Categories
Foundations
Fields and Extensions
Three important functors
The $\spec$ of a ring
Noetherian rings and modules
Graded and filtered rings
Integrality and valuation rings
Unique factorization and the class group
Dedekind domains
Dimension theory
Completions
Regularity, differentials, and smoothness
Various topics
Homological Algebra
Flatness revisited
Homological theory of local rings
\'Etale, unramified, and smooth morphisms
Complete local rings
Homotopical algebra
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