Representations of elementary Abelian p-groups and vector bundles / David J. Benson, University of Aberdeen.

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Bibliographic Details
Main Author:	Benson, D. J. (David J.), 1955-
Format:	Book
Language:	English
Published:	Cambridge, United Kingdom ; New York, NY, USA : Cambridge University Press, [2017]
Series:	Cambridge tracts in mathematics ; 208.
Subjects:	Abelian p-groups. Abelian groups. Vector bundles.

Table of Contents:

1. Modular representations and elementary abelian groups
2. Cyclic groups of order p
3. Background from algebraic geometry
4. Jordan type
5. Modules of constant Jordan type
6. Vector bundles on projective space
7. Chern classes
8. Modules of constant Jordan type and vector bundles
9. Examples
10. Restrictions coming from Chern numbers
11. Orlov's correspondence
12. Phenomenology of modules over elementary Abelian p-groups.
Machine generated contents note: 1. Modular Representations and Elementary Abelian Groups
1.1. Introduction
1.2. Representation Type
1.3. Shifted Subgroups
1.4. The Language of π-Points
1.5. The Stable Module Category
1.6. The Derived Category
1.7. Singularity Categories
1.8. Cohomology of Elementary Abelian p-Groups
1.9. Chouinard's Theorem, Dade's Lemma and Rank Varieties
1.10. Carlson's Lζ Modules, and a Matrix Version
1.11. Diagrams for Modules
1.12. Tensor Products
1.13. Duality
1.14. Symmetric and Exterior Powers
1.15. Schur Functions
1.16. Schur Functors
1.17. Radical Layers of k E
1.18. Twisted Versions of k E
2. Cyclic Groups of Order p
2.1. Introduction
2.2. Modules for Z/p
2.3. Tensor Products
2.4. Gaussian Polynomials
2.5. Generalised Gaussian Polynomials and a Hook Formula
2.6. λ-Rings and Representations of SL(2, C)
2.7. The Representation Ring of Z/p
2.8. Symmetric and Exterior Powers of Jordan Blocks
2.9. Schur Functors for SL(2, C) and Z/p
3. Background from Algebraic Geometry
3.1. Affine Space and Affine Varieties
3.2. Generic Points and Closed Points
3.3. Projective Space and Projective Varieties
3.4. Tangent Spaces
3.5. Presheaves and Sheaves
3.6. Stalks and Sheafification
3.7. The Language of Schemes
3.8. Sheaves of Modules
3.9. Coherent Sheaves on Projective Varieties
3.10. Cohomology of Sheaves
4. Jordan Type
4.1. Nilvarieties
4.2. Matrices and Tangent Spaces
4.3. A Theorem of Gerstenhaber
4.4. Dominance Order and Nilpotent Jordan Types
4.5. Generic and Maximal Jordan Type
4.6. Tensor Products
5. Modules of Constant Jordan Type
5.1. Introduction and Definitions
5.2. Homogeneous Modules
5.3. An Exact Category
5.4. Endotrivial Modules
5.5. Wild Representation Type
5.6. The Constant Image Property
5.7. The Generic Kernel
5.8. The Subquotient Rad-1R(M)/Rad2R(M)
5.9. The Constant Kernel Property
5.10. The Generic Image
5.11. W-Modules
5.12. Constant Jordan type with One Non-Projective Block
5.13. Rickard's Conjecture
5.14. Consequences and Variations
5.15. Further Conjectures
6. Vector Bundles on Projective Space
6.1. Definitions and First Properties
6.2. Tests for Vector Bundles
6.3. Vector Bundles on Projective Space
6.4. The Tangent Bundle and the Euler Sequence
6.5. Homogeneity and Uniformity
6.6. Monads and Subquotients
6.7. The Null Correlation Bundle
6.8. The Examples of Tango
6.9. Cohomology of Projective Space
6.10. Differential Forms and Bott's Theorem
6.11. Simplicity
6.12. Hilbert's Syzygy Theorem
7. Chern Classes
7.1. Chern Classes of Graded Modules
7.2. Chern Classes of Coherent Sheaves on Pr-1
7.3. Some Computations
7.4. Restriction of Vector Bundles
7.5. Chern Numbers of Twists and Duals
7.6. Chern Roots
7.7. Power Sums
7.8. The Hirzebruch--Riemann--Roch Theorem
7.9. Chern Numbers and the Frobenius Map
8. Modules of Constant Jordan Type and Vector Bundles
8.1. Introduction
8.2. The Operator θ
8.3. The Action of θ on Fibres
8.4. The Functors Fi and Fi,j
8.5. Twists and Syzygies
8.6. Chern Numbers of Fi(M)
8.7. The Construction: p = 2
8.8. The Construction: p Odd
8.9. Proof of the Realisation Theorem
8.10. Functoriality
8.11. Tensor Products
8.12. Negative Tate Cohomology
8.13. The BGG Correspondence
9. Examples
9.1. Modules for (Z/2)2
9.2. Modules for (Z/p)2
9.3. Larger Rank
9.4. Nilvarieties
9.5. The Tangent and Cotangent Bundles
9.6. The Null Correlation Bundle, p = 2
9.7. The Null Correlation Bundle, p Odd
9.8. Instanton Bundles
9.9. Schwarzenberger's Bundles
9.10. The Examples of Tango
9.11. The Horrocks--Mumford Bundle
9.12. Automorphisms of the Horrocks--Mumford Bundle
9.13. Realising the Horrocks--Mumford Bundle
9.14. The Horrocks Parent Bundle and the Tango Bundle
10. Restrictions Coming from Chern Numbers
10.1. Matrices of Constant Rank
10.2. Congruences on Chern Numbers
10.3. Restrictions on Stable Jordan Type, p Odd
10.4. Eliminating More Stable Jordan Types
10.5. Restrictions on Jordan Type for p = 2
10.6. Applying Hirzebruch--Riemann--Roch for p = 2: The Case m = 0
10.7. Bypassing Hirzebruch--Riemann--Roch
10.8. Applying and Bypassing Hirzebruch--Riemann--Roch for p = 2: The Case 1 [≤] m [≤] r - 3
10.9. Nilvarieties of Constant Jordan Type [p]n for p [≥] 3
10.10. Nilvarieties with a Single Jordan Block
10.11. Babylonian Towers
11. Orlov's Correspondence
11.1. Introduction
11.2. Maximal Cohen--Macaulay Modules
11.3. The Orlov Correspondence
11.4. The Functors
11.5. An Example
11.6. The Bidirectional Koszul Complex
11.7. A Bimodule Resolution
11.8. The Adjunction
11.9. The Equivalence
11.10. The Trivial Module
11.11. Computer Algebra
11.12. Cohomology
11.13. Twisted Versions of kE
12. Phenomenology of Modules over Elementary Abelian p-Groups
12.1. Introduction
12.2. Module Constructions
12.3. Odd Primes Are More Difficult
12.4. Relative Cohomology
12.5. Small Modules for Quadrics, p = 2
12.6. Small Modules for Quadrics, p Odd
12.7. Trying to Understand the Specht Module S(33)
12.8. Modules with Small Loewy Length
12.9. Small Modules for (Z/p)2
12.10. The Bound is Close to Sharp.

Representations of elementary Abelian p-groups and vector bundles / David J. Benson, University of Aberdeen.

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