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Deformation of formal schemes through local homology

Marta Pérez Rodríguez

Abstract

Deformation theory is treated for locally notherian formal schemes (non necessarily smooth). The cotangent complex is defined in the derived category through the homology localization functor. The basic properties and results of a deformation theory are proved. And the complex is described for regular closed immersions and complete intersection morphisms of formal schemes.

Deformation of formal schemes through local homology

Abstract

Deformation theory is treated for locally notherian formal schemes (non necessarily smooth). The cotangent complex is defined in the derived category through the homology localization functor. The basic properties and results of a deformation theory are proved. And the complex is described for regular closed immersions and complete intersection morphisms of formal schemes.
Paper Structure (5 sections, 30 theorems, 46 equations)

This paper contains 5 sections, 30 theorems, 46 equations.

Table of Contents

  1. Preliminaries on formal schemes
  2. The affine case
  3. Definition and properties
  4. Deformation of formal schemes
  5. Cotangent complex and infinitesimal conditions

Key Result

Proposition 2.5

If $f \colon \mathop{\mathrm{Spf}}\nolimits(B) \to \mathop{\mathrm{Spf}}\nolimits(A)$ is an adic smooth morphism in $\mathsf {FS}_{\mathsf {af}}$, then the augmentation map (aug) gives an isomorphism $\widehat{L}_{B/A} \simeq \widehat{\Omega}^{1}_{B/A}[0]$ and $\widehat{L}_{B/A} \in \boldsymbol{\mat

Theorems & Definitions (69)

  • Proposition 2.5
  • proof
  • Corollary 2.6
  • proof
  • Proposition 2.7
  • proof
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • proof
  • ...and 59 more