Alphabeta MathScheme Theory
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
1 pageThis page fixes the standard sheaf vocabulary on a topological space and keeps the route deliberately concrete.
13 definitions, 9 lemmas, 5 theorems, 1 corollary, 1 remarkExamples & counterexamples →
Level 1
1 pageThis page collects the standard sheaf-theoretic operations attached to a continuous map and then adds the ringed-space structure needed for module pullback.
16 definitions, 6 lemmas, 7 theorems, 1 remarkExamples & counterexamples →
Level 2
2 pages- Affine Schemes and the Structure Sheaf30 results
An affine scheme is a prime spectrum equipped with the sheaf obtained by localizing the coordinate ring on distinguished opens.
12 definitions, 5 lemmas, 7 theorems, 3 corollaries, 2 examples, 1 remarkExamples & counterexamples → This page develops sheaf cohomology on a topological space from a supplied functorial injective resolution datum, as the right derived functors of the global-sections…
15 definitions, 33 lemmas, 13 theorems, 2 remarksExamples & counterexamples →
Level 3
1 pageThis page builds the affine-local language of schemes: open charts, gluing, morphisms, reductions, closed subschemes, and the two finiteness conditions.
19 definitions, 4 lemmas, 6 theorems, 1 example, 1 remarkExamples & counterexamples →
Level 4
1 pageFibre products organize compatible morphisms from every test scheme.
9 definitions, 25 lemmas, 4 theorems, 2 corollaries, 2 remarksExamples & counterexamples →
Level 5
3 pagesSeparatedness is a condition on a morphism, not on the point-set topology of its source: the diagonal Δ X/S:X→ X× SX must be a closed immersion.
5 definitions, 15 lemmas, 4 theorems, 3 corollaries, 2 remarksExamples & counterexamples →A group scheme of finite type over a field k is a finite-type k-scheme equipped with multiplication, identity and inverse morphisms satisfying the group-object identities.
2 definitions, 1 lemmaExamples & counterexamples →Kähler differentials linearise derivations. The page begins with derivations and the universal property that defines Ω B/A, proves existence by generators and relations for…
9 definitions, 10 lemmas, 10 theorems, 2 corollaries, 2 remarksExamples & counterexamples →
Level 6
2 pagesAn affine group scheme of finite type over a field k is the same data as a finitely generated commutative Hopf algebra with its comultiplication, counit and antipode: the…
3 definitions, 7 lemmas, 3 theoremsExamples & counterexamples →- Finite Proper and Projective Morphisms49 results
This page develops the affine-local algebra and valuative tools used to prove that finite morphisms are proper and projective morphisms are proper.
11 definitions, 25 lemmas, 8 theorems, 3 corollaries, 2 remarksExamples & counterexamples →
Level 7
2 pages- Flat Smooth and Etale Morphisms77 results
This page develops flat, smooth, and étale morphisms and their local criteria.
11 definitions, 45 lemmas, 18 theorems, 1 corollary, 1 example, 1 remarkExamples & counterexamples → This page develops quasi-coherent and coherent module sheaves on a scheme, their basic operations, the vector-bundle dictionary, quasi-coherent ideals, closed subschemes, and scheme-theoretic images.
13 definitions, 12 lemmas, 13 theorems, 1 corollary, 1 remarkExamples & counterexamples →
Level 8
3 pagesCartier and Weil divisors are the two complementary descriptions of codimension-one data on a scheme, and this page develops both of them and the comparison between them.
15 definitions, 12 lemmas, 6 theorems, 2 corollaries, 1 remarkExamples & counterexamples →The Lie algebra of a group scheme of finite type over a field is its tangent space at the identity, recorded in the two equivalent forms used throughout: the dual of the…
1 definition, 6 lemmas, 3 theoremsExamples & counterexamples →This page constructs Proj of a nonnegatively graded ring, develops the associated sheaves of graded modules and the twisting sheaves O X(n), and defines ampleness, relative…
12 definitions, 13 lemmas, 12 theorems, 1 remarkExamples & counterexamples →
Level 9
2 pagesThis page develops actions of finite-type group schemes on schemes, their orbit maps and stabilizers, and the two controlled settings in which quotient sheaves are representable by schemes.
2 definitions, 5 lemmas, 1 proposition, 2 theorems, 1 remarkExamples & counterexamples →This page collects the cohomology, finiteness, Hilbert-polynomial and base-change results listed above.
5 definitions, 31 lemmas, 12 theorems, 7 corollaries, 1 remarkExamples & counterexamples →
Level 10
4 pagesThis page builds the standard fppf foundations for algebraic spaces and algebraic stacks and the first layer of the cotangent complex.
21 definitions, 25 lemmas, 4 theoremsExamples & counterexamples →- Compact Support Sheaf Calculus11 results
This page develops the compact-support calculus for sheaves of abelian groups (in particular of vector spaces) on locally compact Hausdorff spaces, over the Axiom of Choice, and isolates the exact points where choice enters.
3 definitions, 8 lemmasExamples & counterexamples → For a locally Noetherian scheme S and a projective finitely presented S-scheme X carrying a relatively ample invertible sheaf L, this page defines the Hilbert functor Hilb…
3 definitions, 18 lemmas, 1 theoremExamples & counterexamples →Smooth proper curves are the objects of this page. A curve over a field k is a geometrically integral, separated k-scheme of finite type whose underlying topological space…
15 definitions, 13 lemmas, 10 theorems, 2 corollariesExamples & counterexamples →
Level 11
2 pagesDeformation theory measures the flat families through which a scheme moves.
5 definitions, 9 lemmas, 2 theorems, 2 corollariesExamples & counterexamples →This page proves the Riemann–Roch theorem for a smooth proper geometrically integral curve C over a field k in its Euler-characteristic form, ℓ(D) - i(D) = deg k(D) + 1 - g…
4 definitions, 14 lemmas, 6 theorems, 9 corollaries, 1 remarkExamples & counterexamples →
Level 12
2 pagesThis page develops the blowup of a scheme along a quasi-coherent ideal sheaf of finite type, together with its exceptional subscheme, its functorial universal property, its…
8 definitions, 22 lemmas, 13 theorems, 4 corollaries, 2 remarksExamples & counterexamples →This page develops residues of rational differentials on a smooth curve, the residue pairing on principal parts, Serre duality for curves over an arbitrary field, and the…
5 definitions, 18 lemmas, 12 theorems, 10 corollaries, 2 remarksExamples & counterexamples →
Level 13
1 pageThis page builds the ordinary six-operation calculus for sheaves of vector spaces on analytic spaces of complex algebraic varieties, over an arbitrary field and with the…
2 definitions, 4 lemmas, 4 theoremsExamples & counterexamples →