ALGEBRAIC GEOMETRY SEMINAR @SISSA - Thursday 20 February, at 16:30 (Hayato Morimura)

[ICTP Math Section]

ALGEBRAIC   GEOMETRY   SEMINAR


Venue: SISSA, room 133


Thursday 20 February, at 16:30 hrs.


Speaker: Hayato Morimura (SISSA)


Title: Introduction to stacks


Abstract: The goal of this talk is to construct the quotient stack by a 
smooth group scheme and to see that it is an Artin stack. The talk will 
be divided into four parts. First, we define the algebraic space over a 
fixed scheme S. Algebraic spaces are sheaves on the big étale site Sch/S 
with the representable diagonal morphism and an étale covering. Then, we 
give the original definition of the stack. Stacks are categories fibered 
in groupoid over a site for which every covering is of effective descent.
We also provide a necessary and sufficient condition for a category 
fibered in groupoid to be a stack. Third, we define the Artin stack. 
Artin stacks are stacks over the big étale site Sch/S with the 
representable diagonal morphism and an smooth covering. Here, algebraic 
spaces play a role to describe property of morphisms of stacks. Finally, 
given a smooth group scheme G acting on a scheme X over S, we construct 
the quotient [X/G] and check that [X/G] satisfies the all conditions to 
be a stack, moreover an Artin stack. If time allows, we will provide 
some examples of quotient stacks.

For further information about the seminar please visit:
http://users.ictp.it/~arincon/AGS.html