Chapter I. Algebraic Geometry
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 36 definitions, 21 theorems, 17 propositions, 4 lemmas, 5 corollaries, 1 example, 5 constructions, 1 convention and 5 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Convention: base field and rings
K is algebraically closed unless a smaller field is explicitly introduced; rings in §0 are commutative with identity.
Definition/result: noetherian rings and Hilbert basis theorem
A ring is noetherian iff its ideals satisfy the standard finite-generation/ACC/maximal-element conditions; quotients are noetherian, and R noetherian implies R[T] noetherian.
Recorded result: polynomial rings over a field are UFDs
For a field K, K[T_1,…,T_n] is a unique factorization domain.
Theorem/recorded result: weak Nullstellensatz
If a finitely generated K-algebra K[x_1,…,x_n] is a field, then every x_i is algebraic over K.
Definition/result: algebraic independence and transcendence degree
A transcendence basis is a maximal algebraically independent subset; its cardinality is tr.deg. For finitely generated field extensions a transcendence basis may be chosen among the generators and the residual extension is finite algebraic.
Theorem/recorded result: Lüroth theorem
Every proper intermediate field K⊊L⊆K(T) is itself a simple purely transcendental extension of K.
Definition/result: norm of a finite field extension
For finite E/F, N_{E/F}(a) is the determinant of multiplication by a on the F-vector space E; it is multiplicative and agrees with the stated power/minimal-polynomial formula.
Definition/result: integral elements, integral closure, integrally closed domains
x is integral over S iff it satisfies a monic polynomial, equivalently S[x] is finite as S-module; integral closure and integrally closed domains are defined accordingly, and polynomial rings over integrally closed domains remain integrally closed.
Theorem/recorded result: Noether normalization
A finitely generated integral K-domain R of transcendence degree d is integral over a polynomial subalgebra K[y_1,…,y_d] on algebraically independent elements of R.
Theorem/recorded result: going up
For an integral extension R/S, prime and maximal ideals of S lift to prime and maximal ideals of R.
Theorem/recorded result: going down
If R/S is integral and S integrally closed, a chain P_1⊂P_2 under a prime Q_2 admits Q_1⊂Q_2 lying over P_1.
Theorem/recorded result: extension theorem for maps from integral extensions
A K-valued homomorphism on S extends across an integral extension R/S; the source also records the criterion for prescribing the value of one integral element.
Lemma/recorded result: finite prime avoidance
If an ideal lies in a finite union of prime ideals, it lies in one of them.
Construction/result: generalized rings of quotients and local rings
For a multiplicative set S, S^{-1}R is formed from fraction classes; primes correspond to primes of R disjoint from S. Localization R_P is local, preserves noetherianity and integral closedness in the stated cases, and gives the standard residue/tangent-quotient identifications.
Lemma/recorded result: Nakayama lemma
If V is finitely generated and V=MV for a maximal ideal M, some x∉M annihilates V; over a local ring this forces V=0.
Definition/result: M-adic topology, Krull dimension, regular local rings
The powers of the maximal ideal define the separated M-adic topology; Krull dimension is maximal prime-chain length; a local ring is regular when this equals the minimal number of maximal-ideal generators. Regular local rings are integrally closed domains.
Recorded result: minimal primes and nilpotent nilradical
For I in a noetherian ring, the intersection of primes minimal over I maps to the nilradical of R/I, which is nilpotent.
Definition/results: separable and separably generated field extensions
The source defines separability in arbitrary characteristic via p-power linear independence and separably generated extensions; for finitely generated extensions the two notions agree, perfect bases give separability, and separability is transitive in towers.
Definition/result: field derivations and separability criterion
An F-derivation E→L is additive, Leibniz, and vanishes on F. For separably generated E/F, Der_F(E,L) has dimension tr.deg_F E; separability is characterized by extension of derivations, and derivations vanish on pth powers in characteristic p.
