Gorenstein star modules and  Gorenstein tilting modules

Peiyu Zhang

Displaying similar documents to “Gorenstein star modules and Gorenstein tilting modules”

Commutative rings whose certain modules decompose into direct sums of cyclic submodules

Farid Kourki, Rachid Tribak (2023)

Czechoslovak Mathematical Journal

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We provide some characterizations of rings $R$ for which every (finitely generated) module belonging to a class $𝒞$ of $R$ -modules is a direct sum of cyclic submodules. We focus on the cases, where the class $𝒞$ is one of the following classes of modules: semiartinian modules, semi-V-modules, V-modules, coperfect modules and locally supplemented modules.

$n$ -strongly Gorenstein graded modules

Zenghui Gao, Jie Peng (2019)

Czechoslovak Mathematical Journal

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Let $R$ be a graded ring and $n \geq 1$ an integer. We introduce and study $n$ -strongly Gorenstein gr-projective, gr-injective and gr-flat modules. Some examples are given to show that $n$ -strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules need not be $m$ -strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules whenever $n > m$ . Many properties of the $n$ -strongly Gorenstein gr-injective and gr-flat modules are discussed, some known results are generalized....

On the structure of sequentially Cohen-Macaulay bigraded modules

Leila Parsaei Majd, Ahad Rahimi (2015)

Czechoslovak Mathematical Journal

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Let $K$ be a field and $S = K [x_{1}, ..., x_{m}, y_{1}, ..., y_{n}]$ be the standard bigraded polynomial ring over $K$ . In this paper, we explicitly describe the structure of finitely generated bigraded “sequentially Cohen-Macaulay” $S$ -modules with respect to $Q = (y_{1}, ..., y_{n})$ . Next, we give a characterization of sequentially Cohen-Macaulay modules with respect to $Q$ in terms of local cohomology modules. Cohen-Macaulay modules that are sequentially Cohen-Macaulay with respect to $Q$ are considered.

Top-stable and layer-stable degenerations and hom-order

S. O. Smalø, A. Valenta (2007)

Colloquium Mathematicae

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Using geometrical methods, Huisgen-Zimmermann showed that if M is a module with simple top, then M has no proper degeneration $M <_{d e g} N$ such that $^{t} M /^{t + 1} M ≃^{t} N /^{t + 1} N$ for all t. Given a module M with square-free top and a projective cover P, she showed that $d i m_{k} H o m (M, M) = d i m_{k} H o m (P, M)$ if and only if M has no proper degeneration $M <_{d e g} N$ where M/M ≃ N/N. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from...

Non-weight modules over the super Schrödinger algebra

Xinyue Wang, Liangyun Chen, Yao Ma (2024)

Czechoslovak Mathematical Journal

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We construct a family of non-weight modules which are free $U (𝔥)$ -modules of rank 2 over the $N = 1$ super Schrödinger algebra in $(1 + 1)$ -dimensional spacetime. We determine the isomorphism classes of these modules. In particular, free $U (𝔥)$ -modules of rank 2 over $𝔬𝔰𝔭 (1 | 2)$ are also constructed and classified. Moreover, we obtain the sufficient and necessary conditions for such modules to be simple.

Gorenstein projective complexes with respect to cotorsion pairs

Renyu Zhao, Pengju Ma (2019)

Czechoslovak Mathematical Journal

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Let $(𝒜, ℬ)$ be a complete and hereditary cotorsion pair in the category of left $R$ -modules. In this paper, the so-called Gorenstein projective complexes with respect to the cotorsion pair $(𝒜, ℬ)$ are introduced. We show that these complexes are just the complexes of Gorenstein projective modules with respect to the cotorsion pair $(𝒜, ℬ)$ . As an application, we prove that both the Gorenstein projective modules with respect to cotorsion pairs and the Gorenstein projective complexes with respect to cotorsion...

On commutative rings whose maximal ideals are idempotent

Farid Kourki, Rachid Tribak (2019)

Commentationes Mathematicae Universitatis Carolinae

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We prove that for a commutative ring $R$ , every noetherian (artinian) $R$ -module is quasi-injective if and only if every noetherian (artinian) $R$ -module is quasi-projective if and only if the class of noetherian (artinian) $R$ -modules is socle-fine if and only if the class of noetherian (artinian) $R$ -modules is radical-fine if and only if every maximal ideal of $R$ is idempotent.

