Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras (original) (raw)

Relative Igusa-Todorov Functions and Relative Homological Dimensions

Algebras and Representation Theory, 2016

We develope the theory of the E-relative Igusa-Todorov functions in an exact IT-context (C, E) (see Definition 2.1). In the case when C = mod (Λ) is the category of finitely generated left Λ-modules, for an artin algebra Λ, and E is the class of all exact sequences in C, we recover the usual Igusa-Todorov functions [33]. We use the setting of the exact structures and the Auslander-Solberg relative homological theory to generalise the original Igusa-Todorov's results. Furthermore, we introduce the E-relative Igusa-Todorov dimension and also we obtain relationships with the relative global and relative finitistic dimensions and the Gorenstein homological dimensions. Contents 26 6. Relative n-Igusa-Todorov categories 31 7. Examples 32 References 36

Igusa-Todorov functions for Artin algebras

Journal of Pure and Applied Algebra, 2017

In this paper we study the behaviour of the Igusa-Todorov functions for Artin algebras A with finite injective dimension, and Gorenstein algebras as a particular case. We show that the φ-dimension and ψ-dimension are finite in both cases. Also we prove that monomial, gentle and cluster tilted algebras have finite φ-dimension and finite ψ-dimension.

Igusa–Todorov functions for radical square zero algebras

Journal of Algebra

In this paper we study the behaviour of the Igusa-Todorov functions for radical square zero algebras. We show that the left and the right φ-dimensions coincide, in this case. Some general results are given, but we concentrate more in the radical square zero algebras. Our study is based on two notions of hearth and member of a quiver Q. We give some bounds for the φ and the ψ-dimensions and we describe the algebras for which the bound of ψ is obtained. We also exhibit modules for which the φ-dimension is realised.

The Phi-dimension: A new homological measure

arXiv: Representation Theory, 2013

K. Igusa and G. Todorov introduced two functions phi\phiphi and psi,\psi,psi, which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become into a powerful tool to understand better the finitistic dimension conjecture. In this paper, for an artin RRR-algebra AAA and the Igusa-Todorov function phi,\phi,phi, we characterise the phi\phiphi-dimension of AAA in terms either of the bi-functors mathrmExtiA(−,−)\mathrm{Ext}^{i}_{A}(-, -)mathrmExtiA(−,−) or Tor's bi-functors mathrmTorAi(−,−).\mathrm{Tor}^{A}_{i}(-,-).mathrmTorAi(−,−). Furthermore, by using the first characterisation of the phi\phiphi-dimension, we show that the finiteness of the phi\phiphi-dimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz's result as follows: For an artin algebra A,A,A, a tilting AAA-module TTT and the endomorphism algebra B=mathrmEndA(T)op,B=\mathrm{End}_A(T)^{op},B=mathrmEndA(T)op, we have that $\mathrm{Fidim}\,(A)-\mathrm{pd}\,T\leq \mathrm{Fidim...

The Φ-Dimension: A New Homological Measure

2016

In [15], K. Igusa and G. Todorov introduced two functions φ and ψ, which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become into a powerful tool to understand better the finitistic dimension conjecture. In this paper, for an artin R-algebra A and the Igusa-Todorov function φ, we characterise the φ-dimension of A in terms either of the bi-functors Ext i A (−, −) or Tor's bi-functors Tor A i (−, −). Furthermore, by using the first characterisation of the φ-dimension, we show that the finiteness of the φdimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz's result [3, Corollary 1] as follows: For an artin algebra A, a tilting A-module T and the endomorphism algebra B = End A (T) op , we have that φ dim (A) − pd T ≤ φ dim (B) ≤ φ dim (A) + pd T.

Igusa-Todorov for radical square zero algebras

2015

Igusa-Todorov ϕ Function for Truncated Path Algebras

Algebras and Representation Theory, 2019

Given a truncated path algebra A = kQ J k we prove that φdimA = φdimA op. We also compute the φ-dimension of A in function of the φdimension of kQ J 2 when Q has no sources nor sinks. This allows us to bound the φ-dimension for truncated path algebras. Finally, we characterize A when its φ-dimension is equal to 1.

Idempotent Ideals and the Igusa-Todorov Functions

Algebras and Representation Theory, 2016

Let Λ be an artin algebra and A a two-sided idempotent ideal of Λ, that is, A is the trace of a projective Λ-module P in Λ. We consider the categories of finitely generated modules over the associated rings Λ/A, Λ and Γ = End Λ (P) op and study the relationship between their homological properties via the Igusa-Todorov functions.

Self-injective Right Artinian Rings and Igusa Todorov Functions

Algebras and Representation Theory, 2013

We show that a right artinian ring R is right self-injective if and only if ψ(M) = 0 (or equivalently φ(M) = 0) for all finitely generated right R-modules M , where ψ, φ : mod R → N are functions defined by Igusa and Todorov. In particular, an artin algebra Λ is self-injective if and only if φ(M) = 0 for all finitely generated right Λ-modules M .

Algebra of dimension theory

2004

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes dim(A)\dim(A)dim(A) of graded groups AAA. There are two geometric interpretations of those equivalence classes: \linebreak 1. For pointed CW complexes KKK and LLL, dim(Hast(K))=dim(Hast(L))\dim(H_\ast(K))=\dim(H_\ast(L))dim(Hast(K))=dim(Hast(L)) if and only if the infinite symmetric products SP(K)SP(K)SP(K) and SP(L)SP(L)SP(L) are of the same extension type (i.e., SP(K)inAE(X)SP(K)\in AE(X)SP(K)inAE(X) iff SP(L)inAE(X)SP(L)\in AE(X)SP(L)inAE(X) for all compact XXX). \linebreak 2. For pointed compact spaces XXX and YYY, dim(calH−ast(X))=dim(calH−ast(Y))\dim(\cal H^{-\ast}(X))=\dim(\cal H^{-\ast}(Y))dim(calH−ast(X))=dim(calH−ast(Y)) if and only if XXX and YYY are of the same dimension type (i.e., dimG(X)=dimG(Y)\dim_G(X)=\dim_G(Y)dimG(X)=dimG(Y) for all Abelian groups GGG). Dranishnikov's version of Hurewicz Theorem in extension theory becomes dim(piast(K))=dim(Hast(K))\dim(\pi_\ast(K))=\dim(H_\ast(K))dim(piast(K))=dim(Hast(K)) for all simply connected KKK. The concept of cohomological dimension $\...

Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras (original) (raw)

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