Contents of the course
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