Кто читает:: Факультет математики

Статус:: Дисциплина общефакультетского пула

Когда читается:: 1, 2 модуль

Преподаватель

Павлов Александр Борисович

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Abstract

This course offers an introduction to the theory of algebraic spaces and stacks, a cornerstone of modern algebraic geometry. Born from the study of moduli spaces and the need to construct quotients by group actions, these concepts were formalized in the foundational work of Deligne, Mumford, and Artin. The first part of the course that covers Grothendieck topologies, fibered categories, descent and stacks is based only on the category theory. But algebraic spaces/stacks requires working knowledge of algebraic geometry (e.g. Hartshorn’s textbook). One of the goals of the course is to include as many examples as time permits, in particular applications to constructions of various moduli spaces.

Learning Objectives

To develop a systematic understanding of the foundations of modern algebraic geometry related to Grothendieck topologies, sheaves, fibered categories, descent theory, algebraic spaces, and algebraic stacks.

Expected Learning Outcomes

To know the definition of a Grothendieck topology and a site.
To understand how Grothendieck topologies generalize ordinary topological coverings.
To know the definitions of presheaves and sheaves on a site.
To understand the main examples of Grothendieck topologies used in algebraic geometry.
To be able to verify the sheaf condition in basic examples.
To know the definition of a category fibered over a base category.
To understand the idea of objects and morphisms varying over a base.
To know the role of pullbacks and cartesian morphisms.
To understand the relation between fibered categories and pseudofunctors.
To be able to recognize moduli problems naturally described by fibered categories.
To understand the general principle of descent and local-to-global reconstruction.
To know the definition of descent data for objects and morphisms.
To understand the notions of effective descent and descent for morphisms.
To know the definition of a stack and the stack condition.
To be able to check the stack condition in standard examples.
To know the definition of an algebraic space.
To understand why algebraic spaces generalize schemes.
To know the role of étale coverings in the definition of algebraic spaces.
To understand how algebraic spaces arise from quotient constructions.
To be able to distinguish between schemes and algebraic spaces in basic examples.
To know the definition of an algebraic stack.
To understand the role of automorphisms in the theory of stacks.
To know the notions of atlas and representable morphism.
To understand the difference between Deligne–Mumford stacks and Artin stacks.
To be able to recognize standard examples of algebraic stacks and quotient stacks.

Course Contents

Sheaves in Grothendieck topologies
Fibered categories.
Descent and the stack condition.
Algebraic spaces
Algebraic stacks

Assessment Elements

Assignment
10 problems
Final Exam
Final exam

Interim Assessment

2026/2027 2nd module
0.6 * Assignment + 0.4 * Final Exam

Bibliography

Recommended Core Bibliography

Algebraic spaces, Knutson, D., 1971
Edidin, D. (2010). Equivariant geometry and the cohomology of the moduli space of curves. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1006.2364
Fundamental algebraic geometry : Grothendieck's FGA explained, Fantechi, B., 2006
Martin Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces, Theoretical and Mathematical Physics (Springer, Berlin Heidelberg 2007) DOI 10.1007/b11501497

Recommended Additional Bibliography

Categories and sheaves, Kashiwara, M., 2006

Authors

Pavlov Aleksandr Borisovich

Бакалаврская программа «Математика»

Introduction to Algebraic Spaces and Stacks