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) Brauer-Severi 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 `non-separated 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.
We will show how specialization of points can look very different in this algebraic space X, which illustrates the need for modifying the valuative criterion for universal closedness of morphisms when we wish to generalize from schemes to algebraic spaces. Further generalization to algebraic stacks needs a further modification in the valuative criteria, to be  addressed later.