Definition: affine space and affine variety
An=Kn; an affine algebraic set/variety is a common zero locus of finitely many polynomials, with V(I) and I(X) giving the ideal–zero-set correspondence.
Definition/result: radical ideals and vanishing ideals
The radical √I consists of f with some power in I; I(V(I))=√I over K.
Theorem: Hilbert Nullstellensatz
For every ideal I⊆K[T_1,…,T_n], I(V(I))=√I.
Definition/result: Zariski topology on affine space
Closed sets are algebraic zero loci; V converts sums to intersections and products/intersections to unions, giving the Zariski topology. Principal open sets form a useful basis.
Definition/result: noetherian topological spaces
A space is noetherian when closed subsets satisfy DCC; affine varieties are noetherian because their coordinate rings are.
Definition: irreducible space
A nonempty space is irreducible when it is not a union of two proper closed subsets; equivalently any two nonempty opens meet, or every nonempty open is dense.
Proposition: closure and images preserve irreducibility
A subspace is irreducible iff its closure is, and the continuous image of an irreducible space is irreducible.
Proposition/definition: finite irreducible decomposition in a noetherian space
Every nonempty noetherian space is a finite union of maximal irreducible closed subsets, uniquely determined up to order; these are its irreducible components.
Proposition: irreducible affine sets and prime ideals
A closed affine set X is irreducible iff I(X) is prime.
Construction/result: product of affine varieties
For X⊆Am,Y⊆An, X×Y is affine with the Zariski product construction; K[X×Y] identifies with K[X]⊗_K K[Y].
Definition: affine algebra and polynomial functions
K[X]=K[T]/I(X) is the affine coordinate algebra; its elements are polynomial functions on X.
Definition/result: morphisms and comorphisms of affine varieties
A map X→Y is regular when its coordinates are polynomial functions. It induces a contravariant K-algebra map K[Y]→K[X], and every such K-algebra map arises uniquely this way.
Comparison: affine varieties and affine K-algebras
The source identifies affine varieties contravariantly with reduced finitely generated K-algebras and notes that affine products correspond to tensor-product coproducts.
Definition: projective space and homogeneous coordinates
P^n is the set of one-dimensional subspaces of K^{n+1}, with homogeneous coordinates modulo nonzero scalar; P(V) denotes the projectivization of a vector space V.
Definition: projective algebraic sets and varieties
Projective closed sets are common zero loci of homogeneous polynomials; standard affine charts U_i≅A^n define their local affine structure.
Proposition: projective zero sets and homogeneous radical ideals
Projective closed subsets correspond inclusion-reversingly to homogeneous radical ideals other than the irrelevant ideal.
Proposition/construction: Segre embedding and product of projective varieties
The Segre map Pm×Pn→P^{mn+m+n} is a bijection onto a closed subset and takes products of projective varieties to projective varieties.
Construction: Grassmann varieties via Plücker coordinates
The d-dimensional subspaces of V embed in P(∧^dV) by the Plücker map; its image is closed, giving the Grassmann variety G_d(V).
Definition/construction: flags and flag variety
A flag is a strictly increasing chain of subspaces, full when successive dimensions increase by one. Full flags form a closed subvariety of a product of Grassmann varieties.
Definition/result: local ring of a point
For irreducible X and x∈X, O_x consists of rational functions defined at x; it is the localization K[X]_{I(x)}, with maximal ideal of functions vanishing at x.
Proposition: global affine functions as intersection of local rings
For irreducible affine X, K[X]=∩_{x∈X}O_x inside K(X).
Construction: regular-function sheaf on affine varieties
Regular functions on opens are locally quotients of polynomial functions with nonvanishing denominators; these assignments satisfy gluing and restriction and define the structure sheaf.
Definition: prevariety and affine open covering
A prevariety is a noetherian topological space with a sheaf of K-valued functions locally isomorphic to affine varieties; affine open subsets form a basis.
Definition: subprevariety and quasiprojective variety
Locally closed subsets inherit prevariety structures; open subsets of projective varieties are quasiprojective.
