On Finiteness Properties of Local Cohomology Modules

This thesis has its foundations in the fields of homological and commutative algebra, in particular in the study of local cohomology. Local cohomology was introduced by Grothendieck in the early 1960s, in part to answer a conjecture of Pierre Samuel about when certain types of commutative rings are unique factorization domains. In the area of algebraic geometry, local cohomology modules arise in a natural way. Indeed, given a function defined on an open subset of an algebraic variety, they measure the obstruction to extending that function to a larger domain. But in addition to this, local cohomology allows us to answer many seemingly difficult problems such as the minimum number of defining equations of algebraic sets or their connectedness properties. However, our development of the theory will be largely independent from this point of view. There are many equivalent ways to define local cohomology. From the homological algebra perspective, one of the easiest definitions is the following: given an ideal a in a Noetherian commutative ring R, for each module we consider the submodule of elements annihilated by some power of a. This operation is functorial but is not exact and so the theory of derived functors leads us to define the local cohomology functors H^i_a(_), which measure the failure of this exactness. Typically, local cohomology modules are not finitely generated, and in this sense they may seem "big" and difficult to work with. However, they frequently have properties that make them manageable objects of study. Our main interest is on their finiteness properties. For a Noetherian local ring, with maximal ideal m, it is well known that all the local cohomology modules H^i_m(M) are Artinian when the module M is finitely generated. The main problem we consider in this thesis is how to extend this result to the non-local case and to non-maximal ideals. Local cohomology modules are torsion, and for m-torsion modules N, one may show that being Artinian is equivalent to Ext^i_R(R/m,N) being finitely generated for all i. Modules which satisfies the latter property with respect to an ideal a are called a-cofinite modules, and the above led Grothendieck to conjecture that H^i_a(M) is a-cofinite whenever M is a finitely generated module over a Noetherian ring. Hartshorne has showed that this conjecture is false, and was able to construct a counterexample. We will study these counterexamples to Grothendieck's conjecture, and then describe a way to correct it, using the language of derived categories. The idea is that instead of considering the classical derived functors H^i_a(M), we will consider the total right derived functor R\Gamma_a(M), which is the complex whose i-th cohomology is equal to H^i_a(M). We will follow papers of Porta-Shaul-Yekutieli to show that the complex R\Gamma_a(M) is always a-cofinite and discuss relations between the category of a-cofinite complexes and the category of complexes with finitely generated cohomologies. This thesis is being developed under the supervision of Prof. Liran Shaul (University of Prague) and Prof. Jorge Vitoria (University of Padova).