The Course topics will be sheaf theory (on topological and ringed spaces), homological algebra in abelian categories, derived functors, sheaf cohomology and applications.
- B. R. Tennison, Sheaf theory, LMS
- B. Iversen, Cohomology of sheaves, Springer Verlag
- M. Kashiwara, P. Schapira, Sheaves on manifolds, Springer Verlag
- C. Weibel, an introduction to homological algebra, CUP
- Stacks Project: https://stacks.math.columbia.edu
Learning Objectives
Knowledge acquired: Basics of category theory, pre sheaves and sheaves on a topological space, ringed space and modules over them, basics of sheaf cohomology and homological algebra Competence acquired: Basic tools in sheaf cohomology and homological algebra. Skills acquired (at the end of the course): Ability to apply understanding and knowledge of basics of cohomology to geometry and analysis.
Prerequisites
Courses to be used as requirements (required and/or recommended) Courses required: none Courses recommended: any course in basic general topology and algebra.
Teaching Methods
CFU: 9 Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 72 Hours reserved to private study and other indivual formative activities: 102 Hours for lectures: 72 Hours for laboratory: 0 Hours for laboratory-field/practice: 0 Seminars (hours): 0 Stages (hours): 0 Intermediate examinations (hours): 0
Further information
Attendance of lectures, practice and lab: Not mandatory Teaching tools: Handwritten Notes by the teacher. Office hours: To be announced, upon request by e-mail. Contact: E-mail: gabriele.vezzosi@unifi.it luigi.verdiani@unifi.it Web: http://www.dma.unifi.it/~vezzosi/
Type of Assessment
Oral exam consisting of a short lecture on course's topics or nearby ones (to be assigned specifically to the student), together with a couple of exercises from course's topics.
Both the exercises and a list of seminar topics are available at the teacher's personal webpage.
Course program
- Basic category theory - Presehaves and sheaves. definition and basic operations; ringed spaces. Examples - Homological algebra in an abelian category. Derived functors. - Cohomology of sheaves. - Definition of Cech cohomology. Comparison with sheaf cohomology. De Rham theorem. Examples and computations.