Krull-Gabriel Dimension Research Articles

Krull–Gabriel dimension of Cohen–Macaulay modules over hypersurfaces of countable Cohen-Macaulay representation type

We calculate the Krull–Gabriel dimension of the functor category of the (stable) category of maximal Cohen–Macaulay modules over complete Cohen-Macaulay local rings of countable Cohen-Macaulay representation type. We show that the Krull–Gabriel dimension is 0 if and only if the ring is of finite Cohen–Macaulay representation type and that is 2 if the ring is a hypersurface of countable but not finite Cohen–Macaulay representation type.

On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras

Assume that K is an algebraically closed field and denote by KG(R) the Krull-Gabriel dimension of R, where R is a locally bounded K-category (or a bound quiver K-algebra). Assume that C is a tilted K-algebra and Cˆ,Cˇ,C˜ are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that KG(C˜)=KG(Cˇ)≤KG(Cˆ). Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that KG(C˜)=KG(Cˇ)=KG(Cˆ)∈{0,2,∞}. Finally, in the Appendix Grzegorz Bobiński presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle.

On Krull-Gabriel dimension and Galois coverings

Assume that K is an algebraically closed field, R a locally support-finite locally bounded K-category, G a torsion-free admissible group of K-linear automorphisms of R and A=R/G. We show that the Krull-Gabriel dimension KG(R) of R equals the Krull-Gabriel dimension KG(A) of A. Furthermore, we determine the Krull-Gabriel dimension of locally support-finite repetitive K-categories. These results are applied to determine the Krull-Gabriel dimension of standard selfinjective algebras of polynomial growth. Finally, we show that there are no super-decomposable pure-injective modules over standard selfinjective algebras of domestic type.

The Ziegler spectrum for derived-discrete algebras

Let Λ be a derived-discrete algebra. We show that the Krull–Gabriel dimension of the homotopy category of projective Λ-modules, and therefore the Cantor–Bendixson rank of its Ziegler spectrum, is 2, thus extending a result of Bobiński and Krause [8]. We also describe all the indecomposable pure-injective complexes and hence the Ziegler spectrum for derived-discrete algebras, extending a result of Z. Han [17]. Using this, we are able to prove that all indecomposable complexes in the homotopy category of projective Λ-modules are pure-injective, so obtaining a class of algebras for which every indecomposable complex is pure-injective but which are not derived pure-semisimple.

Representation embeddings, interpretation functors and controlled wild algebras

We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First we show that representation embeddings between categories of modules of finite-dimensional algebras induce embeddings of lattices of pp formulas and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width $\infty$ and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: that a representation embedding from ${\rm Mod}\mbox{-}S$ to ${\rm Mod}\mbox{-}R$ admits an inverse interpretation functor from its image and hence that, in this case, ${\rm Mod}\mbox{-}R$ interprets ${\rm Mod}\mbox{-}S$. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings. Finally we prove that if $R,S$ are finite dimensional algebras over an algebraically closed field and $I:{\rm Mod}\mbox{-}R\rightarrow{\rm Mod}\mbox{-}S$ is an interpretation functor such that the smallest definable subcategory containing the image of $I$ is the whole of ${\rm Mod}\mbox{-}S$ then, if $R$ is tame, so is $S$ and similarly, if $R$ is domestic, then $S$ also is domestic.

A Note on Generic Objects and Locally Finite Triangulated Categories

For a finite dimensional algebra A, we prove that the homotopy category of injective A-modules is generically trivial if and only if the derived category of all A-modules is generically trivial. Moreover we show some connections between the generic objects, locally finiteness and Krull-Gabriel dimension.

The Krull-Gabriel Dimension of Cycle-Finite Artin Algebras

We determine the Krull-Gabriel dimension of the cycle-finite categories of finitely generated modules over artin algebras and derive some consequences.

The Krull–Gabriel dimension of discrete derived categories

We compute the Krull–Gabriel dimension of the category of perfect complexes for finite dimensional algebras which are derived discrete.

Krull–Gabriel dimension and Cantor–Bendixson rank of 1-domestic string algebras

Krull–Gabriel dimension and Cantor–Bendixson rank of 1-domestic string algebras

SUPERDECOMPOSABLE PURE INJECTIVE MODULES OVER COMMUTATIVE NOETHERIAN RINGS

We investigate width and Krull–Gabriel dimension over commutative Noetherian rings which are "tame" according to the Klingler–Levy analysis in [4–6], in particular over Dedekind-like rings and their homomorphic images. We show that both are undefined in most cases.

