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Academic literature on the topic 'Projective modules (Algebra)'

Author: Grafiati

Published: 4 June 2021

Last updated: 31 July 2024

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Journal articles on the topic "Projective modules (Algebra)"

1

FRISK, ANDERS, and VOLODYMYR MAZORCHUK. "PROPERLY STRATIFIED ALGEBRAS AND TILTING." Proceedings of the London Mathematical Society 92, no. 1 (2005): 29–61. http://dx.doi.org/10.1017/s0024611505015431.

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We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

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2

Asefa, Dadi. "Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras." Journal of Mathematics 2021 (November 19, 2021): 1–8. http://dx.doi.org/10.1155/2021/8127282.

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Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A , how to construct all the Gorenstein-projective A -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ = A M B A 0 B . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ = A M B A 0 B .

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3

Li, Huanhuan. "The Projective Leavitt Complex." Proceedings of the Edinburgh Mathematical Society 61, no. 4 (2018): 1155–77. http://dx.doi.org/10.1017/s001309151800007x.

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AbstractFor a finite quiverQwithout sources, we consider the corresponding radical square zero algebraA. We construct an explicit compact generator for the homotopy category of acyclic complexes of projectiveA-modules. We call such a generator the projective Leavitt complex ofQ. This terminology is justified by the following result: the opposite differential graded endomorphism algebra of the projective Leavitt complex ofQis quasi-isomorphic to the Leavitt path algebra ofQop. Here,Qopis the opposite quiver ofQ, and the Leavitt path algebra ofQopis naturally${\open Z}$-graded and viewed as a differential graded algebra with trivial differential.

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4

Asefa, Dadi. "Gorenstein-Projective Modules over Morita Rings." Algebra Colloquium 28, no. 03 (2021): 521–32. http://dx.doi.org/10.1142/s1005386721000407.

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Let [Formula: see text] be a Morita ring which is an Artin algebra. In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring [Formula: see text] and the algebras [Formula: see text] and [Formula: see text]. We prove that if [Formula: see text] is a Gorenstein algebra and both [Formula: see text] and [Formula: see text] (resp., both [Formula: see text] and [Formula: see text]) have finite projective dimension, then [Formula: see text] (resp., [Formula: see text]) is a Gorenstein algebra. We also discuss when the CM-freeness and the CM-finiteness of a Morita ring [Formula: see text] is inherited by the algebras [Formula: see text] and [Formula: see text].

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5

Popescu, Dorin. "Polynomial rings and their projective modules." Nagoya Mathematical Journal 113 (March 1989): 121–28. http://dx.doi.org/10.1017/s0027763000001288.

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Let R be a regular noetherian ring. A central question concerning projective modules over polynomial R-algebras is the following.(1.1) BASS-QUILLEN CONJECTURE ([2] Problem IX, [10]). Every finitely generated projective module P over a polynomial R-algebra R[T], T = (T1,…, Tn) is extended from R, i.e.P≊R[T]⊗R P/(T)P.

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6

Sheu, Albert Jeu-Liang. "Projective Modules Over Quantum Projective Line." International Journal of Mathematics 28, no. 03 (2017): 1750022. http://dx.doi.org/10.1142/s0129167x17500227.

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Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces [Formula: see text] constructed from the multipullback quantum spheres introduced by Hajac and collaborators, we analyze the structure of the C*-algebra [Formula: see text] realized as a concrete groupoid C*-algebra, and find its [Formula: see text]-groups. Furthermore, after a complete classification of the unitary equivalence classes of projections or equivalently the isomorphism classes of finitely generated projective modules over the C*-algebra [Formula: see text], we identify those quantum principal [Formula: see text]-bundles introduced by Hajac and collaborators among the projections classified.

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7

Maulana, Iqbal. "Schanuel's Lemma in P-Poor Modules." Jurnal Matematika "MANTIK" 5, no. 2 (2019): 76–82. http://dx.doi.org/10.15642/mantik.2019.5.2.76-82.

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Modules are a generalization of the vector spaces of linear algebra in which the “scalars” are allowed to be from a ring with identity, rather than a field. In module theory there is a concept about projective module, i.e. a module over ring R in which it is projective module relative to all modules over ring R. Next, there is the fact that every module over ring R is projective module relative to all semisimple modules over ring R. If P is a module over ring R which it’s projective relative only to all semisimple modules over ring R, then P is called p-poor module. In the discussion of the projective module, there is a lemma associated with the equivalence of two modules K1 and K2 provided that there are two projective modules P1 and P2 such that is isomorphic to . That lemma is known as Schanuel’s lemma in projective modules. Because the p-poor module is a special case of the projective module, then in this paper will be discussed about Schanuel’s lemma in p-poor modules

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8

Wang, Ren. "The MCM-approximation of the trivial module over a category algebra." Journal of Algebra and Its Applications 16, no. 06 (2017): 1750109. http://dx.doi.org/10.1142/s0219498817501092.

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For a finite free EI category, we construct an explicit module over its category algebra. If in addition the category is projective over the ground field, the constructed module is a maximal Cohen–Macaulay approximation of the trivial module and is the tensor identity of the stable category of Gorenstein-projective modules over the category algebra. We give conditions on when the trivial module is Gorenstein-projective.

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9

BRZEZIŃSKI, TOMASZ. "DIVERGENCES ON PROJECTIVE MODULES AND NON-COMMUTATIVE INTEGRALS." International Journal of Geometric Methods in Modern Physics 08, no. 04 (2011): 885–96. http://dx.doi.org/10.1142/s0219887811005440.

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A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first-order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence are studied, in particular the formula of integration by parts is derived. Specific examples include inner calculi on a non-commutative algebra, the Berezin integral on the supercircle and integrals on Hopf algebras.

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10

Zhang, Aiping. "Endomorphism algebras of Gorenstein projective modules." Journal of Algebra and Its Applications 17, no. 09 (2018): 1850177. http://dx.doi.org/10.1142/s0219498818501773.

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Let [Formula: see text] be an Artin algebra, [Formula: see text] be a Gorenstein projective [Formula: see text]-module and [Formula: see text]. We give a characterization of modules on [Formula: see text] and show that if [Formula: see text] is [Formula: see text]-representation-finite, then [Formula: see text] is also [Formula: see text]-representation-finite. As an application, we prove if [Formula: see text] is a CM-finite [Formula: see text]-Gorenstein algebra, then [Formula: see text] is a [Formula: see text]-Igusa-Todorov algebra.

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