Notes on Piecewise-Koszul Algebras

The relationships between piecewise-Koszul algebras and other ＂Koszul-type＂ algebras are discussed. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A^1 to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary ＂period＂ and piecewise-Koszul algebras with arbitrary ＂jump-degree＂.

Selfinjective Koszul Algebras of Finite Complexity

In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Yt of modules with complexity less or equal to t, is resolving and coresolving. We show that for each 0 ≤ 1 ≤ m there exist a family of modules of complexity 1 parameterized by G（l, m）, the Grassmannian of l-dimensional subspaces of an m-dimensional vector space V, for the exterior algebra of V. Using complexity, we also give a new approach to the representation theory of a tame symmetric algebra with vanishing radical cube over an algebraically closed field of characteristic 0, via skew group algebra of a finite subgroup of SL（2, C） over the exterior algebra of a 2-dimensional vector space.

One-Point Extensions of t-Koszul Algebras

We introduce the category of t-fold modules which is a full subcategory of graded modules over a graded algebra. We show that this subcategory and hence the subcategory of t-Koszul modules are both closed under extensions and cokernels of monomorphisms. We study the one-point extension algebras, and a necessary and sufficient condition for such an algebra to be t-Koszul is given. We also consider the conditions such that the category of t-Koszul modules and the category of quadratic modules coincide.

Trivial extension of Koszul algebras

Let A be a Koszul algebra, and let M be a graded A-bimodule. We prove that the trivial extension algebra of A by M is also a Koszul algebra whenever M is Koszul as a left A-module. Applications and examples are also provided.

Clifford Deformations of Koszul Frobenius Algebras and Noncommutative Quadrics

A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z_(2)-graded algebra E(θ),which corresponds to a noncommutative quadric hypersurface E^(!)/(z)for some central regular element z∈E_(2)^(!).It turns out that the bounded derived category D^(b)(gr_(Z_(2))E(θ))is equivalent to the stable category of the maximal Cohen-Macaulay modules over E^(!)/(z)provided that E!is noetherian.As a consequence,E^(!)/(z)is a noncommutative isolated singularity if and only if the corresponding Clifford deformation E(θ)is a semisimple Z_(2)-graded algebra.The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces.As an application,we recover Knörrer's periodicity theorem without using matrix factorizations.

Koszul property of a class of graded algebras with nonpure resolutions

Given any integers a,b,c, and d with a 〉 1, c ≥ 0, b ≥ a＋c, and d ≥ b ＋ c, the notion of （a, b, c, d）-Koszul algebra is introduced, which is another class of standard graded algebras with ＂nonpure＂ resolutions, and includes many Artin-Schelter regular algebras of low global dimension as specific examples. Some basic properties of （a, b, c, d）-Koszul algebras/modules are given, and several criteria for a standard graded algebra to be （a, b, c, d）- Koszul are provided.

Koszulity and Koszul modules of dual extension algebras

Let A and B be algebras, and let T be the dual extension algebra of A and B. We provide a different method to prove that T is Koszul if and only if both A and B are Koszul. Furthermore, we prove that an algebra is Koszul if and only if one of its iterated dual extension algebras is Koszul, if and only if all its iterated dual extension algebras are Koszul. Finally, we give a necessary and sufficient condition for a dual extension algebra to have the property that all linearly presented modules are Koszul modules, which provides an effective way to construct algebras with such a property.

An Embedding for the E_2-term of the Adams Spectral Sequence at Odd Primes

Let p be an odd prime. In this paper we introduce a quadratic linear Fp-algebra Q1 obtained by suitably changing the generators of Q, tile homogeneous quadratic algebra of cohomology operations in the category of H∞-ring spectra, and study the map induced on cohomology by the quotient π ： Q1 → φp. Like in the case p = 2, it turns out that π is injective. Thus, its target contains the E2-term of the classical Adams spectral sequence as subalgebra. An explicit description of ExtQ1 （Fp, Fp） is given under the reasonable assumption on Q to be a Koszul algebra.

Hochschild cohomology of Zn-Galois coverings of an algebra

We consider the Zn-Galois covering An of the algebra A introduced by F. Xu [Adv. Math., 2008, 219： 1872-1893]. We calculate the dimensions of all Hochschild cohomology groups of An and give the ring structure of the Hochschild cohomology ring modulo nilpotence. As a conclusion, we provide a class of counterexamples to Snashall-Solberg＇s conjecture.