Dirac geometry I: Commutative algebra

Abstract

Introduction 1 2. Dirac schemes 8 2.1. Dirac rings and ideals 8 2.2. Localization 11 2.3. The Dirac-Zariski space 13 2.4. The category of Dirac schemes 17 2.5. Evenness 21 3. Commutative algebra 28 3.1. Properties of modules 28 3.2. Properties of algebras 32 3.3. Faithfully flat descent 35 3.4. Integrality and Zariski's main theorem 38 4. Étale morphisms 44 4.1. Differentials 44 4.2. Smooth, unramified, and étale morphisms 46 4.3. Evenness of étale extensions 49 4.4. Étale maps of ring spectra 52 Appendix A. Geometric realization in an n-category 58 Appendix B. Conservativity and limits 59 References 60