Integer torsion in local cohomology, and questions on tight closure theory

Grothendieck's theory of local cohomology has applications to basic questions such as determining the minimal number of polynomial equations needed to define an algebraic set. These modules are typically not finitely generated, and a question of Huneke asks whether they have finitely many associated prime ideals. This was settled in the negative by Singh, who constructed a local cohomology module that has prime-torsion for each prime integer. We extend this work in Chapter 1 by showing that the module in question contains a copy of each finitely generated abelian group. Moreover, the module has a natural fine grading, and we are able to show that each finitely generated abelian group embeds into a single graded component. In Chapter 2 we study F-injectivity, i.e., the property that the Frobenius action on local cohomology modules is injective. We obtain an effective criterion to determine if a diagonal subalgebra of a bigraded hypersurface is F-injective; such subalgebras provide a rich source of examples of various F-singularities, and thus form a natural testing ground for various questions and conjectures related to classes of singularities defined via the Frobenius map. Chapter 3 is an investigation of the tight closure properties of rings of invariants of finite groups acting linearly on polynomial rings over fields of positive characteristic.