Finitistic Dimension Conjecture Research Articles

Filtrations for the Roots of Ext

We survey the set–theoretic methods of module theory that make it possible to equip roots of the contravariant Ext functor with filtrations built from the small roots. The power of these methods is illustrated by several applications: a solution to the Kaplansky problem on Baer modules and some of the related problems for relative Baer modules, the structure of tilting modules and classes, the structure of Matlis localizations of commutative rings, and in particular cases, proofs of the finitistic dimension conjectures, and of the telescope conjecture for module categories.

Tilting modules and Gorenstein rings

In his pioneering work [30], Miyashita extended classical tilting theory to the setting of finitely presented modules of finite projective dimension over an arbitrary ring. More recently, tilting theory has been generalized to arbitrary modules of finite projective dimension [2], [3], [6], et al. Encouraged by this developement, in the present paper, we apply infinite dimensional tilting theory to the setting of modules over Iwanaga-Gorenstein rings. We start with pointing out the central role played by resolving subcategories of modR for an arbitrary ring R. In Theorem 2.2 we prove that these subcategories correspond bijectively to tilting classes of finite type in ModR, as well as to cotilting classes of cofinite type in RMod. This generalizes a classical result of Auslander-Reiten [9] characterizing cotilting classes in modR over an artin algebra R. Moreover, it provides for a bijection between certain cotilting left modules and tilting right modules which extends the duality between finitely generated cotilting and tilting modules over artin algebras. We then consider Iwanaga-Gorenstein rings, that is (not necessarily commutative) two-sided noetherian rings of finite selfinjective dimension on both sides. We show that the first finitistic dimension conjecture holds true for any Iwanaga-Gorenstein ring (Theorem 3.2). We prove that Gorenstein injectivity, and ∗This work was started while the first author was a Ramon y Cajal Fellow at the Universitat Autonoma de Barcelona. The research of the second author was partially supported by the DGESIC (Spain) through the project PB98-0873, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. Third author supported by grant SAB2001-0092 of Secretaria de Estado de Educacion y Universidades MECD at CRM IEC Barcelona, and by MSM 113200007.

On the finitistic dimension conjecture I: related to representation-finite algebras

We use the class of representation-finite algebras to investigate the finitistic dimension conjecture. In this way we obtain a large class of algebras for which the finitistic dimension conjecture holds. The main results in this paper are: (1) Let A be an artin algebra and let I j ,1⩽ j⩽ n be a family of ideals in A with I 1 I 2⋯ I n =0, such that proj.dim( A I j )<∞ and proj.dim( I j ) A =0 for all j⩾3. If A/ I 1 and A/ I 2 are representation-finite and if A/ I j has finite finitistic dimension for j⩾3, then the finitistic dimension of A is finite. In particular, the finitistic dimension conjecture is true for algebras obtained from representation-finite algebras by forming dual extensions, trivially twisted extensions, Hochschild extensions, matrix algebras and tensor products with algebras of radical-square-zero. (2) Let A, B and C be three artin algebras with the same identity such that (i) C⊆ B⊆ A, and (ii) the Jacobson radical of C is a left ideal of B and the Jacobson radical of B is a left ideal of A. If A is representation-finite, then C has finite finitistic dimension. We also provide a way to construct algebras satisfying all conditions in (2), and this leads to a new reformulation of the finitistic dimension conjecture.

Tilting theory and the finitistic dimension conjectures

Let R R be a right noetherian ring and let P &gt; ∞ \mathcal {P}^{&gt;\infty } be the class of all finitely presented modules of finite projective dimension. We prove that findim R = n &gt; ∞ R = n &gt; \infty iff there is an (infinitely generated) tilting module T T such that pd T = n T = n and T ⊥ = ( P &gt; ∞ ) ⊥ T ^\perp = (\mathcal P^{&gt;\infty })^\perp . If R R is an artin algebra, then T T can be taken to be finitely generated iff P &gt; ∞ \mathcal P^{&gt;\infty } is contravariantly finite. We also obtain a sufficient condition for validity of the First Finitistic Dimension Conjecture that extends the well-known result of Huisgen-Zimmermann and Smalø.

