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Strictly atomic modules in definable categories

Mike Prest

Abstract

If ${\cal D}$ is a definable category then it may contain no nonzero finitely presented modules but, by a result of Makkai, there is a $\varinjlim$-generating set of strictly ${\cal D}$-atomic modules. These modules share some key properties of finitely presented modules. We consider these modules in general and then in the case that ${\cal D}$ is the category of modules of some fixed irrational slope over a tubular algebra.

Strictly atomic modules in definable categories

Abstract

If is a definable category then it may contain no nonzero finitely presented modules but, by a result of Makkai, there is a -generating set of strictly -atomic modules. These modules share some key properties of finitely presented modules. We consider these modules in general and then in the case that is the category of modules of some fixed irrational slope over a tubular algebra.

Paper Structure

This paper contains 31 sections, 60 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Mittag-Leffler and relatively atomic modules
  4. Strictly atomic modules
  5. Strictly atomic generators
  6. Constructing strictly atomic models
  7. The ring of definable scalars
  8. Tilting and silting classes
  9. Modules of irrational slope
  10. Pp formulas near an irrational and definable closures
  11. Purity in ${\cal D}_r$
  12. Non-pure morphisms in ${\cal D}_r$
  13. Background from Model Theory
  14. Pp formulas
  15. Solution sets of pp formulas
  16. ...and 16 more sections

Key Result

Theorem 3.1

Suppose that $M$ is a right $R$-module. Then the following conditions are equivalent. (i) $M$ is Mittag-Leffler. (ii) For every set $\{ L_i\}_{i\in I}$ of left $R$-modules, the canonical map $M\otimes_R \, (\prod_{i\in I} \, L_i) \to \prod_{i\in I} \, (M\otimes_R L_i)$ is monic. (iii) Every pp-type

Theorems & Definitions (68)

  • Theorem 3.1
  • Theorem 3.2
  • Definition 3.3
  • Lemma 3.4
  • Definition 3.5
  • Lemma 3.6
  • Proposition 3.7
  • Lemma 3.8
  • Lemma 3.9
  • Lemma 3.10
  • ...and 58 more