Definition: function field of an irreducible prevariety
The affine charts of an irreducible prevariety have canonically identified fraction fields; this common field is K(X).
Definition/result: morphisms of prevarieties
A morphism is continuous and pulls regular functions back to regular functions; it can be checked on affine covers and agrees with the affine comorphism notion.
Proposition: local criterion for a map to be a morphism
Given compatible affine covers on source and target, a map is a morphism when pullbacks of regular functions are regular on the chosen source opens.
Construction: rational functions as maps on domains of definition
A rational function is regular on an open domain of definition and thereby gives a morphism to A^1; zero and nonzero loci are defined on that domain.
Definition: birational morphism and birational equivalence
For irreducible prevarieties, a morphism is birational when the induced map on function fields is an isomorphism; varieties with isomorphic function fields are birationally equivalent.
Proposition/construction: product of affine prevarieties
For affine X,Y the cartesian product with tensor-product coordinate algebra has the expected regular functions and projections.
Theorem: existence of products of prevarieties
Products exist in the category of prevarieties by gluing products of affine charts and satisfy the categorical universal property.
Definition: variety via the diagonal/Hausdorff axiom
A prevariety is a variety when its diagonal is closed in X×X; equivalently morphisms into it have closed equality loci.
Lemma: affine-neighborhood criterion for a prevariety to be a variety
If every pair of points of a prevariety lies in a common affine open subset, then it satisfies the diagonal axiom and is a variety.
Proposition: closed graphs and uniqueness on dense subsets
For a morphism X→Y with Y a variety the graph is closed; two morphisms to a variety agreeing on a dense subset agree everywhere.
Definition/result: dimension of an irreducible variety
For irreducible X, dim X is tr.deg_K K(X); arbitrary varieties have dimension equal to the maximum over irreducible components.
Proposition: dimension of a product
For irreducible X,Y, dim(X×Y)=dim X+dim Y.
Definition: codimension
For irreducible closed Y⊆X, codim_X Y=dim X−dim Y.
Proposition: proper irreducible closed subsets have smaller dimension
If X is irreducible and Y⊊X is closed irreducible, then dim Y<dim X.
Corollary: codimension-one affine subvariety is a component of one equation
If X is irreducible affine and Y⊆X is closed irreducible of codimension one, then Y is an irreducible component of V(f) for some f∈K[X].
Definition: hypersurface
A hypersurface in affine space is the zero set of one nonscalar polynomial; more generally a nonzero nonunit f∈K[X] defines a hypersurface in an affine variety X.
Proposition: components of an affine-space hypersurface have codimension one
Every irreducible component of a hypersurface in affine space A^n has codimension one.
Theorem: dimension theorem for a principal hypersurface
If X is irreducible affine, f is a nonzero nonunit, and Y is an irreducible component of V(f), then codim_X Y=1.
Corollary: chains realizing codimension
An irreducible closed subset of codimension r admits a chain of irreducible closed subsets increasing codimension by one at each step.
Corollary: codimension bound for common zero sets
Each irreducible component of the common zero set of r regular functions has codimension at most r.
Corollary: codimension-r subvariety as a component of r equations
A closed irreducible affine subset of codimension r is an irreducible component of the zero set of r functions.
Definition: fibres and dominant morphisms
The fibre of φ:X→Y over y is φ^{-1}(y); a morphism is dominant when its image is dense, and a component over W dominates W when its image is dense in W.
Theorem: lower bound on fibre dimensions
For dominant φ:X→Y irreducible, r=dim X−dim Y, any component over irreducible W that dominates W has dimension at least dim W+r; fibre components have dimension at least r.
Definition: finite morphism
For affine X,Y, φ is finite when K[X] is integral over φ^*K[Y], equivalently a finite module over the image algebra.
Proposition: basic properties of finite dominant morphisms
Finite dominant affine morphisms are closed and surjective; restrictions to closed subsets remain finite, and under integral-closedness hypotheses components over irreducible subsets dominate them.
