Yerel kohomoloji ve radikal olarak mükemmel idealler

Local cohomology theory was first introduced by Alexander Grothendieck in 1961 and since then it has been used as a powerful tool to solve many problems in both algebraic geometry and commutative algebra. Basically, local cohomology functors are defined as the right derived functors of a certain torsion functor: For any module M over a commutative ring R, set \Gamma_I(M)= {x\in M :there exists an n\in \mathbb{N} such that I^nx=0 }. The i^{th} local cohomology of M with respect to the ideal I is the i^{th} cohomology module of the sequence obtained by applying the left exact functor \Gamma_I(-), which is defined above, to an injective resolution of M and this module is denoted by H^{i}_{I}(M). Local cohomology theory has been applied to the study of several conjectures in commutative algebra one of which is related to radically perfect ideals. An ideal I of a commutative (not necessarily Noetherian) ring R is said to be radically perfect if the minimal number of elements of R which generates I up to radical is finite and equals to the height of I. Clearly, when R is Noetherian, the terms radically perfect and set theoretic complete intersection coincide. One of the classical and long-standing problem in commutative algebra and algebraic geometry is to determine whether each height two prime ideal of the polynomial ring K[X,Y,Z] over the field K is set theoretic complete intersection (radically perfect). Although it is shown by Cowsik and Nori that this conjecture has an affirmative answer when K is of characteristic p>0, it still remains as an open problem in the characteristic zero case. But, based on the observations from his several results, Erdo\u{g}du has a foresight that this problem would fail to be true when K is of characteristic zero. Furthermore he raised another conjecture that "If R is a commutative domain (not necessarily Noetherian) containing a field of characteristic zero, then each prime ideal of R[X] is radically perfect implies R is of Krull dimension one." which was proved to be so in many cases but the exact answer of this conjecture is also not known in general. The main purpose of this thesis is to understand certain structures of local cohomology modules and determining their relationship with set theoretic complete intersection ( radically perfect ) ideals and our motivation is suggested by the well-known fact that if I is an ideal of a Noetherian ring R of height n, and if there exists some R-module M such that H^i_I(M)\neq 0 for i>n, then I is not a set theoretic complete intersection. Moreover, Hellus showed the relation between set theoretic complete intersection ideals and Matlis duals of local cohomology modules by proving the fact that if {H}^i_I(R)=0 for all i\neq c and \textbf{\em x}= \{x_1,x_2,...,x_c\} is a regular sequence in I, then I= \sqrt{x_1,x_2,...,x_c} if and only if x_i form a {D}({H}^c_I(R))-regular sequence. In this thesis, motivated by Hellus' result, we first deal with the set of associated prime ideals of Matlis duals of local cohomology modules and show that over a Noetherian regular local ring of characteristic p>0, for any non-zero ideal I of R and for i>0, zero ideal is in the set of associated prime ideals of Matlis dual of any non-zero local cohomology module H^i_I(R). We then determine conditions under which a given positive integer t is a lower bound for the cohomological dimension$cd(I,M):= sup { i \in\mathbb{N} | {H}^i_I(M)\neq 0 }$ of any module M with respect to an ideal I of a Noetherian ring R, and use this to conclude that non-catenary Noetherian integral domains contain prime ideals that are not radically perfect (i.e. set theoretic complete intersection). Bearing in mind that non-catenary rings are of Krull dimension >2, this result is in partial support with Erdo\u{g}du's conjecture. Furthermore if I is any ideal of R and M is any R- module with \operatorname{cd}(I,M)=c>0, we show the existence of a descending chain of ideals $I=I_c\supsetneq I_{c-1}\supsetneq \cdots \supsetneq I_0$ of R such that for each 0\leq i\leq c, cd(I_i,M)=i. In the last chapter of this thesis, we examine the structures of local cohomology modules and show that over a Noetherian unique factorization domain of dimension at most three, top local cohomology module $H^{cd(I,R)}_I(R)$ is Artinian only in the trivial case when \operatorname{cd}(I,R)=\dim R. We then obtain several results on the Artinianness of top local cohomology modules in more general cases. Finally, our study is concerned around the modules of finite length and, in this regard, we first present necessary and sufficient conditions for various modules to be of finite length. We then use our results to give an alternative proof of the well-known result that if R is a Noetherian local ring with maximal ideal \mathfrak{m} and M is a finitely generated R- module of dimension d, then $\operatorname{H}^{d}_{\mathfrak m}(M)$ is finitely generated if and only if d=0. Throughout, R will always denote a commutative ring with identity, the dimension of a ring R will always mean its Krull dimension.