I. Introduction

II. The spectrum of a ring

II.1 Definition, basic properties

  • The prime spectrum of a ring and the Zariski topology
  • Basic properties of the Zariski topology (principal open subsets, correspondence between radical ideals and closed subsets)
  • Irreducible (closed) subsets, irreducible components, generic points

Useful preparation: Recall the notion of topological space. Recall the notion of the prime spectrum of a ring (with the Zariski topology) from the course on commutative algebra. (We will go through he definitions and key results again.)

The notion of irreducible topological space was defined in the course on commutative algebra (and on Problem Sheet 1). Find some examples of (non-)irreducible topological spaces.

II.2 Spec as a functor

  • The continuous map $\mathop{\rm Spec} B\to\mathop{\rm Spec} A$ attached to a ring homomorphism $A\to B$
  • Special cases: $A \to A/\mathfrak a$, $A\to S^{-1}A$.
  • Examples

Useful preparation: Try to determine the space $\mathop{\rm Spec} A$ for some rings $A$. Find some ring homomorphisms such that the corresponding maps between their spectra are injective, surjective, bijective, homeomorphisms, …

III. Sheaves

III.1 Definition and simple properties

  • Presheaves, morphisms of presheaves
  • Sheaves, Examples
  • Inductive limits, stalks
  • The sheaf associated to a presheaf

Useful preparation: Although not strictly necessary, it may be useful to recall the notions of category and functor; they will make a brief appearance at this point of the course. We will need the notion of inductive limit (see Problem Sheets 2, 3).

III.2 Direct and inverse image of sheaves

III.3 Locally ringed spaces

IV. Schemes

IV.1 Affine Schemes

IV.2 Definition of scheme

IV.3 Morphisms

  • Morphisms into affine schemes
  • Morphisms from the spectrum of a field into a scheme

IV.4 Projective space

IV.5 Topological properties of schemes

IV.6 Subschemes

IV.7 Reduced and integral schemes

V. Fiber products, separated and proper morphisms

The functorial point of view

Fiber products of schemes

Separated morphisms

Proper morphisms