h k n and n &gt; 2 and equal to the braid group of the sphere P 1 if n = 2 . Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. k = n - 1 .

The cleanness of (symbolic) powers of Stanley-Reisner ideals

Somayeh Bandari, Ali Soleyman Jahan (2017)

Czechoslovak Mathematical Journal

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Let Δ be a pure simplicial complex on the vertex set [ n ] = { 1 , ... , n } and I Δ its Stanley-Reisner ideal in the polynomial ring S = K [ x 1 , ... , x n ] . We show that Δ is a matroid (complete intersection) if and only if S / I Δ ( m ) ( S / I Δ m ) is clean for all m and this is equivalent to saying that S / I Δ ( m ) ( S / I Δ m , respectively) is Cohen-Macaulay for all m . By this result, we show that there exists a monomial ideal I with (pretty) cleanness property while S / I m or S / I ( m ) is not (pretty) clean for all integer m 3 . If dim ( Δ ) = 1 , we also prove that S / I Δ ( 2 ) ( S / I Δ 2 ) is clean if and only...

Hardness of embedding simplicial complexes in d

Jiří Matoušek, Martin Tancer, Uli Wagner (2011)

Journal of the European Mathematical Society

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Let 𝙴𝙼𝙱𝙴𝙳 k d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k , does there exist a (piecewise linear) embedding of K into d ? Known results easily imply polynomiality of 𝙴𝙼𝙱𝙴𝙳 k 2 ( k = 1 , 2 ; the case k = 1 , d = 2 is graph planarity) and of 𝙴𝙼𝙱𝙴𝙳 k 2 k for all k 3 . We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that 𝙴𝙼𝙱𝙴𝙳 d d and 𝙴𝙼𝙱𝙴𝙳 ( d - 1 ) d are undecidable for each d 5 . Our main result is NP-hardness of 𝙴𝙼𝙱𝙴𝙳 2 4 and, more generally, of 𝙴𝙼𝙱𝙴𝙳 k d for all...