Algebraic Geometry Seminar 

Date 
27 Feb Tuesday, 11.30 am 
Venue 
Room 105 
Host 
Sudarshan Gurjar 
speaker 
Nitin Nitsure 
Affiliation 
Bhaskaracharya Pratishthana 
Title 
Galois descent in topology, algebra, and geometry. 
Abstract: We will begin by introducing Galois descent, and giving diverse examples of effective Galois descent to produce `twisted forms' in topology, algebra and geometry: (1) The Mobius band is a twisted form of a cylinder. (2) The division algebra H of Hamilton quaternions is a twisted form of the 2 x 2 matrix algebra M over real numbers R. (3) The real algebraic groups SO(2,R) and SU(2) are twisted forms of GL(1,R) and SL(2,R) respectively. (4) BrauerSeveri varieties are twisted forms of projective spaces. We will then connect the questions of twisted forms and effective Galois descent to the 1st cohomology set of the Galois group. After this, we will turn to the general problem of effective Galois descent for schemes (of which (3) and (4) are examples). We will show that: (A) Under a certain condition, effective Galois descent holds for schemes. (B) But more generally, effective Galois descent always holds for algebraic spaces. (C) The `nonseparated affine line' over R has as a twisted form an algebraic space X over R that is not a scheme, which shows that Galois descent (and so etale descent or flat descent) is not always effective for schemes. 