Finitistic Dimension Conjecture Research Articles

Generalised Lat-Igusa-Todorov Algebras and Morita Contexts

AbstractIn this paper we define (special) GLIT classes and (special) GLIT algebras. We prove that GLIT algebras, which generalise Lat-Igusa-Todorov algebras, satisfy the finitistic dimension conjecture and give several properties and examples. In addition we show that special GLIT algebras are exactly those that have finite finitistic dimension. Lastly we study Morita algebras arising from a Morita context and give conditions for them to be (special) GLIT in terms of the algebras and bimodules used in their definition. As a consequence we obtain simple conditions for a triangular matrix algebra to be (special) GLIT and also prove that the tensor product of a GLIT $$\mathbb {K}$$ K -algebra with a path algebra of a finite quiver without oriented cycles is GLIT.

Injective generation of the derived category and finitistic dimension conjecture

Injective generation of the derived category and finitistic dimension conjecture

Left–right symmetry of finite finitistic dimension

AbstractWe show that the finitistic dimension conjecture for finite‐dimensional algebras is equivalent to the left–right symmetry of finite finitistic dimension for finite‐dimensional algebras. We also prove the equivalent statement for injective generation.

On Lat-Igusa-Todorov algebras

Lat-Igusa-Todorov algebras are a natural generalization of Igusa-Todorov algebras. They are defined using the generalized Igusa-Todorov functions given in Bravo et al. (J Algebra, 580:63–83, 2021) and also verify the finitistic dimension conjecture. In this article we give new ways to construct examples of Lat-Igusa-Todorov algebras. On the other hand we show an example of a family of algebras that are not Lat-Igusa-Todorov.

Cut cotorsion pairs

AbstractWe present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

Finitistic dimension conjectures via Gorenstein projective dimension

It is a well-known result of Auslander and Reiten that contravariant finiteness of the class P∞fin (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, we examine in this work the Gorenstein counterpart of Auslander–Reiten condition, namely contravariant finiteness of the class GP∞fin (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class GP∞fin implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander–Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in the sense that contravariant finiteness of the class GP∞fin implies contravariant finiteness of the class P∞fin over any Artin algebra, and the converse holds for Artin algebras over which the class GP0fin (of finitely generated Gorenstein projective modules) is contravariantly finite.

The depth, the delooping level and the finitistic dimension

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 main theorem, we prove that these bounds agree whenever the first non-trivial grade conditions are satisfied for simple modules, so that either invariant computes the finitistic dimension in this case. Over Artinian rings, we show that the delooping level also bounds the big finitistic dimension, and we obtain a sufficient cohomological criterion for the first finitistic dimension conjecture to hold. Over commutative local Noetherian rings, these conditions always hold and we obtain a new characterisation of the depth as the delooping level of the ring.

Some new criteria for small Igusa-Todorov algebras

n-Igusa-Todorov algebras are closely related to the finitistic dimension conjecture. Each n-Igusa-Todorov algebra has finite finitistic dimension. In this article, we find some new criteria for algebras being n-Igusa-Todorov algebras (here n = 0, 1). As applications, we extend the proof of the finitistic dimension conjecture to several new classes of algebras.

Infinitely Generated Gorenstein Tilting Modules

The theory of finitely generated relative (co)tilting modules has been established in the 1980s by Auslander and Solberg, and infinitely generated relative tilting modules have recently been studied by many authors in the context of Gorenstein homological algebra. In this work, we build on the theory of infinitely generated Gorenstein tilting modules by developing “Gorenstein tilting approximations” and employing these approximations to study Gorenstein tilting classes and their associated relative cotorsion pairs. As applications of our results, we discuss the problem of existence of complements to partial Gorenstein tilting modules as well as some connections between Gorenstein tilting modules and finitistic dimension conjectures.

Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras

In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of the developed theory are given in order to relate the different homological dimensions which have been discussed through the paper.

THE FINITISTIC DIMENSION AND CHAIN CONDITIONS ON IDEALS

AbstractLet Λ be an artin algebra and $0=I_{0}\subseteq I_{1} \subseteq I_{2}\subseteq\cdots \subseteq I_{n}$ a chain of ideals of Λ such that $(I_{i+1}/I_{i})\rad(\Lambda/I_{i})=0$ for any $0\leq i\leq n-1$ and $\Lambda/I_{n}$ is semisimple. If either none or the direct sum of exactly two consecutive ideals has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. As a consequence, we have that if either none or the direct sum of exactly two consecutive terms in the radical series of Λ has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. Some known results are obtained as corollaries.

