Intro -- Contents -- Introduction -- List of talks and references to publications -- Linearization of local cohomology modules -- Multiplicity-free subvarieties of flag varieties -- Morita contexts, idempotents, and Hochschild cohomology―with applications to invariant rings― -- Introduction -- 1. Morita contexts -- 2. The grade of the stable endomorphism ring -- 3. Equivalences to the generalized Nakayama conjecture -- 4. The comparison homomorphism for Hochschild cohomology -- 5. Hochschild cohomology of Auslander contexts -- 6. Invariant rings: The general case -- 7. Invariant rings: The commutative case -- References -- Some applications of two dimensional complete ideals -- Generically finite morphisms and simultaneous resolution of singularities -- Two results on the number of generators -- When is a binomial ideal equal to a lattice ideal up to radical? -- Depth and amplitude for unbounded complexes -- On the ideal of minors of matrices of linear forms -- Surjectivity of multiplication and F-regularity of multigraded rings -- On the Ext-modules of ideals of Borel type -- Sums of linked tepli ideals -- Cartan involutiveness = Mumford regularity -- The Frobenius endomorphism and homological dimensions -- Characterization of some projective subschemes by locally free resolutions -- 1. Introduction -- 2. Preliminaries -- 3. q-presentations -- 4. Surjective-Buchsbaum modules -- 5. Arithmetically Buchsbaum subschemes -- 6. Quasi-Buchsbaum subschemes -- 7. Surfaces in P4 -- References -- On unique factorization in semigroups of complete ideals -- Modules with strange homological properties and Chebychev polynomials -- Notes on the behavior of the Ratliff-Rush filtration -- On symbolic powers of prime ideals -- Multiplicities and the number of generators of Cohen-Macaulay ideals -- Chains of integrally closed ideals.