Intro; Preface; Contents; Restricting Homology to Hypersurfaces; 1 Differential Graded Algebra; 1.1 Tensor Products; 1.2 Divided Powers; 1.3 The Koszul Complex; 1.4 The Tate Construction; 1.5 Acyclic Closures; 2 Hypersurfaces; 2.1 Degree One Cycles; 2.2 Hypersurfaces; 3 Support Sets; 3.1 Support Sets; 3.2 Projective Dimension; 3.3 Alternative Description of Support; 3.4 Complexes over Regular Rings; 3.5 On Being Closed; 4 Defining Equations; 4.1 Complete Intersections; 5 Group Algebras of Elementary Abelian Groups; 5.1 Flat Dimension; 5.2 Rank Varieties; 5.3 Polynomial Extensions

5.4 Nilpotent OperatorsReferences; Thick Subcategories of the Relative Stable Category; 1 Introduction; 2 Notation and Preliminaries; 3 Annihilators of Cohomology; 4 Additive Tensor Ideals; 5 Non-Noetherian Spectra; 6 Idempotent Modules; 7 Thick Subcategories by Inflation; 8 Empty Thick Tensor Ideals; 9 Examples; 10 One More Question; References; Nilpotent Elements in Hochschild Cohomology; 1 Introduction; 2 Preliminaries; 2.1 The Algebras; 2.2 Hochschild Cohomology; 2.3 Nilpotent Elements; 2.4 Independence of q; 3 A Minimal Bimodule Resolution; 3.1 The problem

3.2 Relating to the one-sided resolution3.3 Minimal generators; 4 Homomorphisms and HHn(A); 4.1 Identities for homomorphisms; 4.2 The Proof of Proposition 4.3 for n+14; 4.3 Small Cases; References; Rational Cohomology and Supports for Linear Algebraic Groups; 1 Introduction; 2 Lecture I: Affine Group Schemes Over k; 2.1 Affine Group Schemes Over k; 2.2 Affine Group Schemes; 2.3 Characteristic p > 0; 2.4 Restricted Lie Algebras; 3 Lecture II: Algebraic Representations; 3.1 Algebraic Actions; 3.2 Examples; 3.3 Weights for G-Modules; 3.4 Representations of Frobenius Kernels

4 Lecture III: Cohomological Support Varieties4.1 Indecomposable Versus Irreducible; 4.2 Derived Functors; 4.3 The Quillen Variety G and the Cohomological Support Variety GM; 5 Lecture IV: Support Varieties for Linear Algebraic Groups; 5.1 1-Parameter Subgroups; 5.2 p-Nilpotent Operators; 5.3 The Support Variety V(G)M; 5.4 Classes of Rational G-Modules; 5.5 Some Questions of Possible Interest; References; Anderson and Gorenstein Duality; 1 Introduction; 1.1 Motivation; 1.2 Description of Contents; 1.3 Conventions; 2 Anderson Duals; 2.1 Construction of Brown-Comenetz Duals

2.2 Properties of Brown-Comenetz Duals2.3 Construction of Anderson Duals; 2.4 Properties of Anderson Duals; 3 The Gorenstein Condition; 3.1 Cellularization; 3.2 Morita Theory; 3.3 The Gorenstein Condition; 3.4 Gorenstein Duality; 3.5 Gorenstein Duality Relative to mathbbFp; 3.6 Mahowald-Rezk Duality; 4 Gorenstein Duality and Anderson Self-duality; 4.1 Nullifying HK; 4.2 Anderson Self-duality from Gorenstein Duality; 4.3 Gorenstein Duality from Anderson Self-duality; 5 Examples with Polynomial or Hypersurface Coefficient Rings; 5.1 The Čech Complex; 5.2 The Algebraic Context