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Left-right symmetry of finite finitistic dimension

Charley Cummings

Abstract

We show that the finitistic dimension conjecture holds for all finite dimensional algebras if and only if, for all finite dimensional algebras, the finitistic dimension of an algebra being finite implies that the finitistic dimension of its opposite algebra is also finite. We also prove the equivalent statement for injective generation.

Left-right symmetry of finite finitistic dimension

Abstract

We show that the finitistic dimension conjecture holds for all finite dimensional algebras if and only if, for all finite dimensional algebras, the finitistic dimension of an algebra being finite implies that the finitistic dimension of its opposite algebra is also finite. We also prove the equivalent statement for injective generation.
Paper Structure (4 sections, 10 theorems, 16 equations)

This paper contains 4 sections, 10 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. The construction of A
  3. The finitistic dimension
  4. Injective generation

Key Result

Theorem A

The finitistic dimension conjecture holds for all finite dimensional algebras if and only if, for all finite dimensional algebras $\Lambda$, the finitistic dimension of $\Lambda$ being finite implies that the finitistic dimension of ${\Lambda}^{\mathop{\mathrm{op}}\nolimits}$ is finite.

Theorems & Definitions (25)

  • Theorem A: Theorem \ref{['thm:little_findim']}
  • Theorem B: Propositions \ref{['prop:findim(A)_less_than_findim(tildeA)']} and \ref{['prop:findim(tildeA^op)_is_zero']}
  • Example 2.2
  • Lemma 2.3
  • proof
  • Definition 3.1: The (little/big) finitistic dimension
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 15 more