Actions for Representations of elementary abelian p-groups and vector bundles
Representations of elementary abelian p-groups and vector bundles / David J. Benson, University of Aberdeen
- Author
- Benson, D. J. (David J.), 1955-
- Published
- Cambridge : Cambridge University Press, 2017.
- Copyright Date
- ©2017
- Physical Description
- xvii, 328 pages ; 24 cm.
- Series
- Contents
- 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 kE -- 1.18.Twisted Versions of kE -- 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.
- Subject(s)
- ISBN
- 9781107174177 hardcover alkaline paper
1107174171 hardcover alkaline paper - Bibliography Note
- Includes bibliographical references and index.
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