Fabric idempotents and higher Auslander-Reiten theory

Over a finite-dimensional algebra A, simple A-modules that have projective dimension one have special properties. For example, Geigle-Lenzing studied them in connection to homological epimorphisms of rings and they have also appeared in work concerning the finitistic dimension conjecture. If we however work in a d-cluster-tilting subcategory, then not all simples are contained in the subcategory. In this context, a replacement might be to work with idempotents instead, and utilise the theory of Auslander-Platzeck-Todorov. We introduce the notion of a fabric idempotent as an analogue of the localisable objects studied by Chen-Krause, and show that they provide rich combinatorial properties to illustrate the theory. In particular, we use fabric idempotents to extend the classification of singularity categories of Nakayama algebras by Chen-Ye to higher Nakayama algebras.

Unbounded derived categories and the finitistic dimension conjecture

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.

Gaps for the Igusa-Todorov function

For a finite dimensional algebra A, with 0<ϕdim(A)=m<∞, we show that there always exist modules M and N, such that ϕ(M)=1, and ϕ(N)=m−1. On the other hand, we give an example of an algebra such that not every value between 1 and its ϕ-dimension is reached by the ϕ function. We call such values gaps, and show that the algebras with gaps verifies the finitistic dimension conjecture.

Finitistic dimension conjecture and extensions of algebras

An extension of algebras is a homomorphism of algebras preserving identities. We use extensions of algebras to study the finitistic dimension conjecture over Artin algebras. Let be an extension of Artin algebras. We denote by the relative finitistic dimension of f, which is defined to be the supremum of relative projective dimensions of finitely generated left A-modules of finite projective dimension. We prove that: (1) If B is representation-finite and , then A has finite finitistic dimension. (2) Suppose that B is representation-finite and . If, for any A-module X with finite projective dimension, has finite projective dimension, then A has finite finitistic dimension. (3) Suppose that the finitistic dimension of B is finite and . If has finite projective dimension and AB is projective, then A has finite finitistic dimension. Also, we prove the following result: Let I, J, K be three ideals of an Artin algebra A such that IJK = 0 and . If both A/I and A/J are A-syzygy-finite, then the finitistic dimension of A is finite.

On the representation dimension and finitistic dimension of special multiserial algebras

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.

Excellent Extensions and Homological Conjectures

In this paper, we introduce the notion of excellent extensions of rings. Let Γ be an excellent extension of an Artin algebra Λ, we prove that Λ satisfies the Gorenstein symmetry conjecture (resp., finitistic dimension conjecture, Auslander–Gorenstein conjecture, Nakayama conjecture) if and only if so does Γ. As a special case of excellent extensions, when G is a finite group whose order is invertible in Λ acting on Λ and Λ is G-stable, we prove that if the skew group algebra ΛG satisfies strong Nakayama conjecture (resp., generalized Nakayama conjecture), then so does Λ.

Dominant dimension and tilting modules

We study which algebras have tilting modules that are both generated and cogenerated by projective–injective modules. Crawley–Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension 2, Auslander algebras are classified by the existence of such tilting modules. In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least 2, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which have a tilting–cotilting module generated–cogenerated by projective–injective modules are precisely 1-minimal Auslander–Gorenstein algebras. When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.

Wakamatsu-silting complexes

ABSTRACTWe introduce Wakamatsu-silting complexes (resp., Wakamatsu-tilting complexes) as a common generalization of both silting complexes (resp., tilting complexes) and Wakamatsu-tilting modules. Characterizations of Wakamatsu-silting complexes are given. In particular, we show that a complex T is Wakamatsu-silting if and only if its dual DT is Wakamatsu-silting. It is conjectured that compact Wakamatsu-silting complexes are just silting complexes. We prove that the conjecture lies under the finitistic dimension conjecture.

Recollements of derived categories III: finitistic dimensions

In this paper, we study homological dimensions of algebras linked by recollements of derived module categories, and establish a series of new upper bounds and relationships among their finitistic or global dimensions. This is closely related to a longstanding conjecture, the finitistic dimension conjecture, in representation theory and homological algebra. Further, we apply our results to a series of situations of particular interest: exact contexts, ring extensions, trivial extensions, pullbacks of rings, and algebras induced from Auslander-Reiten sequences. In particular, we not only extend and amplify Happel’s reduction techniques for finitistic dimenson conjecture to more general contexts, but also generalise some recent results in the literature.