2019/2020

2019/2020

DM270

MAT/02 (ALGEBRA)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

1st semester (30/09/2019 - 10/01/2020)

6

48 lesson hours

Italian

ORAL TEST

CANONACO ALBERTO (titolare) - 6 ECTS

The contents of the courses: Algebra 1, Linear Algebra and Geometry 1.

The aim of the course is to provide an introduction to homological algebra.

(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 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.

Lectures

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.

The exam consists of an oral examination, during which the student must answer some questions, mainly of a theoretical nature.