Theorem: generic image and exact fibre dimension
A dominant morphism of irreducible varieties contains a nonempty open subset of the target in its image; over a suitable open normal locus, components over irreducible subsets have the expected dimension.
Corollary: bijective morphisms are generically finite isomorphism candidates
A bijective morphism of irreducible varieties preserves dimension and restricts on suitable affine opens to a finite morphism.
Definition: constructible subset
A constructible set is a finite union of locally closed subsets.
Theorem: constructibility of morphism images
Morphisms of varieties send constructible sets to constructible sets; in particular, φ(X) is constructible.
Proposition: upper semicontinuity of local fibre dimension
For a dominant morphism, the locus where the maximal dimension of a fibre component through x is at least n is closed.
Theorem: criterion for a morphism to be open
A dominant morphism whose inverse images of every irreducible closed subset have all components of the expected dimension is open.
Theorem: dominant injective morphisms and purely inseparable extensions
A dominant injective morphism of irreducible varieties induces a finite purely inseparable extension of function fields; in characteristic zero it is birational.
Proposition: birational morphisms are isomorphisms on dense opens
A birational morphism between irreducible varieties restricts to an isomorphism over a nonempty open subset of the target.
Definition: geometric Zariski tangent space
For X⊆A^n and x∈X, Tan(X)_x is the common zero space of the differentials d_xf of functions vanishing on X.
Definition/result: intrinsic tangent space
The tangent space T(X)_x is dual to m_x/m_x^2 and canonically identifies with the geometric tangent space; tangent spaces behave compatibly under affine embeddings.
Proposition: tangent space of a product
T(X×Y)_(x,y)≅T(X)_x⊕T(Y)_y.
Definition: simple point and smooth variety
x is simple when dim T(X)_x=dim X; X is smooth/nonsingular when every point is simple.
Theorem: existence and openness of simple points
For irreducible X, dim T(X)_x≥dim X everywhere, with equality on a dense open subset.
Lemma: generators of maximal ideal via m/m²
For a noetherian local ring, elements generate its maximal ideal iff their classes generate m/m².
Theorem: local ring of a simple point is regular and integrally closed
At a simple point of an irreducible variety the local ring is regular, hence an integrally closed domain.
Theorem: pole locus through a point
If O_x is integrally closed and f∉O_x, there is an irreducible codimension-one subvariety through x on which 1/f vanishes wherever defined.
Definition/result: differential of a morphism
A morphism φ:X→Y induces a K-linear map dφ_x:T(X)_x→T(Y)_φ(x), functorial under composition and identity.
Examples: differentials on affine space and GLn
The source computes differentials from coordinate Jacobians and identifies tangent spaces of matrix groups with matrix spaces.
Definition: separable morphism
A dominant morphism of irreducible varieties is separable when the induced function-field extension is separable.
Theorem: differential criterion for separability
If x and φ(x) are simple and dφ_x is surjective for a dominant morphism, then φ is separable.
Recorded result: generic converse to differential separability criterion
The Notes record the converse: for a separable dominant morphism, dφ_x is surjective on a nonempty open set of simple points.
Definition: complete variety
X is complete when projection X×Y→Y is closed for every variety Y.
Proposition: closure properties of complete varieties
Closed subvarieties of complete varieties and products of complete varieties are complete; morphism images of complete varieties are closed and complete; complete subvarieties are closed.
Theorem: projective varieties are complete
Every projective variety is complete.
Theorem: smooth curves dominated by P¹
If X is a smooth curve and P¹→X is dominant, then X≅P¹.
Theorem: automorphism group of P¹
Aut(P¹)=PGL(2,K), via fractional linear transformations.
Result: regular functions on complete irreducible varieties
A morphism from a complete irreducible variety to an affine variety has closed image; in particular global regular functions on complete irreducible varieties are severely constrained, and on projective space are constant.