  In this PhD thesis we will discuss some aspects in Commutative Algebra which
have interactions with Algebraic Geometry, Representation Theory and
Combinatorics. In particular, in the first chapter we will focus on
understanding when certain cohomology modules vanish, a classical problem
raised by Grothendieck. In the second chapter we will use local cohomology to
study the connectedness behavior during a Groebner deformation and the
arithmetical rank of certain varieties. In the third chapter, we will
investigate the relations between the minors of a fixed size of a generic
matrix by using tools from the representation theory of the general linear
group (the results of this chapter will appear in a joint paper with Bruns and
Conca). In the last chapter we will use combinatorial methods to study the
Cohen-Macaulay property of the symbolic powers of Stanley-Reisner ideals. In
the thesis are included five appendixes with some basic needed facts and a
preliminary chapter introducing to local cohomology.
