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Let m ≥ 2 be an integer. By using m submodules of a given module, we construct a certain exact sequence, which is a well known short exact sequence when m = 2. As an application, we compute a minimal projective resolution of the Jacobson radical of a tiled order.

Artin-Schelter regular algebras of dimension five

Gunnar Fløystad, Jon Eivind Vatne (2011)

Banach Center Publications

We show that there are exactly three types of Hilbert series of Artin-Schelter regular algebras of dimension five with two generators. One of these cases (the most extreme) may not be realized by an enveloping algebra of a graded Lie algebra. This is a new phenomenon compared to lower dimensions, where all resolution types may be realized by such enveloping algebras.

BERGMAN under MS-DOS and Anick's resolution.

Cojocaru, S., Ufnarovski, V. (1997)

Discrete Mathematics and Theoretical Computer Science. DMTCS [electronic only]

Bimodule resolution of Möbius algebras.

Generalov, A.I., Kachalova, M.A. (2005)

Zapiski Nauchnykh Seminarov POMI

Bimodule resolution of the Liu-Schulz algebras.

Kosovskaya, N.Yu. (2005)

Zapiski Nauchnykh Seminarov POMI

Cohomology of algebras of semidihedral type. IV.

Generalov, A.I. (2004)

Zapiski Nauchnykh Seminarov POMI

Cohomology of the triangular algebra and applications. (Cohomologie de l'algèbre triangulaire et applications.)

Bendiffalah, B., Guin, D. (2003)

AMA. Algebra Montpellier Announcements [electronic only]

Complexity and periodicity

Petter Andreas Bergh (2006)

Colloquium Mathematicae

Let M be a finitely generated module over an Artin algebra. By considering the lengths of the modules in the minimal projective resolution of M, we obtain the Betti sequence of M. This sequence must be bounded if M is eventually periodic, but the converse fails to hold in general. We give conditions under which it holds, using techniques from Hochschild cohomology. We also provide a result which under certain conditions guarantees the existence of periodic modules. Finally, we study the case when...

Ding projective and Ding injective modules over trivial ring extensions

Lixin Mao (2023)

Czechoslovak Mathematical Journal

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 $M \otimes_{R} M \otimes_{R} X \overset{M \otimes α}{⟶} M \otimes_{R} X \overset{α}{\to} X$ 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.

Existence of Gorenstein projective resolutions and Tate cohomology

Peter Jørgensen (2007)

Journal of the European Mathematical Society

Existence of proper Gorenstein projective resolutions and Tate cohomology is proved over rings with a dualizing complex. The proofs are based on Bousfield Localization which is originally a method from algebraic topology.

Finiteness aspects of Gorenstein homological dimensions

Samir Bouchiba (2013)

Colloquium Mathematicae

We present an alternative way of measuring the Gorenstein projective (resp., injective) dimension of modules via a new type of complete projective (resp., injective) resolutions. As an application, we easily recover well known theorems such as the Auslander-Bridger formula. Our approach allows us to relate the Gorenstein global dimension of a ring R to the cohomological invariants silp(R) and spli(R) introduced by Gedrich and Gruenberg by proving that leftG-gldim(R) = maxleftsilp(R), leftspli(R),...

Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs

Dennis Sullivan (2009)

Banach Center Publications

Using the algebraic theory of homotopies between maps of dga's we obtain a homotopy theory for algebraic structures defined by collections of multiplications and comultiplications. This is done by expressing these structures and resolved versions of them in terms of dga maps. This same homotopy theory of dga maps applies to extract invariants beyond homological periods from systems of moduli spaces that determine systems of chains that satisfy master equations like dX + X*X = 0. Minimal models of...

Koszul duality for N-Koszul algebras

Roberto Martínez-Villa, Manuel Saorín (2005)

Colloquium Mathematicae

The correspondence between the category of modules over a graded algebra and the category of graded modules over its Yoneda algebra was studied in [8] by means of $A_{\infty}$ algebras; this relation is very well understood for Koszul algebras (see for example [5],[6]). It is of interest to look for cases such that there exists a duality generalizing the Koszul situation. In this paper we will study N-Koszul algebras [1], [7], [9] for which such a duality exists.

$N$ -Koszul algebras, a summary.

Green, E.L., Marcos, E.N., Martínez-Villa, R., Zhang, Pu (2003)

AMA. Algebra Montpellier Announcements [electronic only]

$n$ -strongly Gorenstein graded modules

Zenghui Gao, Jie Peng (2019)

Czechoslovak Mathematical Journal

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. Then we investigate...

On finitely generated n-SG-projective modules

Driss Bennis (2010)

Colloquium Mathematicae

We prove that finitely generated n-SG-projective modules are infinitely presented.

On Gorenstein flat preenvelopes of complexes

Gang Yang, Zhongkui Liu, Li Liang (2013)

Rendiconti del Seminario Matematico della Università di Padova

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