We prove that the finitistic dimension conjecture, the Gorenstein Symmetry Conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for artin algebras which can be realized as endomorphism algebras of modules over representation-finite algebras. Note it is a question whether or not all artin algebras have such realizations. It was also shown that if every quasi-hereditary algebras has a left idealized extension which is a monomial algebra or an algebra whose representation dimension is not more than 3, then the finitistic dimension conjecture holds.
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.
We investigate the inequality ${\rm Findim}\ \! Λ^{op} \leq {\rm dell}\ \! Λ$ between the finitistic dimension and the delooping level of an Artin algebra $Λ$, and whether equality holds in general. We prove that equality ${\rm Findim}\ \! Λ^{op} = {\rm dell}\ \! Λ$ always holds for Artin algebras with radical square zero.
For monomial special multiserial algebras, which in general are of wild representation type, we construct radical embeddings into algebras of finite representation type. As a consequence, we show that the representation dimension of monomial and self-injective special multiserial algebras is less or equal to three. This implies that the finitistic dimension conjecture holds for all special multiserial algebras.
We consider the question of whether the injective modules generate the unbounded derived category of a ring as a triangulated category with arbitrary coproducts. We give an example of a non-Noetherian commutative ring where they don't, but prove that they do for any Noetherian commutative ring. For non-commutative finite dimensional algebras the question is open, and we prove that if injectives generate for such an algebra, then the finitistic dimension conjecture holds for that algebra.
It is a well-known result of Auslander and Reiten that contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, in this work we examine the Gorenstein counterpart of Auslander--Reiten condition, namely contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ (of finitely gen…
The notion of Igusa-Todorov classes is introduced in connection with the finitistic dimension conjecture. As application we consider conditions on special ideals which imply the Igusa-Todorov and other finiteness conditions on modules proving the finitistic dimension conjecture and related conjectures in those cases.
The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field should have finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this paper, we investigate those extensions of Artin algebras in which some radical-power of smaller algebras is a one-sided ideal in bigger algebras. Our results, however, are formulated more generally for an arbitrary ideal: Let $B\subseteq A$ be an extension of Artin algebras and $I$ an ideal of $B$ such that the full subcategory of $B/I$-modules is $B$-syzyg…
Let $Λ$ be a finite dimensional algebra and $G$ be a finite group whose elements act on $Λ$ as algebra automorphisms. Under the assumption that $Λ$ has a complete set $E$ of primitive orthogonal idempotents, closed under the action of a Sylow $p$-subgroup $S \leqslant G$. If the action of $S$ on $E$ is free, we show that the skew group algebra $ΛG$ and $Λ$ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra $Λ^S$ is a direct summand of the $Λ^S$-bimodule $Λ$. Using a homological characterization of piecewise hereditary algebras proved by Happel an…
Let $H$ be a finite-dimensional weak Hopf algebra over a field $k$ and $A/B$ be a right faithfully flat weak $H$-Galois extension. We prove that if the finitistic dimension of $B$ is finite, then it is less than or equal to that of $A$. Moreover, suppose that $H$ is semisimple. If the finitistic dimension conjecture holds, then the finitistic dimension of $B$ is equal to that of $A$.
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.
We show that an artin algebra having at most three radical layers of infinite projective dimension has finite finitistic dimension, generalizing the known result for algebras with vanishing radical cube.
Let $A$ be an Artin algebra. We investigate subalgebras of $A$ with certain conditions and obtain some classes of algebras whose finitistic dimensions are finite.
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 ring $B$ such that $\mathrm{H}^0(B) = A$ and such that $\operatorname{FPD}(B) < \infty$. We show that one can always find such a $B$ provided that $A$ is noetherian and has a noncommutative dualizing complex. We then use the intimate relation between $\operatorname{\mathsf{D}}(B)$ and $\operatorname{\mathsf{D}}(\mathrm{H}^0(B))$ to deduce results about $\oper…
In this note, we prove that if $Λ$ is an Artin algebra with a simple module $S$ of finite projective dimension, then the finiteness of the finitistic dimension of $Λ$ implies that of $(1-e)Λ(1-e)$ where $e$ is the primitive idempotent supporting $S$. We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if $Λ$ is the quotient of a path algebra by an admissible ideal $I$ whose defining relations do not involve a certain arrow $α$, then the finitistic dimension of $Λ$ is finite if and only if the finitistic dimension of $Λ/ΛαΛ$ is finite.
