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Formal schemes and infinitesimal neighbourhoods

Federico Bongiorno

Abstract

The aim of this article is to give a rigorous geometric interpretation of the completion of a ring with respect to an ideal. To this end, we define the infinitesimal neighbourhood of an immersion of formal schemes as the largest possible thickening. Further, we show that immersions of formal schemes locally of formal finite presentation admit the existence of infinitesimal neighbourhoods, which are locally given by completing with respect to an ideal.

Formal schemes and infinitesimal neighbourhoods

Abstract

The aim of this article is to give a rigorous geometric interpretation of the completion of a ring with respect to an ideal. To this end, we define the infinitesimal neighbourhood of an immersion of formal schemes as the largest possible thickening. Further, we show that immersions of formal schemes locally of formal finite presentation admit the existence of infinitesimal neighbourhoods, which are locally given by completing with respect to an ideal.
Paper Structure (15 sections, 39 theorems, 38 equations)

This paper contains 15 sections, 39 theorems, 38 equations.

Table of Contents

  1. Formal schemes
  2. Adic rings
  3. Formal schemes
  4. Closed imersions
  5. Surjective adic morphisms
  6. Closed immersions of formal schemes
  7. Thickenings
  8. Thickenings of adic rings
  9. Thickenings of formal schemes
  10. Formal finite presentation
  11. Morphisms of adic rings
  12. Morphisms of formal schemes
  13. Infinitesimal neighbourhoods
  14. Affine infinitesimal neighbourhood
  15. Infinitesimal neighbourhood

Key Result

Proposition 1

Let $f : X \rightarrow Y$ be an immersion of formal schemes locally of formal finite presentation. Then the infinitesimal neighbourhood of $f$ exists.

Theorems & Definitions (86)

  • Definition
  • Proposition
  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 76 more