Supporting readings and prerequisites

Complete proof readings and exact source components accompany the lessons.

• SH02-PREREQ-PROOFS — Supporting verifications for open prerequisites
• Perron's formula and the explicit formula for prime counting
• Counting the zeros: the Riemann–von Mangoldt formula
• Entire functions of order one and the Hadamard product of xi
• The Gamma function and Stirling's formula
• Poisson summation, theta, and the functional equation
• Dirichlet series and Euler products
• Extensions of complete valued fields
• Hensel's lemma, squares and roots of unity in p-adic fields
• Completions, the p-adic numbers and complete discretely valued fields
• Hilbert 90 in Noether's form and Galois descent
• Crossed products and factor systems
• Central simple algebras and the Brauer group
• Representations, characters and the group determinant
• Semisimple rings and Wedderburn's theorem
• Groups with operators
• Sheaves of modules on a ringed space
• Sheaves of modules and their derived categories
• K-injective resolutions in Grothendieck abelian categories
• Complexes, cones and localization
• Injective modules, flasque sheaves and bounded-below derived functors
• Flat modules and K-flat resolutions
• Coherent sheaves on projective schemes: Serre's theorems
• Cohomology of projective space
• Cohomology of affine schemes and Serre's criterion
• Čech cohomology
• Cohomology of sheaves on ringed spaces
• Coherence of higher direct images under proper morphisms
• Ext sheaves and Serre duality on projective space
• Dualizing sheaves and Serre duality for projective schemes
• Effective Cartier divisors and invertible sheaves
• Ample invertible sheaves
• Very ample invertible sheaves, Segre and Veronese embeddings
• Valuation rings and the valuative criterion of separatedness
• Smooth traces, duality and Gysin maps
• Poincaré duality for curves
• Cohomological dimension and the Künneth formula
• Smooth base change and local acyclicity
• Cohomology with compact support
• Constructible sheaves and extension by zero
• The multiplicative group on a curve
• Hypercoverings
• Cohomology on sites
• Topoi, morphisms and points
• Sites and sheaves
• The étale site and its points
• Topologies on schemes
• Brauer groups and Tsen's theorem
• Dimension theory of Noetherian local rings
• The Nullstellensatz and Jacobson rings
• Galois cohomology and the étale cohomology of a field
• Pushforward, pullback and finite morphisms
• Torsion sheaves on curves
• The proper base change theorem
• Quotients and torsors
• Smooth morphisms
• Infinitesimal lifting and the invariance of étale morphisms under thickenings
• Étale neighbourhoods, henselization and quasi-finite morphisms
• Flat morphisms
• Étale morphisms and their local structure
• Faithfully flat descent
• Henselian local rings and henselization
• Completion
• Smooth algebras over a field and the Jacobian criterion
• Formally smooth, unramified and étale ring maps
• Kähler differentials
• Regular local rings
• Regular sequences, depth and Cohen–Macaulay modules
• Simplicial sets, nerves and Kan complexes
• Stacks in groupoids
• Properties and morphisms of algebraic spaces
• Algebraic spaces
• The bootstrap theorem
• The right adjoint of derived pushforward
• Serre's comparison theorems and Chow's theorem
• Comparison with the topological fundamental group over the complex numbers
• Comparison with singular cohomology
• Algebraization of formal schemes
• The theorem on formal functions
• Base change and the Grothendieck complex
• Grothendieck's existence theorem
• Deformations of rings and schemes and the naive cotangent complex
• The étale fundamental group
• Coefficient rings and the Cohen structure theorem
• Hom complexes, internal derived Hom and Ext sheaves
• The derived tensor product and Tor sheaves
• Derived pullback and pushforward
• Purity of the branch locus
• Flatness criteria, dimension and the flat locus
• Krull dimension and Noether normalization
• Noetherian and Artinian rings
• Localization, local properties and support
• Spectra of rings
• Spectral spaces and affine realization
• Associated primes and primary decomposition
• Integral extensions: lying over, going up and going down
• Faithful flatness and the local criterion for flatness
• Tor and flat modules
• Resolutions, Tor and Ext
• Lefschetz theorems for finite étale covers
• Abelian varieties
• The ℓ-adic Tate module of an elliptic curve
• Frobenius elements and determination by traces
• Profinite groups and ℓ-adic representations
• Galois categories
• Descending properties of schemes and morphisms
• Local cohomology
• Formal geometry along a closed subscheme
• Cohomological dimension, vanishing and connectedness
• Local duality and finiteness
• Fundamental groups of proper schemes and the homotopy exact sequence
• Specialization maps and tame ramification
• Weak Lefschetz and complete-intersection ranks
• Determinants, ramification and Tate curves
• Counting over finite fields and the limit q → 1
• Stable pointed-family extension after a separable projective alteration
• Mathematical prerequisites for stable-family extension
• Numerical types, weighted chains and the semistable-reduction bound
• Smooth projective alterations
• Hilbert and Quot schemes
• Moduli stacks are algebraic
• Artin's axioms
• Quotient stacks and Deligne–Mumford stacks
• Algebraic stacks
• Regular surface models and exceptional-curve contraction
• Stable-family proof guide
• Numerical models and corrected weighted calculations
• The stack of curves
• Stable reduction and properness
• Stable pointed-family extension
• Stable-family sources and licence
• The structure of the Picard scheme
• The Picard functor and the Picard scheme of a curve
• Relative divisors and the existence of the Picard scheme