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Frobenius quotients, inflation categories and weighted projective lines

Xiao-Wu Chen, Qiang Dong, Shiquan Ruan

Abstract

We propose the notion of Frobenius quotients between Frobenius exact categories. It turns out that any Frobenius quotient induces Frobenius quotients between the corresponding inflation categories. We obtain an explicit Frobenius quotient from the category of vector bundles on weighted projective lines with three weights to a certain category consisting of monomorphism grids.

Frobenius quotients, inflation categories and weighted projective lines

Abstract

We propose the notion of Frobenius quotients between Frobenius exact categories. It turns out that any Frobenius quotient induces Frobenius quotients between the corresponding inflation categories. We obtain an explicit Frobenius quotient from the category of vector bundles on weighted projective lines with three weights to a certain category consisting of monomorphism grids.

Paper Structure

This paper contains 6 sections, 20 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Pretriangle-equivalences and Frobenius quotients
  3. The $n$-inflation categories
  4. Gorenstein-projective modules and projective-module factorizations
  5. The main result
  6. Maximal Cohen-Macaulay modules and weighted projective lines

Key Result

Theorem A

There is an explicit Frobenius quotient functor whose essential kernel equals $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: The AR quiver of $\textup{vect}\hbox{-}\mathbb{X}(2,2,3)$
  • Figure 2: The AR quiver of $\textup{MCM}^{V_4}(\mathbb{K}[[x, y,z]]/(x^2+y^2+z^3))$
  • Figure 3: The AR quiver of $\textup{mod}\hbox{-}\mathbb{K}[z]/(z^3)$

Theorems & Definitions (41)

  • Theorem A
  • Theorem B
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 31 more