nLab Spectral Algebraic Geometry

This page is to record material related to:

• Jacob Lurie,

  Spectral Algebraic Geometry

  (pdf)

on spectral algebraic geometry.

Contents

• 0 Introduction

I Fundamentals of Spectral Algebraic Geometry

• 1 Schemes and Deligne-Mumford Stacks
• 2 Quasi-Coherent Sheaves
• 3 Spectral Algebraic Spaces

(see also Structured Spaces)

II Proper Morphisms

• 4 Morphisms of Finite Presentation
• 5 Proper Morphisms in Spectral Algebraic Geometry
• 6 Grothendieck Duality
• 7 Nilpotent, Local, and Complete Modules
• 8 Formal Spectral Algebraic Geometry

III Tannaka Reconstruction and Quasi-Coherent Stacks

• 9 Tannaka Duality
• 10 Quasi-Coherent Stacks
• 11 Smooth and Proper Linear $\infty$-Categories

IV Formal Moduli Problems

• 12 Deformation Theories: Axiomatic Approach
• 13 Moduli Problems for Commutative Algebras
• 14 Moduli Problems for Associative Algebras
• 15 Moduli Problems for En-Algebras
• 16 Examples of Formal Moduli Problems

V Representability Theorems

• 17 Deformation Theory and the Cotangent Complex
• 18 Artin’s Representability Theorem
• 19 Applications of Artin Representability

VI Structured Spaces

• 20 Fractured $\infty$-Topoi

  big and little toposes

  fractured $\infty$-topos

• 21 Structured sheaves

VII Variants of Spectral Algebraic Geometry

VIII Higher Algebraic Stacks

IX Rational and $p$-adic Homotopy Theory

X Appendix

• A Coherent $\infty$-Topoi
• B Grothendieck Topologies in Commutative Algebra
• C Prestable $\infty$-Categories
• D Descent for Modules and Linear $\infty$-Categories
• E Profinite Homotopy Theory

category: reference