Please use this identifier to cite or link to this item: http://hdl.handle.net/123456789/1650
Title: Affine Group Schemes
Authors: Lakshmi, R.
Srinivasan, V.R.
Keywords: Affine Varieties and the Zariski Topology
Group Functors
Hopf Algebras
Separable Algebras
Issue Date: Apr-2020
Publisher: IISERM
Abstract: Algebraic geometry is the study of geometric entities through the language of algebra by codifying structures in terms of roots of equations. In this thesis I explore the geometry that corresponds with roots of families of polynomials that form a group under some operation. The relationship between the affine varieties and the polynomials can be extended to a more fundamental relationship between affine group schemes and Hopf algebras. In this thesis I first establish this relationship through the concept of representable functors, and then the reverse relationship via co-algebras. Then, I define comodules, and use this definition to arrive at important finiteness theorems of affine group schemes. Then, I use the concept of separability, and via group action of the Galois group, I prove that separable algebras correspond to finite groups on which the Galois group acts continuously. Lastly, I study matrix groups that correspond to affine group schemes and arrive at results about diagonalisable groups, tori and automorphism groups.
URI: http://hdl.handle.net/123456789/1650
Appears in Collections:MS-15

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