Related papers: A counterexample to the $\phi$-dimension conjectur…

EI categories can be thought of as amalgams of finite posets and finite groups and therefore the associated algebras are built up from incidence algebras and group algebras of finite groups. For this particular class of algebras we present…

In 1989, D. Happel pointed out for a possible connection between the global dimension of a finite-dimensional algebra and its Hochschild cohomology: is it true that the vanishing of Hochschild cohomology higher groups is sufficient to…

We prove a version of Bass' finitistic dimension conjecture for path algebras over arbitrary directed graphs. It is known that the path algebra of a finite directed graph is hereditary, hence it has finite finitistic dimension, when the…

In \cite{SSZ}, the authors proved that an Artin algebra $A$ with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterisation of algebras with finite global…

We construct a nil algebra over a countable field which has finite but non-zero Gelfand-Kirillov dimension.

Given an associative ring $A$, we present a new approach for establishing the finiteness of the big finitistic projective dimension $\operatorname{FPD}(A)$. The idea is to find a sufficiently nice non-positively graded differential graded…

We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal…

In this survey we discuss the results on the finitistic dimension of various stratified algebras. We describe what is already known, present some recent estimates, and list some open problems.

We define a dimension for a triangulated category. We prove a representabilityTheorem for a certain class of functors on finite dimensional triangulatedcategories. We study the dimension of the boundedderived category of an algebra or a…

Given a finite dimensional algebra $\Lambda$, we show that a frequently satisfied finiteness condition for the category ${\cal P}^{\infty}(\Lambda\rm{-mod})$ of all finitely generated (left) $\Lambda$-modules of finite projective dimension,…

For any ring we propose the construction of a cover which increases the finitistic dimension on one side and decreases the finitistic dimension to zero on the opposite side. This complements recent work of Cummings.

It is shown that over an arbitrary countable field, there exists a finitely generated algebra that is nil, infinite dimensional, and has Gelfand-Kirillov dimension at most three.

In this note, we prove that if $\Lambda$ is an Artin algebra with a simple module $S$ of finite projective dimension, then the finiteness of the finitistic dimension of $\Lambda$ implies that of $(1-e)\Lambda(1-e)$ where $e$ is the…

Artin's conjecture is established for all forms that can be realised as a diagonal form on an hyperplane.

This is a brief survey of some recent developments in the study of infinite dimensional Hopf algebras which are either noetherian or have finite Gelfand-Kirillov dimension. A number of open questions are listed.

In [8] V. G\'elinas introduced a homological invariant, called {\it delooping level} (dell), that bounds the finitistic dimension. In this article, we introduce another homological invariant (Dell) related to the delooping level for an…

In a general triangulated category, the finiteness of the finitistic dimension serves as a prerequisite for a categorical obstruction, via the singularity category, to the existence of bounded $t$-structures. In this paper, we investigate…

We investigate the inequality ${\rm Findim}\ \! \Lambda^{op} \leq {\rm dell}\ \! \Lambda$ between the finitistic dimension and the delooping level of an Artin algebra $\Lambda$, and whether equality holds in general. We prove that equality…

Given a geometrically irreducible subscheme X in P^n over F_q of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that the intersection of H and X is geometrically irreducible tends to 1 as d tends to…

We introduce the notion of almost finite dimensionality of algebras and study its connection with the classical finiteness conditions.