On the composition structure of the twisted Verma modules for $𝔰𝔩 (3, ℂ)$

Libor Křižka, Petr Somberg (2015)

Archivum Mathematicum

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We discuss some aspects of the composition structure of twisted Verma modules for the Lie algebra $𝔰𝔩 (3, ℂ)$ , including the explicit structure of singular vectors for both $𝔰𝔩 (3, ℂ)$ and one of its Lie subalgebras $𝔰𝔩 (2, ℂ)$ , and also of their generators. Our analysis is based on the use of partial Fourier tranform applied to the realization of twisted Verma modules as $D$ -modules on the Schubert cells in the full flag manifold for $SL (3, ℂ)$ .

The deformation relation on the set of Cohen-Macaulay modules on a quotient surface singularity

Trond Stølen Gustavsen, Runar Ile (2011)

Banach Center Publications

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Let X be a quotient surface singularity, and define $G^{d e f} (X, r)$ as the directed graph of maximal Cohen-Macaulay (MCM) modules with edges corresponding to deformation incidences. We conjecture that the number of connected components of $G^{d e f} (X, r)$ is equal to the order of the divisor class group of X, and when X is a rational double point (RDP), we observe that this follows from a result of A. Ishii. We view this as an enrichment of the McKay correspondence. For a general quotient singularity X, we prove...

(Strongly) Gorenstein injective modules over upper triangular matrix Artin algebras

Chao Wang, Xiao Yan Yang (2017)

Czechoslovak Mathematical Journal

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Let $Λ = (\begin{matrix} A & M \\ 0 & B \end{matrix})$ be an Artin algebra. In view of the characterization of finitely generated Gorenstein injective $Λ$ -modules under the condition that $M$ is a cocompatible $(A, B)$ -bimodule, we establish a recollement of the stable category $\bar{Ginj (Λ)}$ . We also determine all strongly complete injective resolutions and all strongly Gorenstein injective modules over $Λ$ .

Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup

Jon F. Carlson, Nadia Mazza, Jacques Thévenaz (2013)

Journal of the European Mathematical Society

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Let $G$ be a finite group with a Sylow 2-subgroup $P$ which is either quaternion or semi-dihedral. Let $k$ be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial $k G$ -modules, whose restrictions to $P$ are isomorphic to the direct sum of the known exotic endotrivial $k P$ -modules and some projective modules. This provides a description of the group $T (G)$ of endotrivial $k G$ -modules.

Some Results About Holonomic $ε$ -Modules

Jan-Erik Björk (1983)

Recherche Coopérative sur Programme n°25

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Category $𝒪$ for quantum groups

Henning Haahr Andersen, Volodymyr Mazorchuk (2015)

Journal of the European Mathematical Society

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In this paper we study the BGG-categories $𝒪_{q}$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $𝒪$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $𝒪$ and for finite dimensional $U_{q}$ -modules we are able...

CF-modules over commutative rings

Ahmed Najim, Mohammed Elhassani Charkani (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a commutative ring with unit. We give some criterions for determining when a direct sum of two CF-modules over $R$ is a CF-module. When $R$ is local, we characterize the CF-modules over $R$ whose tensor product is a CF-module.

On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence

Kosar Abolfath Beigi, Kamran Divaani-Aazar, Massoud Tousi (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a local ring and $C$ a semidualizing module of $R$ . We investigate the behavior of certain classes of generalized Cohen-Macaulay $R$ -modules under the Foxby equivalence between the Auslander and Bass classes with respect to $C$ . In particular, we show that generalized Cohen-Macaulay $R$ -modules are invariant under this equivalence and if $M$ is a finitely generated $R$ -module in the Auslander class with respect to $C$ such that $C \otimes_{R} M$ is surjective Buchsbaum, then $M$ is also surjective Buchsbaum. ...

The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras

Piotr Dowbor, Andrzej Mróz (2008)

Colloquium Mathematicae

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Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m (M) = {(m_{x})}_{x \in X} \in ℕ^{X}$ such that $M ≅ ⨁_{x \in X} X_{x}^{m_{x}}$ is studied. A precise formula for $d i m_{k} H o m_{Λ} (M, X)$ , for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors...