MATH 536A: Algebraic Geometry (Fall 2016)

Essential course information is contained in the syllabus.
A summary is below.

Instructor

Jack Hall
email: firstnamelastname at math dot arizona dot edu
office: ENR2 S349

Text

The Rising Sea: Foundations Of Algebraic Geometry Notes by Ravi Vakil.

Class

when: Tuesday and Thursday, 9:30am-10:45am
where: Math Teaching Lab 124

Office hours

Monday, 12:30-2pm; Thursday, 1:30-3pm; and by appointment.

Homework

Due 8/30/2016: all from Vakil, 1.2.B, 1.2.C, 1.2.D 1.3.A, 1.3.B, 1.3.C, 1.3.F, 1.3.Y, 1.3.Z, 1.4.A, 1.4.B, 1.4.C, 1.4.E, 1.4.F, 1.5.E.
Due 9/6/2016: all from Vakil, 3.2.A, 3.2.C, 3.2.D, 3.2.E, 3.2.J, 3.2.K, 3.2.L, 3.2.O, 3.2.P, 3.2.T, 3.4.A, 3.4.B, 3.4.C, 3.4.D, 3.4.E, 3.5.B, 3.5.C.
Due 9/13/2016: all from Vakil, 3.6.F, 3.6.J, 3.6.K, 3.6.L, 3.6.M, 3.6.N, 3.6.S, 3.6.T, 3.6.V, 3.6.W, 3.6.X, 2.1.A, 2.2.B, 2.2.G, 2.2.I, 2.2.J, 2.3.C, 2.3.D, 2.3.E, 2.3.F, 2.3.I, 2.3.J.
Due 9/20/2016: all from Vakil, 2.4.A, 2.4.B, 2.4.E, 2.4.F, 2.4.N, 2.4.O, 2.4.P, 2.5.A, 2.5.B, 2.5.C, 2.5.F, 2.5.H, 2.5.I, 2.5.J, 2.6.C, 2.6.E, 2.6.F, 2.6.G, 2.7.A, 2.7.B, 2.7.C, 2.7.D, 4.1.A, 4.1.B, 4.1.C, 4.1.D.
Due 9/27/2016: all from Vakil, 4.4.A, 4.4.B, 4.4.D, 4.4.E, 4.4.F, 4.5.A, 4.5.B.
Due 10/4/2016: all from Vakil, 5.1.(A+C+D) [counts as one question], 5.1.B, 5.1.E, 5.1.F, 5.1.G, 5.1.I, 5.2.A, 5.2.B, 5.2.C, 5.2.E, 5.2.F, 5.2.G, 5.3.B, 5.3.C, 5.3.D, 5.3.E, 5.3.J.
Due 10/18/2016: all from Vakil, 5.4.A, 5.4.B, 5.4.H, 5.4.I, 5.4.J, 5.4.K, 5.4.L, 5.4.M, 5.4.N.
Due 10/25/2016: all from Vakil, 9.7.A, 9.7.B, 9.7.C, 9.7.D, 9.7.E, 9.7.F, 9.7.G, 9.7.H, 9.7.I, 9.7.J, 9.7.L
Due 11/1/2016: choose your own problems.
Due 11/8/2016: all from Vakil, 13.1.B-I.
Due 11/29/2016: all from Vakil, 10.1.M, 10.1.O, 10.1.P, 10.2.A, 10.2.B, 15.1.B, 15.1.C, 15.1.E.

Topics covered

8/23/2016: Categories, functors, natural transformation, Yoneda, localization of modules.
8/25/2016: Adjoints, tensor products, Sym, limits and colimits, additive and abelian categories.
8/30/2016: Exact sequences, rings, ideals, prime and maximal ideals, prime ideal spectrum, Zariski topology.
9/1/2016: Mumford's picture, extension and contraction of ideals, prime ideals in quotient and localizations, a nullstellensatz.
9/6/2016: Weak and strong nullstellensatz (over uncountable fields), Hilbert basis theorem, definitions and basic examples of presheaves and sheaves.
9/8/2016: Sheafification, pushforward, skyscraper sheaves, inverse image, and stalks.
9/13/2016: Stalks, sheaves on a base, (locally) ringed spaces, sheaves of modules, quasi-coherent and coherent sheaves.
9/15/2016: Pullback, definition of affine schemes, and examples.
9/20/2016: Maps to affine schemes from locally ringed spaces, definition of schemes, examples of non-affine schemes.
9/22/2016: Proj of a graded ring.
9/27/2016: Absolute properties of schemes and the affine communication lemma.
10/4/2016: Blow-ups.
10/6/2016: More blow-ups.
10/11/2016: Integral extensions and normal domains.
10/13/2016: Normalization.
10/18/2016: Noether normalization and applications.
10/20/2016: Nullstellensatz and fiber products.
10/25/2016: Existence of fiber products of schemes.
10/27/2016: The diagonal morphism, Quasi-compactness, quasi-separatedness.
11/1/2016: Quasi-coherent sheaves, pushforward and pullback.
11/3/2016: Closed immersions, immersions, schematic image.
11/8/2016: Separated, universally closed, and proper morphisms.
11/15/2016: Valuative criteria.
11/17/2016: Valuative criteria continued, quasi-coherent sheaves on Proj
11/22/2016: Ampleness.
11/29/2016: Krull dimension, codimension, and some pathologies.
12/1/2016: Dimension and transcendence degree.