A Homological Bridge Between Finite and Infinite-Dimensional Representations of Algebras ?

Given a finite-dimensional algebra Λ, we show that a frequently satisfied finiteness condition for the category $$P^\infty$$ Λ-mod) of all finitely generated (left) Λ-modules of finite projective dimension,namely contravariant finiteness of $$P^\infty$$ (Λ-mod) in Λ-mod, forces arbitrary modules of finite projective dimension to be direct limits of objects in $$P^\infty$$ (Λ-mod). Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimensionconjecture, that is, for the little finitistic dimension of Λ to coincide with the big (this is well known to fail overfinite-dimensional algebras in general).

A geometric approach to the finitistic dimension conjecture

A geometric approach to the finitistic dimension conjecture

Integral Versions of the Nakayama and Finitistic Dimension Conjectures

Integral Versions of the Nakayama and Finitistic Dimension Conjectures

A note on the finitistic dimension conjecture

(1994). A note on the finitistic dimension conjecture. Communications in Algebra: Vol. 22, No. 7, pp. 2525-2528.

Homological domino effects and the first Finitistic Dimension Conjecture

Counterexamples to the first Finitistic Dimension Conjecture are constructed. In fact, for any fieldk and any integerm≧2, there exist finite dimensionalk-algebras Λ such that the little finitistic dimension of Λ ism, while the big finitistic dimension ism+1. Our examples are monomial relation algebras; within this class of algebras the big and little finitistic dimensions cannot differ by more than 1. The analysis of the examples is based on a sharp picture of arbitrary second syzygies over monomial relation algebras.

On the finitistic dimension conjecture for artinian rings

The most frequently quoted finitistic dimension conjecture states that every finite dimensional algebra (or more generally, left artinian ring) has a finitely generated module of largest finite projective dimension. The historical origins of this conjecture are outlined in [7] where the conjecture was verified for monomial algebras. (Another proof was presented in [9].) So far the other most significant contribution regarding the finitistic dimension conjecture is Green and Zimmermann Huisgen's theorem establishing it for left artinian rings with radical cubed zero. In this paper we present a simplification of their proof. The methods used also yield a simple proof of the module theoretic version [3, Corollary 4.21.1] of an old theorem of Fossum, Griffith and Reiten, and verifications of the conjecture for one---sided serial rings and some rings related to, but not covered by, the results of [8]. In each case involving finitistic dimension the basic method used is a reduction of the problem to a ring whose simple modules all have infinite projective dimension. We shall use the terminology R Mod and R mod for the categories of arbitrary and, respectively, finitely generated left R-modules. Similarly, direct summands of arbitrary direct sums and, respectively, finite direct sums of copies of

Graded artin algebras, rational series, and bounds for homological dimensions

is a minimal injective resolution of A, then all the I,, are projective. Then n is selfinjecrive. It is of interest to note that the Nakayama conjecture follows in a trivial way from the finitistic dimension conjecture. This one says the following: Let n be an artin algebra. Then sup{proj dim M( proj dimM<oc$ is finite where M ranges over the finitely generated Amodules. The finitistic projective dimension has its origins in the early 1960s and was formulated from results of Rosenberg and Zelinsky. In this paper we study the behavior or certain rational power series associated to every graded module over a graded finite dimensional algebra, series which are slightly different from the usual Poincare-Betti series. It is interesting to note that rationality is enough to enable us to prove that over a graded finite dimensional algebra, the linitistic projective dimension is finite over the class of finitely generated graded modules whose nonprojective syzygies have no projective summands. This implies the Nakayama conjecture for graded algebras. (See also Wilson [3].) Also using rationality we find bounds for global dimension in terms of graded length and the number of nonisomorphic simple modules. Finally let A be an algebra of finite representation type having n nonisomorphic indecomposable modules. We prove that every composition of more than 2n 2 nonisomorphisms of Amodules is zero. This is done by using the result of Auslander-Reiten which states that every nonzero homomorphism which is not an isomorphism is a sum of compositions of irreducible morphisms.