Berkeley MATH 274 - Lectures on Deformation Theory (169 pages)

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Lectures on Deformation Theory



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Lectures on Deformation Theory

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Lecture Notes


Pages:
169
School:
University of California, Berkeley
Course:
Math 274 - Topics in Algebra

Unformatted text preview:

Math 274 Lectures on Deformation Theory Robin Hartshorne c 2004 Preface My goal in these notes is to give an introduction to deformation theory by doing some basic constructions in careful detail in their simplest cases by explaining why people do things the way they do with examples and then giving some typical interesting applications The early sections of these notes are based on a course I gave in the Fall of 1979 Warning The present state of these notes is rough The notation and numbering systems are not consistent though I hope they are consistent within each separate section The cross references and references to the literature are largely missing Assumptions may vary from one section to another The safest way to read these notes would be as a loosely connected series of short essays on deformation theory The order of the sections is somewhat arbitrary because the material does not naturally fall into any linear order I will appreciate comments suggestions with particular reference to where I may have fallen into error or where the text is confusing or misleading Berkeley September 6 2004 i ii CONTENTS iii Contents Preface i Chapter 1 Getting Started 1 1 2 3 4 5 Introduction 1 Structures over the dual numbers 4 The T i functors 11 The infinitesimal lifting property 18 Deformation of rings 25 Chapter 2 Higher Order Deformations 33 6 7 8 9 Higher order deformations and obstruction theory Obstruction theory for a local ring Cohen Macaulay in codimension two Complete intersections and Gorenstein in codimension three 10 Obstructions to deformations of schemes 11 Dimensions of families of space curves 12 A non reduced component of the Hilbert scheme 33 39 43 54 58 64 68 Chapter 3 Formal Moduli 75 13 14 15 16 17 18 19 20 21 Plane curve singularities 75 Functors of Artin rings 82 Schlessinger s criterion 86 Fibred products and flatness 91 Hilb and Pic are pro representable 94 Miniversal and universal deformations of schemes 96 Deformations of sheaves and the Quot



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