Professor: Geoffroy Horel

Website of the course

The lectures (by Geoffroy Horel) will take place on Mondays 8h30 - 10h30, Université Paris Cité, salle 378 F Halle aux Farines, and Wednesdays 8h30 - 10h30, Université Paris Cité, salle 1009 Sophie Germain.

The TDs (by me) take place on Fridays 9h - 11h, Université Paris Cité, salle 1021 Sophie Germain.

TD sheet A (updated on January 23rd, 2024)
TD sheet B (updated on January 31st, 2024)
TD sheet C (updated on February 6th, 2024)

TD sessions (click to expand):

A good reference for exercises: the book Modern Classical Homotopy Theory by Jeffrey Strom.
A good reference for theory and exercises: Notes on homotopical algebra by Emanuele Pavia.

• Recall: Endomorphisms of a vector space over a field.
• Recall: Notion of $k$-algebra. Mentions: euclidean, principal, factorial ring (without definitions).
• Recall: Prime elements and irreducible elements in a commutative unital ring.
• Example: Irreducible elements in C[x].
• Exercise: Irreducible elements in R[x].

• Recall: $k$-algebra generated by an endomorphism, minimal polynomial of an endomorphism.
• Exercise 4.3 of Chapter 1 of the polycopie.

• Recall: lemma of the kernels (lemme des noyaux).
• Exercises 4.1, 4.2 of Chapter 1 of the polycopie.
• Exercise 4.4 points 1,2 of Chapter 1 of the polycopie.

• Cyclic subspaces and cyclic endomorphisms.
• Cayley-Hamilton Theorem.
• Statement of the Frobenius decomposition theorem (Theorem 3.5 of the polycopie).
• Similar endomorphisms.
• Euclidean rings.
• Reduction algorithm (page 25 in the polycopie).

• Exercises from Chapter 3, Section 5.