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In this paper we show that Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra.

It has been asked whether there is a version of the tensor product property for support varieties over finite dimensional algebras defined in terms of Hochschild cohomology. We show that in general no such version can exist. In particular, we show that for certain quantum complete intersections, there are modules and bimodules for which the variety of the tensor product is not even contained in the variety of the one-sided module., Comment: 9 pages

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples., Comment: 37 pages

We generalize the notions of $n$-cluster tilting subcategories and $\tau$-selfinjective algebras into $n$-precluster tilting subcategories and $\tau_n$-selfinjective algebras, where we show that a subcategory naturally associated to $n$-precluster tilting subcategories has a higher Auslander--Reiten theory. Furthermore, we give a bijection between $n$-precluster tilting subcategories and $n$-minimal Auslander--Gorenstein algebras, which is a higher dimensional analog of Auslander--Solberg correspondence (Auslander--Solberg, 1993) as well as a Gorenstein analog of $n$-Auslander correspondence (Iyama, 2007). The Auslander--Reiten theory associated to an $n$-precluster tilting subcategory is used to classify the $n$-minimal Auslander--Gorenstein algebras into four disjoint classes. Our method is based on relative homological algebra due to Auslander--Solberg., Comment: 32 pages, final published version. The terminology `n-minimal Auslander-Gorenstein algebra' is used

Let $R$ be a Koszul algebra over a field $k$ and $M$ be a linear $R$-module. We study a graded subalgebra $\Delta_M$ of the Ext-algebra $\operatorname{Ext}_R^*(M,M)$ called the diagonal subalgebra and its properties. Applications to the Hochschild cohomology ring of $R$ and to periodicity of linear modules are given. Viewing $R$ as a linear module over its enveloping algebra, we also show that $\Delta_R$ is isomorphic to the graded center of the Koszul dual of $R$.

Given an artin algebra $\Lambda$ with an idempotent element $a$ we compare the algebras $\Lambda$ and $a\Lambda a$ with respect to Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology. In particular, we identify assumptions on the idempotent element $a$ which ensure that $\Lambda$ is Gorenstein if and only if $a\Lambda a$ is Gorenstein, that the singularity categories of $\Lambda$ and $a\Lambda a$ are equivalent and that Fg holds for $\Lambda$ if and only if Fg holds for $a\Lambda a$. We approach the problem by using recollements of abelian categories and we prove the results concerning Gorensteinness and singularity categories in this general setting. The results are applied to stable categories of Cohen-Macaulay modules and classes of triangular matrix algebras and quotients of path algebras., Comment: 42 pages, no figures

One of our main results is a classification all the weakly symmetric radical cube zero finite dimensional algebras over an algebraically closed field having a theory of support via the Hochschild cohomology ring satisfying Dade's Lemma. Along the way we give a characterization of when a finite dimensional Koszul algebra has such a theory of support in terms of the graded centre of the Koszul dual.

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension $n$ to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin-Schelter regular algebra or a category of dimension $n$ is shown to have common properties with the classical Artin-Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the $\Ext$-category of nice sets of simple objects of maximal projective dimension $n$ is a finite length Frobenius category., Comment: 29 pages

Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there is always a naturally associated Koszul theory. To obtain this, the notions of Koszul algebras, linear modules and Koszul duality are extended to additive (graded) categories over a field. The main focus of this paper is to provide these generalizations and the necessary preliminaries., Comment: 34 pages

We define relative support varieties with respect to some fixed module over a finite dimensional algebra. These varieties share many of the standard properties of classical support varieties. Moreover, when introducing finite generation conditions on cohomology, we show that relative support varieties contain homological information on the modules involved. As an application, we provide a new criterion for a selfinjective algebra to be of wild representation type., Comment: 10 pages

We provide an algorithmic method for constructing projective resolutions of modules over quotients of path algebras. This algorithm is modified to construct minimal projective resolutions of linear modules over Koszul algebras.

We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can classify ideals in terms of their support. Applications arise from studying abelian or triangulated tensor categories. Specific examples from algebraic geometry and modular representation theory are discussed, illustrating the power of this approach which is inspired by recent work of Balmer., Comment: 25 pages

Let $\Lambda=kQ/I$ be a Koszul algebra over a field $k$, where $Q$ is a finite quiver. An algorithmic method for finding a minimal projective resolution $\mathbb{F}$ of the graded simple modules over $\Lambda$ is given in Green-Solberg. This resolution is shown to have a "comultiplicative" structure in Green-Hartman-Marcos-Solberg, and this is used to find a minimal projective resolution $\mathbb{P}$ of $\Lambda$ over the enveloping algebra $\Lambda^e$. Using these results we show that the multiplication in the Hochschild cohomology ring of $\L$ relative to the resolution $\mathbb{P}$ is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of $\L$ with respect to a canonical basis over $k$ associated to the resolution $\mathbb{F}$. The natural map from the Hochschild cohomology to the Koszul dual of $\Lambda$ is shown to be surjective onto the graded centre of the Koszul dual., Comment: 13 pages

Support varieties for any finite dimensional algebra over a field were introduced by Snashall-Solberg using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webb's theorem is true.