Lecture course, winter term 2026/27
This course continues the lecture Commutative Algebra. After a short introduction to category theory, we will develop the theory of schemes, including central properties of schemes and morphisms of schemes, as well as quasi-coherent and locally free sheaves. This will lead to the construction of the Grassmannian, which parametrizes the d-dimensional linear subspaces of an n-dimensional vector space.
Time and place
- Lectures: Tuesdays and Wednesdays, 16:00–18:00, M3 (Einsteinstraße 64).
Course materials
Exercises and independent study
For this course, I have developed a format for weekly independent work and tutorials that I call RECAP. The sheets guide students through the first four stages and help them prepare for the final one:
- Recall: recall central definitions and results.
- Explore: investigate examples, counterexamples, and variations.
- Construct: develop or reconstruct proofs and arguments.
- Apply: apply the material to new problems.
- Present: in the tutorials, explain the mathematics in your own words and respond to questions.
The prompts on the sheets help students prepare for this mathematical conversation; the Present stage itself takes place in the tutorial.
Students may use AI tools while working on the RECAP sheets, and a course-specific AI tutor will be available. It is intended to support exploration, feedback, and preparation, not to replace independent understanding. In the tutorials, students are expected to explain and discuss the mathematics themselves.
Topics
- Category theory
- Affine schemes and schemes
- Properties of schemes and morphisms of schemes
- Quasi-coherent and locally free sheaves
- The Grassmannian
What is the idea of algebraic geometry?
Algebraic geometry begins with systems of polynomial equations, but its aim is not merely to list their solutions. It studies the geometry carried by those solutions and uses geometric intuition to answer algebraic questions—and conversely.
- Linear algebra studies systems of linear equations. Their solution sets form vector subspaces.
- Algebra studies polynomial equations and the arithmetic and algebraic structures encoded by them.
- Algebraic geometry studies the spaces defined by several polynomial equations and the interaction between their geometry and algebra.
For example, we think of x2 + y2 = 1 as a circle, not merely as a set of pairs. As in differential geometry, spaces arise locally as zero loci of functions. In algebraic geometry the functions are polynomial, and we allow singularities. The theory of schemes records the additional algebraic information needed to make this viewpoint work over general fields and rings.
Literature
Textbooks in algebraic geometry
- S. Bosch, Algebraic Geometry and Commutative Algebra, Springer, 2013.
A self-contained treatment of commutative algebra and the theory of schemes.
- D. Eisenbud and J. Harris, The Geometry of Schemes, Graduate Texts in Mathematics 197, Springer, 2000.
Detailed, illustrative, and rich in examples.
- U. Görtz and T. Wedhorn, Algebraic Geometry I: Schemes, Vieweg+Teubner, 2010.
A comprehensive introduction with many examples and substantial motivation.
- U. Görtz and T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Spektrum, 2023.
A comprehensive continuation devoted to cohomological methods.
- R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, 1977.
- Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford University Press, 2002.
More on commutative algebra
- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1969.
- D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer, 1995.
Category theory
- M. Brandenburg, Einführung in die Kategorientheorie, Springer Spektrum, 2nd ed., 2017.
- S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer, 2nd ed., 1998.
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