Lecture course, winter term 2026/27
This course continues the lecture Commutative Algebra. We begin with affine algebraic varieties and the Zariski topology and then develop the language and geometry of schemes, sheaves, projective geometry, and vector bundles. Categorical concepts are introduced or recalled as needed.
Time and place
- Lectures: Tuesdays and Wednesdays, 16:00–18:00, M3 (Einsteinstraße 64).
- Language: English.
Course materials
RECAP sheets
The weekly RECAP sheets will be posted here as the course progresses.
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 easy examples, counterexamples, and variations. Think of examples yourself.
- Connect: understand how concepts and results relate to, imply, and depend on one another, and use these connections to develop or reconstruct proofs and arguments.
- Apply: apply the material to new problems. This is closest to traditional exercise problems.
- Present: explain the mathematics in your own words and respond to questions.
The tasks on the sheets help students prepare for this mathematical conversation; the Present stage itself takes place in the tutorial. Students meet with a tutor in groups of four to explain ideas and arguments, describe how different concepts and results are related, and respond to follow-up questions.
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
- Affine algebraic varieties and the Zariski topology
- Spectra of rings, sheaves, and locally ringed spaces
- Schemes, morphisms, and fibre products
- Projective schemes and the Proj construction
- Quasi-coherent sheaves, line bundles, and vector bundles
Motivation
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
The following references complement the lecture notes and provide different perspectives on the material; they are not intended as a required reading list.
- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1969.
- A. Grothendieck, with the collaboration of J. Dieudonné, Éléments de géométrie algébrique, I–IV, Publications Mathématiques de l’IHÉS 4, 8, 11, 17, 20, 24, 28, 32, 1960–1967. Part I
- U. Görtz and T. Wedhorn, Algebraic Geometry. Part I: Schemes. With Examples and Exercises, Advanced Lectures in Mathematics, Vieweg+Teubner, 2010. DOI
- R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, 1977. DOI
- D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Mathematics 1358, Springer-Verlag, 1988; 2nd expanded ed., 1999. First edition; second edition
- P. Scholze, Algebraic Geometry I, University of Bonn, winter term 2016/17, lecture notes by J. Davies. Lecture notes
- P. Scholze, Algebraic Geometry II, University of Bonn, summer term 2017, lecture notes by J. Davies.
- The Stacks Project Authors, The Stacks Project, 2026.
← Back to teaching