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Delooping levels

Marcos Barrios, Marcelo Lanzilotta, Gustavo Mata

Abstract

In [8] V. Gélinas introduced a homological invariant, called {\it delooping level} (dell), that bounds the finitistic dimension. In this article, we introduce another homological invariant (Dell) related to the delooping level for an Artin algebra. We compare this new tool with other dimensions as the finitistic dimension or the $φ$-dimension (where $φ$ is the first Igusa-Todorov function), and we also generalize Theorem 4.3. from [9] to truncated path algebras (Theorem 4.18). Finally, we show that for a monomial algebra $A$ the difference dell($A$) - Findim($A$) can be arbitrarily large (Example 4.22).

Delooping levels

Abstract

In [8] V. Gélinas introduced a homological invariant, called {\it delooping level} (dell), that bounds the finitistic dimension. In this article, we introduce another homological invariant (Dell) related to the delooping level for an Artin algebra. We compare this new tool with other dimensions as the finitistic dimension or the -dimension (where is the first Igusa-Todorov function), and we also generalize Theorem 4.3. from [9] to truncated path algebras (Theorem 4.18). Finally, we show that for a monomial algebra the difference dell() - Findim() can be arbitrarily large (Example 4.22).

Paper Structure

This paper contains 10 sections, 28 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Igusa-Todorov functions and Igusa-Todorov algebras
  4. Delooping levels
  5. Relations between $\hbox{\rm{dell}}$ and other homological dimensions
  6. Dell on Gorenstein algebras
  7. Monomial algebras
  8. Radical square zero algebras
  9. Truncated path algebras
  10. Example

Key Result

Proposition 2.3

(HLM, IT) If $M,N\in\hbox{\rm{mod}} A$, then we have the following.

Theorems & Definitions (63)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 2.7: BLMV
  • Definition 2.8: BLMV
  • Remark 2.9: Definition 2.2 from W
  • Theorem 2.10
  • ...and 53 more