We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call the "delooping level". These give lower and upper bounds for the finitistic dimension, respectively. As first theorems, we give a necessary and sufficient criterion for the delooping level to be finite in terms of the splitting of the unit map of a related adjunction, and use this to give a sufficient torsionfreeness criterion for finiteness. We further relate these invariants to the Auslander-Bridger grade conditions for modules, which are vanishing conditions on double Ext duals. As ma…
In 2005, the second author and Todorov introduced an upper bound on the finitistic dimension of an Artin algebra, now known as the ϕ-dimension. The ϕ-dimension conjecture states that this upper bound is always finite, a fact that would imply the finitistic dimension conjecture. In this paper, we present a counterexample to the ϕ-dimension conjecture and explain where it comes from. We also discuss implications for further research and the finitistic dimension conjecture.
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 a construction of an upper bound for the finitistic dimension which is due to Lueck.
Let $Λ$ be an artinian ring. Generalizing the Loewy length, we propose the layer length associated with a torsion theory, which is a new measure for finitely generated $Λ$-modules.
We introduce the notion of a directed stratification for a finite-dimensional algebra. For algebras that admit such a stratification we characterise the projective resolutions of finitely generated modules and obtain a result for the finitistic dimension, which is an inductive version of a result of Fossum, Griffith and Reiten. With the developed techniques, which are adopted from the theory of EI-category algebras, we gain deeper insight in the combinatorial nature of this result. A characterisation of algebras which do not admit a directed stratification is given in terms of the Ext-quiver.
If A is an artin algebra, Gélinas has introduced an interesting upper bound for the finitistic dimension of A, namely the delooping level del A. We assert that for any Nakayama algebra, its finitistic dimension is equal to del A. This yields also a new proof that the finitistic dimension of A and of its opposite algebra are equal, as shown recently by Sen. If S is a simple module, let e(S) be the minimum of the projective dimension of S and of its injective envelope (one of these numbers has to be finite); and e*(S) the minimum of the injective dimension of S and of its projective cover. Then …
Let $\cQ$ be a quiver and $K$ a field. We study the interrelationship of homological properties of algebras associated to convex subquivers of $\cQ$ and quotients of the path algebra $K\cQ$. We introduce the homological heart of $\cQ$ which is a particularly nice convex subquiver of $\cQ$. For any algebra of the form $K\cQ/I$, the algebra associated to $K\cQ/I$ and the homological heart have similar homological properties. We give an application showing that the finitistic dimension conjecture need only be proved for algebras with path connected quivers.
Let $Λ$ be an artin algebra and let $\mathcal{P}^{<\infty}_Λ$ the category of finitely generated right $Λ$-modules of finite projective dimension. We show that $\mathcal{P}^{<\infty}_Λ$ is contravariantly finite in $\rm mod\,Λ$ if and only if the direct sum $E$ of the indecomposable Ext-injective modules in $\mathcal{P}^{<\infty}_Λ$ form a tilting module in $\rm mod\,Λ$. Moreover, we show that in this case $E$ coincides with the direct sum of the minimal right $\mathcal{P}^{<\infty}_Λ$-approximations of the indecomposable $Λ$-injective modules and that the projective dimension of $E$ equal to …
Recently, Chen and Koenig in \cite{CheKoe} and Iyama and Solberg in \cite{IyaSol} independently introduced and characterised algebras with dominant dimension coinciding with the Gorenstein dimension and both dimensions being larger than or equal to two. In \cite{IyaSol}, such algebras are named Auslander-Gorenstein algebras. Auslander-Gorenstein algebras generalise the well known class of higher Auslander algebras, where the dominant dimension additionally coincides with the global dimension. In this article we generalise Auslander-Gorenstein algebras further to algebras having the property th…
Let $R$ be a finite dimensional $k$-algebra over an algebraically closed field $k$ and $\mathrm{mod} R$ be the category of all finitely generated left $R$-modules. For a given full subcategory $\mathcal{X}$ of $\mathrm{mod} R,$ we denote by $\pfd \mathcal{X}$ the projective finitistic dimension of $\mathcal{X}.$ That is, $\pfd \mathcal{X}:=\mathrm{sup} \{\pd X : X\in\mathcal{X} \text{and} \pd X<\infty\}.$ \ It was conjectured by H. Bass in the 60's that the projective finitistic dimension $\pfd (R):=\pfd (\mathrm{mod} R)$ has to be finite. Since then, much work has been done toward the proof o…
