MAT 663 Advanced Topics in Algebra, Spring 2004

Sheaves on manifolds: introduction to derived categories

Academic Calendar.

Important: I shall be absent due to professional meetings on TU FEB 24, TH Feb 26, TU Mar 30, TU Apr 27, TH Apr 29. Please check the web page before those dates in case, my trips are canceled. In that case the classes WILL TAKE PLACE.

Lectures: TU and TH 9:50am - 11:10 am, Physics P-123.

Instructor: Mark Andrea de Cataldo. Office hours: by appointment.

Tentative Syllabus. I use derived categories frequently, but have not yet understood them the way I would like to. In this course I plan to explain what I hope to understand in an accessible way. I shall use several sources, such as: Intersection cohomology (A. Borel), Cohomology of sheaves (Iversen), Sheaves on manifolds (Kashiwara-Shapira), Homological algebra (Gelfand-Manin, Cartan-Eilenberg) ... Topics to be discussed: - Categories of complexes: the cone construction, homotopy, localization, derived categories, derived functors. - Sheaves: the formalism of the six functors and Poincare'-Verdier Duality. - (If time allows) Perverse sheaves on complex manifolds: intersection cohomology.

Syllabus. Lecture 1: Categories and functors. Leture 2: Mono, epi, zero. Lecture 3,4: Sheaves. Lecture 5: (f^*,f_*), (pre-)additive. Lecture 6: Representable. Abelian. Lecture 7: Homotopy category of complexes. Lecture 8: Cones. Lecture 9: Triangulated categories. Lecture 10: Localization. Lecture 11: null systems; derived categories. Lecture 12: Injective objects; equivalence K^+(I)=D^+(A); further description of morphisms. Lecture 13: Derived functors; some formulae. Lecture:14-20: More formulae. Lecture 21: Verdier Dality. Lecture 22-23: Example: contraction of curves on surfaces. No time for perverse sheaves.

Grade: based on in-class participation. The will be no homework and no final exam.

Special needs. If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. All information and documentation of disability is confidential. Arrangements should be made early in the semester (before the first exam) so that your needs can be accommodated.

  • Mark Andrea de Cataldo's homepage.
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