Exercise Sessions in Algebra 1
Tuesdays 10-12 (Room 0.007), Summer term 2023, Uni Bonn
Solutions to the exercise sheets
Warning: These are non-official, there might be errors. If you find errors, feel free to write me :)
- Sheet 1 Calculating Nilradicals, Jaobson radicals and Units of a few rings; The Chinese remainder theorem; Idempotents.
- Sheet 2 Algebraic rings of integers, PIDs, minimal primes.
- Sheet 3 Prime ideals of polynomial rings of PIDs, a first version of Hilbert’s Nullstellensatz, Krull dimension of polynomial rings, localization and saturation.
- Sheet 4 The ring of power series over noetherian rings is noetherian, holomorphic functions, stuff with modules.
- Sheet 5 Noetherian rings, elementary divisors, universal properties. (Corrected an error!).
- Sheet 6 More on tensor products.
- Sheet 7 Hom-Tensor adjunction and other tensor identites, Support of modules, localization and Hom, Example for a non-flat ideal.
- Sheet 8 Finitely presented algebras, integral extensions, finite morphisms have finite fibers on spectra, no proof of the 5-Lemma.
- Sheet 9 Random number theory exercise, more fibers on spectra (and remarks on how to calculate and think about them), invariants of a group action, finitely presentedness does not depent on the choice of generators.
- Sheet 10 something something finite $k$-algebras, algebraic subsets, Jacobson rings, Finitely presented over local rings implies free (also some Tor).
- Sheet 11 Krull dimension of tensor products of $k$-algebras,normalization of the cusp, minimal ideals and irreducible components.
- Sheet 12 Integrality of invariants under finite group action, normality of the quadratic cone, Galois actions on the ring of integers of number fields, normalization of the nodal curve.
Here are all the solutions in a single file.
Thank you for the fun exercise sessions, I learned a lot. And good luck with the exams!
Max von Consbruch, email: s6mavonc(at)uni-bonn(dot)de