MTH437 An introduction to schemes

[Credits:4, Lectures:3, Tutorials:1, Laboratory:0]

Course Outline

• Hilbert’s zero theorem, localization, Spectrum of a ring Spec(R): the underlying topological space, Finite spectrum, reducedness and irreducibility, Spectrum of a ring: structure sheaf OX, morphisms of affine schemes, Kronecker’s big picture (about 7 lectures)
• Ringed and locally ringed spaces, the notion of a scheme, morphism of schemes, example of Absolute Frobenius, Gluing : example of non-separated scheme and projective scheme, Classical varieties as schemes: Projective space, Elliptic curves, group schemes (about 5 lectures)
• The notion of subschemes, primary decomposition and closed subchemes, Schematic image ( 3 lectures)
• Finiteness conditions: noetherian schemes, generic points of noetherian schemes, associated points of a noetherian scheme, schemes of finite type over a field, Separated schemes, morphisms of finite type, proper morphisms, complete varieties, finite morphisms (8 lectures)
• Presheaves, separated presheaves, sheaves, dagger construction †, Sheafification functor as left adjoint to inclusion functor of sheaves in presheaves, ideal subsheaves, homogenous ideal of a closed subset of ℙAn, Quasi-coherent sheaves, the tilde construction M˜, Operations on quasi-coherent sheaves: direct image, pull-back, adjunction between f* and f*, tensor products, Hom, projection formula, Coherent sheaves (7 lectures)
• Yoneda Lemma, representable functors, Fiber Product construction, Relative frobenius, The functor of points: projective spaces, grassmannians, Examples of relative constructions: Geometric vector bundle, Defining schemes as functors and morphisms between them as natural transformations (7 lectures)
• The functor Proj(R): Invertible sheaves and twists, Quasi-coherent sheaves on Proj(R), ℙAn is proper, Constructible sets and Chevalley’s theorem (4 lectures)

Recommended Reading

• William Fulton: Algebraic Curves, http://www.math.lsa.umich.edu/ wfulton/
• C.Musili: Algebraic Geometry for beginners, Texts and Readings in Mathematics 20, Hindustan Book Agency.
• Joe Harris: Algebraic Geometry - A first course Graduate Texts in Mathematics, Springer Verlag.
• David Mumford: Red books of varieties and schemes, Lecture notes in Mathematics, Springer.
• David Mumford, Tadao Oda: Algebraic Geometry II, Texts and Readings in Mathematics 73, Hindustan Book Agency.
• Igor Shafarevich: Varieties in Projective Space I, Springer Verlag 1994