## Introduction to schemes

### Prof. Barbara Fantechi

Number of cycles: 2

Second semester: Starting in March

The course will cover chapter 2 and part of chapter 3 of Hartshorne's Algebraic Geometry. The focus will not be on the proofs but on motivating the definitions, providing concrete examples and counterexamples for theorems, working through the most important exercises and developing an intuition for the material; we will not follow the book closely, but occasionally refer to proofs in it. Having a background knowledge in classical algebraic geometry (quasi-projective varieties) will be useful but not strictly necessary.

Contents

• Sheaves on a topological space.
• Locally ringed spaces.
• Affine schemes and their morphisms.
• Schemes and their morphisms; closed and open subschemes.
• Fiber products of schemes and tensor products of rings.
• Reduced schemes, reduction, universal properties.
• Separatedness and properness.
• Irreducibility and dimension theory.
• Coherent and quasicoherent sheaves.
• Spec of a sheaf of algebras, proj of a sheaf of graded algebras.
• Projective morphisms.
• Cotangent sheaf and its properties.
• Smooth schemes and morphisms; universal properties.
• Informal introduction to abelian categories and derived functors.
• Injective sheaves and how to use them without knowing any example.
• Higher push-forwards of abelian sheaves, Leray spectral sequence.
• Higher push-forwards of coherent sheaves via projective morphisms.
• Ext groups, local-to-global spectral sequence.
• Serre vanishing theorem, Serre duality.
Moreover, these topics will be treated or not depending on the students' interests and background:
• Castelnuovo-Mumford regularity and the Hilbert polynomial.
• Flatness of morphisms.
• Statement of the theorem of cohomology and base change, and examples of applications.
• Hilbert and Quot schemes.