Krull-Gabriel dimension and the model-theoretic complexity of the category of modules over group rings of finite groups

We classify group rings of finite groups over a field F according to the model-theoretic complexity of the category of their modules. For instance, we prove that, if F contains a primitive cubic root of 1, then the Krull–Gabriel dimension of such rings is 0, 2, or undefined.

Krull–Gabriel Dimension of 1-domestic String Algebras

We classify the indecomposable pure injective modules over a wide class of $1$ -domestic string algebras and calculate the Krull–Gabriel dimension of these algebras.

IDEALS IN MOD-$R$ AND THE $\omega$-RADICAL

Let be an artin algebra, and let mod- denote the category of finitely presented right -modules. The radical of this category and its finite powers play a major role in the representation theory of . The intersection of these finite powers is denoted , and the nilpotence of this ideal has been investigated, in , for instance. In , arbitrary transfinite powers, , of rad were defined and linked to the extent to which morphisms in may be factorised. In particular, it has been shown that if is an artin algebra, then the transfinite radical, , the intersection of all ordinal powers of rad, is non-zero if and only if there is a 'factorisable system' of morphisms in rad and, in that case, the Krull-Gabriel dimension of equals (that is, is undefined). More precise results on the index of nilpotence of rad for artin algebras were proved in.

The Krull–Gabriel Dimension of a Serial Ring

We prove that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair. Using this we calculate the Krull–Gabriel dimension of the module category over serial rings. For instance, we show that this dimension cannot be equal to 1.

The Ziegler and Zariski Spectra of Some Domestic String Algebras

It was a conjecture of the second author that the Cantor–Bendixson rank of the Ziegler spectrum of a finite-dimensional algebra is either less than or equal to 2 or is undefined. Here we refute this conjecture by describing the Ziegler spectra of some domestic string algebras where arbitrary finite values greater than 2 are obtained. We give a complete description of the Ziegler and Gabriel–Zariski spectra of the simplest of these algebras. The conjecture has been independently refuted by Schroer who, extending his work (1997) on these algebras, computed their Krull–Gabriel dimension.

The spectrum of a module category

Introduction The functor category Definable subcategories Left approximations duality Ideals in the category of finitely presented modules Endofinite modules Krull-Gabriel dimension The infinite radical Functors between module categories Tame algebras Rings of definable scalars Reflective definable subcategories Sheaves Tame hereditary algebras Coherent rings Appendix A. Locally coherent Grothendieck categories Appendix B. Dimensions Appendix C. Finitely presented functors and ideals Bibliography.

On the Krull-Gabriel dimension of an algebra

On the Krull-Gabriel dimension of an algebra

The Endomorphism Ring of a Localized Coherent Functor

LetCbe a commutative artinian ring and Λ an artinC-algebra. The category of coherent additive functorsA: mod-Λ→Ab on the finitely presented right Λ-modules will be denoted by Ab(Λ). This category is equivalent to the free abelian category over the ring Λ. If S0⊆Ab(Λ) is the Serre subcategory of the finite length objects of Ab(Λ) andA∈Ab(Λ), it is proved that the endomorphism ring EndAb(Λ)/S0AS0of the localized objectAS0is a locally artinC-algebra. This is used to show that the Krull-Gabriel dimension of the category Ab(Λ) cannot equal 1. In particular, this holds for finite rings.

The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences

For an Artin Algebra Λ of finite representation type, the category Λ-mod, considered as a ring with several objects, has Krull dimension zero. Contrary, for a wild hereditary Artin Algebra this dimension does not exist. In this paper we show that the Krull dimension of Λ-mod for an Artin Algebra Λ of tame representation type is two. The corresponding Krull-Gabriel filtration by Serre subcategories of the category F of finitely presented contravariant functors on Λ-mod leads to a hierarchy of exact sequences in Λ-mod. Influenced by the functorial approach to almost split sequences by M. Auslander and I. Reiten, we investigate the exact sequences whose corresponding functors become simple in one of the successive quotient categories of the Krull-Gabriel filtration of F.

Krull Dimension and Gabriel Dimension of Idealizers of Semimaximal Left Ideals

Krull Dimension and Gabriel Dimension of Idealizers of Semimaximal Left Ideals