Algebraic Geometry

Job: geometric study of solution sets of polynomial equations via schemes, sheaves, and cohomology — the language of arithmetic geometry and moduli.

Core

Classical varieties over algebraically closed fields become a special case of schemes (glued spectra of rings). Morphisms, fiber products, and coherent cohomology replace point-set topology and de Rham cohomology as the primary toolkit.

Layer Content
Varieties / \(\operatorname{Spec}\) affine sets, Zariski topology, structure sheaf sketch
Schemes gluing, ringed spaces, fiber products, \(\operatorname{Proj}\)
Morphisms proper, separated, flat, smooth — one-liners that matter
Cohomology coherent sheaves, Čech / derived, Serre duality, RR

Reading map

Page Focus
Foundations varieties vs schemes, \(\operatorname{Spec}\), Zariski, morphisms, \(\mathcal{O}_X\)
Schemes affine/gluing, (locally) ringed spaces, fiber products, Proj, morphism types
Cohomology coherent sheaves, \(H^i\), Serre duality pointer, classical RR

Dependency order

  1. Commutative algebra: localization, Spec intuition, Noetherian, Nakayama — commutative algebra, rings, modules.
  2. Sheaves as a categorical/topological gadget — category theory, topology.
  3. Foundations \(\to\) schemes \(\to\) cohomology; then special topics (curves, surfaces, stacks) off-map.

Interfaces

Area Contact
Abstract algebra \(R\mapsto\operatorname{Spec} R\); modules \(\leftrightarrow\) quasi-coherent sheaves
Homological algebra derived functors for \(H^i(X,-)\); Ext sheaves
Category theory \(\mathbf{Sch}\), fiber products as limits, Yoneda for moduli
Algebraic topology comparison theorems (étale, singular); not the same \(H^i\)
Representation theory geometric representation theory, flag varieties
Prismatic cohomology p-adic cohomology unifying crystalline / de Rham / étale

Notation (paper-standard)

  • \(\operatorname{Spec} R\), \(\operatorname{Proj} S\), \(X\times_S Y\), \(f_*\), \(f^*\), \(\mathcal{O}_X\), \(\widetilde{M}\)
  • \(\operatorname{QCoh}(X)\), \(\operatorname{Coh}(X)\), \(H^i(X,\mathcal{F})\), \(\chi(\mathcal{F})\)
  • étale / fppf / Zariski topologies as sites (pointer only here)

See it

Arithmetic of elliptic curves. An elliptic curve \(E/\mathbb{Q}\) is a scheme (smooth proper curve of genus 1 with section). Mordell–Weil, \(L\)-functions, and modularity are statements about morphisms, cohomology of \(\ell\)-adic sheaves, and moduli stacks — all scheme language. Even classical Weierstrass models are best read as \(\operatorname{Proj}\) of graded rings with open affines \(\operatorname{Spec}\mathbb{Z}[x,y]/(y^2-x^3-ax-b)\) after base change.

Check

Why is the Zariski topology on \(\mathbb{A}^1_k\) not Hausdorff for \(k\) infinite alg. closed?

Closed sets are finite (zeros of polynomials) or all of \(\mathbb{A}^1\). Any two nonempty opens meet (complements of finite sets), so distinct points cannot be separated by disjoint opens.

In one sentence: what extra data does a scheme carry beyond a topological space?

A sheaf of rings \(\mathcal{O}_X\) (with local stalks) so that open sets look like \(\operatorname{Spec} A\) with \(\mathcal{O}\) recovering \(A\) on affines — geometry of rings, not just closed sets of zeros.

Pitfalls

  • “Points of \(\operatorname{Spec} R\)” include primes, not just maximals; generic points and specialization are essential.
  • Varieties \(\neq\) schemes: nilpotents, non-reduced structure, arithmetic base schemes \(\operatorname{Spec}\mathbb{Z}\) have no classical-variety avatar.
  • Sheaf cohomology \(\neq\) singular cohomology of the complex analytic space without comparison theorems (GAGA, etc.).

Refs

  • Hartshorne, Algebraic Geometry
  • The Stacks Project (online)
  • EGA / Vakil, The Rising Sea (foundations)
  • Mumford, The Red Book of Varieties and Schemes

Next

Foundations · Schemes · Cohomology · Commutative algebra · Rings and fields · Modules · Homological algebra · Category theory