Abstract & Pure

Learn Commutative Algebra & Algebraic Geometry

Rings as geometry: ideals, Noetherian conditions, the Nullstellensatz, and the dictionary between algebra and varieties.

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Free to start · adaptive placement finds your level · reviews timed to your own forgetting.

What you'll learn

34 lessons in Commutative Algebra & Algebraic Geometry

Noetherian rings & Hilbert basisPrime & maximal idealsLocalizationModules over a ringExact sequencesStructure theorem over a PIDThe prime spectrumThe Zariski topologyHilbert's NullstellensatzAffine varieties & dimensionTensor products of modulesIntegral extensions & Noether normalizationSchemes (a glimpse)Primary decomposition (Lasker–Noether)Krull dimension & the principal ideal theoremHilbert series & Hilbert polynomialsDedekind domains & discrete valuation ringsRegular local rings & smoothnessGoing-up & going-downGröbner basesProjective varieties & graded ringsCompletion & the Krull intersection theoremFlat & faithfully flat modulesDepth & Cohen–Macaulay ringsHomological dimension & Auslander–BuchsbaumTor, Ext & free resolutionsKähler differentialsLocal cohomologyAssociated graded rings, Rees algebra & blowupsAffine varieties & the Nullstellensatz correspondenceProjective space & projective varietiesDivisors & the divisor class groupSheaves & sheaf cohomology (the idea)The Riemann–Roch theorem for curves
How Erudia teaches

Built to be understood — and remembered.

Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the day its model predicts you would forget it. A short placement check finds what you already know, so you start Commutative Algebra & Algebraic Geometry exactly where it's useful.

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Related Abstract & Pure subjects
Abstract AlgebraRepresentation TheoryLie GroupsUniversal AlgebraAlgebraic TopologyHomological Algebra