The aim of the course is to provide an introduction to homological algebra. The students are expected to obtain a good understanding (both theoretical and practical) of some homological tools commonly used in modern algebra, like abelian categories and derived functors.
Course Prerequisites
The contents of the courses: Algebra 1, Algebra 2, Linear Algebra and Geometry 1.
Teaching Methods
Lectures and exercise sessions
Assessment Methods
The exam consists of an oral examination which aims to evaluate the understanding of the theoretical topics and the ability to use them in concrete examples.
Texts
P. Aluffi, "Algebra: chapter 0", Graduate Studies in Mathematics 104, American Mathematical Society, 2009. S. Bosch, "Algebraic Geometry and Commutative Algebra", Universitext, Springer, 2013. R. Godement, "Topologie algébrique et théorie des faisceaux", Hermann, 1973 P.J. Hilton, U. Stammbach, "A Course in Homological Algebra", second edition, Graduate Texts in Mathematics 4, Springer-Verlag, 1997. S. Mac Lane, "Categories for the Working Mathematician", second edition, Graduate Texts in Mathematics 5, Springer-Verlag, 1998. M.S. Osborne, "Basic Homological Algebra", Graduate Texts in Mathematics 196, Springer-Verlag, 2000. C.A. Weibel, "An Introduction to Homological Algebra", Cambridge University Press, 1994.
Contents
(Left or right) modules over a (noncommutative) ring; bimodules; operations on modules; tensor product of modules. Categories, functors and natural transformations; (co)limits in a category; adjoint functors. (Pre)additive categories and (pre)abelian categories; (left and/or right) exact functors. Injective and projective objects in an abelian category; resolutions; derived functors. Injective, projective and flat modules; Ext and Tor functors; dimension theory for modules and rings. Cohomology of groups. Sheaves on a topological space and cohomology of sheaves.
Course Language
Italian
More information
Students in the categories identified by the project on innovative teaching will have the opportunity to have receptions also electronically and by appointment at times to be agreed with the teacher, or to view the recordings of lessons from previous years.
