The Rising Sea: Foundations of Algebraic Geometry
Read the structure-sheaf construction alongside localization calculations. Keep an affine example at hand when the definition becomes global.
Read: Chapters 2–4; especially §§3.5 and 4.1–4.4.
Algebraic Geometry
An equation gives a first picture of a space. Algebraic geometry asks what survives a change of coordinates, a change of field, or passage to a family. The six sections below collect a few places to read about those questions, with links back to the relevant foundations.
Algebraic geometry learning route · First-course reading and exercises · Algebraic geometry in the library
For a first course, use the varieties and schemes section and the existing learning route. The later sections are separate branches: choose one for a question you want to understand, rather than reading every reference in order. The links open the authors’ or projects’ full texts; the catalogue remains the place for the longer booklist.
Varieties and schemes
On the open set where a polynomial f does not vanish, why should f become invertible in the ring of functions? Work through that question before treating a scheme as a list of definitions. Localization explains the affine pieces; compatible functions on overlaps explain how those pieces fit together.
Before reading: Commutative algebra · Localization of a Commutative Ring · Prime Spectrum of a Commutative Ring · Sheaf
Milne is the shorter route through varieties; Vakil is the main choice here for schemes. If you can construct the structure sheaf on distinguished opens and glue affine charts, go on to the next section without repeating the entire first course.
Algebraic Geometry
Use affine varieties and local rings to give the algebra geometric meaning before switching to schemes. This is an alternative starting point, not an extra prerequisite book.
Read: Chapters 2, 4 and 5; §4c on local rings and §5f on patching.
Related concepts: Localization of a Commutative Ring · Prime Spectrum of a Commutative Ring · Scheme
Sheaves and cohomology
Local solutions need not give a global solution. For example, local trivializations of a line bundle can differ by invertible functions on overlaps. Sheaves record the local data; cohomology helps organize the remaining global information. Start with a calculation, not with every possible cohomological formalism.
Before reading: Scheme · Sheaf · Chain Complex of Modules · Homological algebra
Try a two-chart calculation on the projective line. If the connecting homomorphism is the obstacle, pause for Baker’s short notes; if the algebra is familiar, go straight to the geometric calculation.
The Rising Sea: Foundations of Algebraic Geometry
Follow the cohomology of line bundles on projective space. Return to the derived-functor construction when you need to compare the two definitions.
Read: §§18.1–18.3; then §§23.4–23.5 for the comparison with derived functors.
Notes on Basic Homological Algebra
Use the short exact sequence of complexes to construct a connecting map. These fifteen pages supply algebraic practice, not a course on sheaf cohomology.
Read: §§1–3; Theorem 2.2 and the following cochain-complex convention.
The Stacks Project: first cohomology and torsors
Read how first cohomology classifies torsors for an abelian sheaf. Compare the statement with transition functions rather than treating cohomology only as a group to compute.
Read: Tag 02FN, Definition 20.4.1 and Lemma 20.4.3.
Related concepts: Sheaf · Short Exact Sequence of Modules · Ext Group of Modules · Spectral Sequence
Descent, algebraic spaces and stacks
A moduli problem asks for families, not just a set of isomorphism classes. What happens to an object’s automorphisms if we keep only that set? Work from a family of curves or bundles toward groupoids and descent; algebraic spaces and stacks then answer different parts of the problem.
Before reading: Scheme · Sheaf · Representable Functor · Categorical Pullback
First distinguish gluing objects from gluing their morphisms. A quotient stack is not in general the same object as the orbit space. Read the definitions of algebraic spaces and stacks side by side after the moduli examples.
Stacks and Moduli
Start with families and automorphisms, then read the stack condition and scheme atlases. The examples give a reason for the extra categorical structure.
Read: §§1.3–1.6; then §§3.1, 3.5 and 4.1.
The Stacks Project: fpqc descent
Work from the affine statement to descent of quasi-coherent sheaves. Keep both effectivity and full faithfulness in view; they are separate assertions.
Read: Tag 023R, Lemma 35.5.1 and Proposition 35.5.2.
