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A.Y. 2020 / 2021

First semester
Type of Learning Activity 
Related/additional subjects
Study Path 
[PDS0-2018 - Ord. 2018] common
Teaching language 


Learning objectives 

Knowledge and understanding: the students will know the basic theory of
schemes and their cohomology.
Ability to apply knowledge and understanding: the students will be able
to apply the theory to the solution of standard problems and exercises,
and to formulate and prove simple theoretical results.
Judgement autonomy: the students will have a clear idea of the basic
techniques and methods needed to work in modern algebraic geometry.
Communication skills: the students will master the standard terminology
in modern algebraic geometry and will be able to express in a clear
manner the basic results in the theory of schemes.
Learning skills: the students will be able to advantageously read the basic
literature about the theory of schemes and varieties, and will be able to
start approaching the advanced literature.


Basic notions in commutative and homological algebra, basic theory of
algebraic varieties. Some knowledge of differential geometry will help


The course is addressed to students who already had a basic course in
algebraic geometry, and aims at providing an introduction to the theory
of schemes. The section on cohomology will also include a basic
introduction to the theory of derived functors.

Teaching format 

The course will consist of 36 hours of theory and a minimum of 12 of
exercises, with traditional lectures at the blackboard, and an emphasis on
the interaction with the students.

Extended Programme 

1. Schemes: Introduction to sheaves. Schemes. Projective schemes.
Morphisms of schemes. Sheaves of modules. Divisors. Projective
morphisms. Differentials.
2. Cohomology: Derived functors. Sheaf cohomology. Ext groups and
sheaves. Higher direct images. Flat and smooth morphism. The
semicontinuity theorem.

End-of-course test 

A one hour seminar talk on a topic agreed with the student

Other information 



R. Hartshorne, Algebraic Geometry, Springer-Verlag, Chaps. II and III

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