SYSTEMATIC MATHEMATICS

Algebraic Geometry

How can polynomial equations become spaces, how do functions and local rings reveal structure that a plot cannot show, why are projective points needed at infinity, and how do schemes keep nilpotents, residue fields, and base change honest? This course answers those questions in fifty-nine visible bilingual chapters across eleven content-sized units of lengths 6, 5, 6, 6, 5, 6, 5, 6, 5, 5, and 4. It develops affine algebraic sets and coordinate rings, morphisms and rational maps, projective space and projective varieties, schemes and the functor of points, sheaves and quasi-coherent modules, fiber products and base change, separated, proper, and projective morphisms, divisors, line bundles and Picard groups, differentials, tangent spaces, smooth and étale maps, ramification, curves and Riemann–Roch, sheaf cohomology and projective tools, birational geometry, blowups, singularity boundaries, and a complete worked dossier. Every base field or base scheme, topology, point convention, ring-map direction, locality claim, finite-type hypothesis, and geometric-versus-computational boundary is named. The course introduces only the category and homological language it actually uses; it does not claim to replace full courses in category theory, homological algebra, complex or arithmetic geometry, intersection theory, étale cohomology, stacks and moduli, the minimal model program, or derived algebraic geometry.

Before this course: Completed Commutative Algebra, Abstract Algebra, Proof, Logic & Set Theory, and Linear Algebra. Students should know ideals and quotients, prime and maximal ideals, localization, modules, tensor products, Noetherian rings, integral dependence, primary decomposition, Krull dimension, local rings, Nakayama’s lemma, completion, regular local rings, polynomial rings, fields, vector spaces, bases, dual maps, equivalence relations, and proof by contradiction and induction. Point-set Topology is useful but not required; the needed topology is rebuilt. No category theory, homological algebra, scheme theory, complex analysis, or differential geometry is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Affine space and common zero sets turn equations into geometric objects

Objective: Why does V(xy)=V(x)∪V(y) need the assumption that k is a field or at least has no zero divisors?

Algebraic geometry begins with a deliberately simple question: which coordinate points make all the named polynomial equations equal to zero? The answer is a set of points. It may look like a curve or surface over the real numbers, but the definition works over any field and does not require distance, angle, or a drawing. Before calculating, name the base field, ambient space, coordinate variables, point set, defining ideal, and every map being used. The exact result to understand is: If (S) denotes the ideal generated by S, then V(S)=V((S)). Moreover V(S∪T)=V(S)∩V(T), while V(ST)=V(S)∪V(T), where ST={fg:f∈S,g∈T}. Thus adding equations intersects solution sets, whereas multiplying equations takes a union.

The new object means: Let k be a field. Affine n-space over k, written A^n_k and read “affine n-space over k,” is the set k^n of ordered n-tuples a=(a_1,…,a_n). For S⊆k[x_1,…,x_n], define V(S)={a∈A^n_k:f(a)=0 for every f∈S}; V(S) is read “the common zero set of S.” An affine algebraic set is any subset of A^n_k equal to V(S) for some S. Read each displayed formula as a sentence about points, functions, or neighborhoods; do not manipulate an unnamed symbol merely because it resembles earlier algebra.

Rebuild the worked problem “In A^2_k with coordinates x and y, determine V(xy), prove it is the union of the two coordinate axes, and compare it with V(x,y).” from the definitions. Check both directions of every set equality, identify the corresponding ring map, and keep this failure boundary visible: A^n_k is a set of k-valued coordinate tuples, not Euclidean n-space unless k=R and one deliberately adds Euclidean structure. Over a finite field it has finitely many points, and over C a single equation can describe four real dimensions rather than an ordinary plotted surface.

If (S) denotes the ideal generated by S, then V(S)=V((S)). Moreover V(S∪T)=V(S)∩V(T), while V(ST)=V(S)∪V(T), where ST={fg:f∈S,g∈T}. Thus adding equations intersects solution sets, whereas multiplying equations takes a union.

Every element h of (S) is a finite sum h=Σr_if_i with f_i∈S. If a makes every f_i zero, then h(a)=Σr_i(a)f_i(a)=0; hence V(S)⊆V((S)). The reverse inclusion holds because S⊆(S).

A point lies in V(S∪T) exactly when it kills every polynomial in S and every polynomial in T, which says exactly that it lies in both V(S) and V(T).

If every product fg vanishes at a, then either all f∈S vanish at a or all g∈T vanish at a: otherwise choose f(a)≠0 and g(a)≠0, whose product is nonzero in the field k. This proves V(ST)=V(S)∪V(T).

In A^2_k with coordinates x and y, determine V(xy), prove it is the union of the two coordinate axes, and compare it with V(x,y).

  1. A point (a,b) lies in V(xy) exactly when the evaluated product ab equals 0 in k. Because a field has no nonzero zero divisors, ab=0 means a=0 or b=0.
  2. The condition a=0 describes V(x), the y-axis, and b=0 describes V(y), the x-axis. Therefore V(xy)=V(x)∪V(y).
  3. By contrast, V(x,y)=V(x)∩V(y) requires a=0 and b=0 simultaneously, so it is only the origin {(0,0)}.

Result: V(xy)=V(x)∪V(y) is the union of the two coordinate axes, whereas V(x,y)={(0,0)} is their intersection.