Given a finite dimensional algebra $Λ$, we show that a frequently satisfied finiteness condition for the category ${\cal P}^{\infty}(Λ\rm{-mod})$ of all finitely generated (left) $Λ$-modules of finite projective dimension, namely contravariant finiteness of ${\cal P}^{\infty}(Λ\rm{-mod})$ in $Λ\rm{-mod}$, forces arbitrary modules of finite projective dimension to be direct limits of objects in ${\cal P}^{\infty}(Λ\rm{-mod})$. Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimension conjecture, that is, for the little finit…
Let $A$ be an Artin algebra, $M$ be a Gorenstein projective $A$-module and $B =$ End$_A M$, then $M$ is a $A$-$B$-bimodule. We use the restricted flat dimension of $M_B$ to give a characterization of the homological dimensions of $A$ and $B$, and obtain the following main results: (1) if $A$ is a CM-finite algebra with $\cal GP$($A$) = add$_AE$ and fin.dim $A \geq 2,$ then ${\rm fin.dim}\ B \leq {\rm fin.dim}\ A + {\rm rfd}(M_B) +{\rm pd}_B{\rm Hom}_A(M, E);$ (2) If $A$ is a CM-finite $n$-Gorenstein algebra with $\cal GP$($A$) = add$_AE$ and $n \geq 2$, then gl.dim $B \leq n + {\rm pd}_B{\rm H…
Self-Orthogonal $τ$-Tilting Modules and Tilting Modules III: Finitistic Dimension
School of Mathematics and Statistics, Jiangsu Normal University（江苏师范大学数学与统计学院）；School of Mathematics and Statistics, Changsha University of Science and Technology（长沙理工大学数学与统计学院）
Birge Huisgen-Zimmermann calls a finite dimensional algebra homologically tame provided the little and the big finitistic dimension are equal and finite. The question formulated in the title has been discussed by her in the paper "Representation-tame algebras need not be homologically tame", by looking for any $r > 0$ at a sequence of algebras $Λ_m$ with big finitistic dimension $r+m$. As we will show, also the little finitistic dimension of $Λ_m$ is r+m. It follows that contrary to her assertion, all her algebras $Λ_m$ are homologically tame.
Let A be an Artin algebra and e an idempotent in A. It is an interesting topic to compare the homological dimension of the algebras A,A/AeA and eAe. For example, in [2], the relation among the global dimension of these algebras is discussed under the condition that AeA is a strongly idempotent ideal. Motivated by this, we try to compare the finitistic dimension of these algebras under certain homological conditions on AeA. In particular, under the condition that AeA is a strongly idempotent ideal with finite projective dimension, we prove that if the finitistic projective (or injective) dimens…
In this paper, we construct derived equivalences between matrix subrings. As applications, we calculate the global dimensions and the finitistic dimensions of some matrix subrings. And we show that the finitistic dimension conjecture holds for a class of Harada algebras and a class of tiled triangular algebras.
We give another proof of the recent result of Ringel, which asserts equality between the finitistic dimension and delooping level of Nakayama algebras. The main tool is syzygy filtration method introduced in \cite{sen2019}. In particular, we give characterization of the finitistic dimension one Nakayama algebras.
For any cyclic Nakayama algebra $Λ$, we construct \emph{syzygy filtered algebra} $\bm\varepsilon(Λ)$ which corresponds to various syzygy modules as the name suggests. We prove that the category of modules over the syzygy filtered algebra $\bm\varepsilon(Λ)$ is equivalent to the wide subcategory cogenerated by projective-injective modules of the original algebra $Λ$ along with other categorical equivalences. In terms of this new algebra, we interpret the following homological invariants of $Λ$: left and right finitistic dimension, left and right $φ$-dimension, Gorenstein dimension, dominant dim…
We show that, also within the class of representation-tame finite dimensional algebras $Λ$, the big left finitistic dimension of $Λ$ may be strictly larger than the little. In fact, the discrepancies $Findim Λ- findim Λ$ need not even be bounded for special biserial algebras which constitute one of the (otherwise) most thoroughly understood classes of tame algebras. More precisely: For every positive integer $r$, we construct a special biserial algebra $Λ$ with the property that $findim Λ= r + 1$, while $Findim Λ= 2r + 1$. In particular, there are infinite dimensional representations of $Λ$ wh…
Repetitiveness in projective and injective resolutions and its influence on homological dimensions are studied. Some variations on the theme of repetitiveness are introduced, and it is shown that the corresponding invariants lead to very good -- and quite accessible -- upper bounds on various finitistic dimensions in terms of individual modules. These invariants are the `repetition index' and the `syzygy type' of a module $M$ over an artinian ring $Λ$. The repetition index measures the degree of repetitiveness among non-projective direct summands of the syzygies of $M$, while the syzygy type o…
In this paper we consider several homological dimensions of crossed products $A _α ^σ G$, where $A$ is a left Noetherian ring and $G$ is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of $A ^σ _α G$ are classified: global dimension of $A ^σ _α G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^σ _α G$ coincides with that of $A$. A criterion for skew group rings to have finite global dimensions is deduced. Under the hy…
