### A construction of complex syzygy periodic modules over symmetric algebras

We construct arbitrarily complicated indecomposable finite-dimensional modules with periodic syzygies over symmetric algebras.

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We construct arbitrarily complicated indecomposable finite-dimensional modules with periodic syzygies over symmetric algebras.

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.

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.

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...

Let $R\u22c9M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$ such that ${M}_{R}$, ${}_{R}M$, ${(R,0)}_{R\u22c9M}$ and ${}_{R\u22c9M}(R,0)$ have finite flat dimensions. We prove that $(X,\alpha )$ is a Ding projective left $R\u22c9M$-module if and only if the sequence $M{\otimes}_{R}M{\otimes}_{R}X\stackrel{M\otimes \alpha}{\u27f6}M{\otimes}_{R}X\stackrel{\alpha}{\to}X$ is exact and $\mathrm{coker}\left(\alpha \right)$ is a Ding projective left $R$-module. Analogously, we explicitly describe Ding injective $R\u22c9M$-modules. As applications, we characterize Ding projective and Ding injective modules over Morita context rings with zero bimodule homomorphisms.

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.

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),...

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...

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.

Let $R$ be a graded ring and $n\ge 1$ 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...

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