Università di Pavia - Offerta formativa

HIGHER ALGEBRA

Enrollment year

2018/2019

Academic year

2019/2020

Regulations

DM270

Academic discipline

MAT/02 (ALGEBRA)

Department

DEPARTMENT OF PHYSICS

Course

Curriculum

Fisica teorica

Year of study

2°

Period

1st semester (30/09/2019 - 17/01/2020)

ECTS

6

Lesson hours

48 lesson hours

Language

Italian

Activity type

ORAL TEST

Teacher

CANONACO ALBERTO (titolare) - 6 ECTS

Prerequisites

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

Learning outcomes

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

Course 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 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.

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.

Teaching methods

Lectures

Reccomended or required readings

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.

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.

Assessment methods

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

Further information

Sustainable development goals - Agenda 2030