Related concepts: Fpqc Descent for Quasi-Coherent Sheaves · Algebraic Space · Algebraic Stack · Quotient Stack · Grothendieck Site · Grothendieck Topos
Étale methods and arithmetic geometry
Over a finite field there is no complex topology to supply the usual Betti numbers. Which coverings should replace ordinary open sets? Étale morphisms provide a way to ask cohomological questions over other fields; arithmetic enters when those invariants interact with Galois actions and Frobenius.
Before reading: Scheme · Sheaf · Galois Extension · Homological algebra
Begin with finite étale covers and the spectrum of a field. The pro-étale paper is a later branch for infinite constructions and coefficient issues, not a shorter substitute for a first étale course.
Lectures on Étale Cohomology
Connect étale morphisms, the fundamental group and sites before the cohomology construction. These course notes are distinct from Milne’s 1980 book.
Read: §§2–3 and 5–10; return to §26 for the Weil-conjecture motivation.
The pro-étale Topology for Schemes
Read the introduction’s coefficient problem first, then compare the pro-étale and étale sites. This explains a use for the new topology before its technical development.
Read: §1, especially Definitions 1.1–1.2; §§4–5 for the topology and comparison.
Related concepts: Étale Fundamental Group · Étale Cohomology · Grothendieck Site · Crystalline Cohomology
Derived geometry and higher categories
An ordinary fiber product can forget higher Tor groups in an intersection. Derived geometry retains this information in the geometric object. Higher categories provide a language for keeping equivalences and their compatibilities as well. These are related needs, but learning a model of higher categories is not the same task as learning derived geometry.
Before reading: Scheme · Tensor Product of Modules · Chain Complex of Modules · Category theory · Simplicial Set
Start with Toën’s intersection example. Use HTT to learn the language you actually encounter, then consult SAG for spectral geometry. Simplicial commutative rings and commutative ring spectra are not interchangeable in arbitrary characteristic.
Derived Algebraic Geometry
Use the intersection discussion to understand what is added to an ordinary scheme. Then compare the definition of a derived scheme with its classical truncation.
Read: Introduction; §2.2, Definition 2.5 and the truncation discussion; §4.1 for the cotangent complex.
Higher Topos Theory
Read the inner-horn definition with nerves of ordinary categories as examples. Consult the topos chapter when descent begins to involve space-valued sheaves.
Read: §1.1.2, Definition 1.1.2.4; §6.1, especially Definition 6.1.0.4.
Spectral Algebraic Geometry
Use the introduction to compare spectral and classical geometry, then identify the prerequisites you need. This unfinished draft is for selective reference, not a promised complete course.
Read: §§0.1–0.2, motivation and prerequisites; Chapter 1 is the subsequent scheme and Deligne–Mumford-stack reference.
Related concepts: Cotangent Complex · Derived Scheme · Simplicial Set · Stable Infinity-Category · Infinity-Topos · Higher Categories and the Topos Bridge
Perfectoid spaces and prismatic cohomology
A p-adic field has characteristic zero, while its residue field has characteristic p. Tilting relates geometric settings across this divide. Prismatic cohomology addresses a different question: how several integral p-adic cohomology theories can arise from a common construction. Read the connection without conflating the two theories.
Before reading: Scheme · Étale Cohomology · Homological algebra · Perfectoid Field
You also need complete nonarchimedean fields and some adic geometry. Begin with the Berkeley lectures if that language is new. The papers below are research sources; understanding the examples and comparison statements is a sensible first pass.
Berkeley Lectures on p-adic Geometry
Learn adic spaces, perfectoid rings and tilting in a lecture sequence. Read the adic examples before trying to use the perfectoid-space equivalence.
Read: Lectures 2–4 and 6–7; §§6.1–6.2 and 7.1–7.3.
Perfectoid Spaces
Compare the field and algebra definitions with the geometric tilting theorem. Use the paper to understand the role of almost mathematics, not as a replacement for valuation-theoretic preparation.
Read: §§3–5; then §6 for analytic topology and §7 for the étale topology.
Prisms and Prismatic Cohomology
Work through the prism examples before the site. Use the stated comparison theorems to identify what is being compared and under which hypotheses.
Read: §1, Definition 1.1 and Example 1.3; §§2–4; §9 for the étale comparison.
Related concepts: Perfectoid Field · Perfectoid Space · Tilting of Perfectoid Spaces · Prism in p-adic Geometry · Prismatic Cohomology