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On the equivalence between the existence of $n$-kernels and $n$-cokernels

Vitor Gulisz, Wolfgang Rump

Abstract

We give an elementary proof of the statement that if an idempotent complete additive category has weak kernels and weak cokernels, then it has $n$-kernels if and only if it has $n$-cokernels, where $n$ is a nonnegative integer. As a consequence, elementary proofs of two results concerning the equality between the global dimensions of certain right and left module categories are obtained.

On the equivalence between the existence of $n$-kernels and $n$-cokernels

Abstract

We give an elementary proof of the statement that if an idempotent complete additive category has weak kernels and weak cokernels, then it has -kernels if and only if it has -cokernels, where is a nonnegative integer. As a consequence, elementary proofs of two results concerning the equality between the global dimensions of certain right and left module categories are obtained.

Paper Structure

This paper contains 2 sections, 4 theorems.

Table of Contents

  1. Introduction
  2. The results and their proofs

Key Result

Theorem 1

Let $\mathcal{C}$ be an idempotent complete preadditive category that has weak kernels and weak cokernels, and let $n$ be a nonnegative integer. Then $\mathcal{C}$ has $n$-kernels if and only if $\mathcal{C}$ has $n$-cokernels.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • proof
  • Corollary 4
  • proof