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This thesis originated from the investigation of the following question: If R is a complete local domain containing a field k with the characteristic of k equal to p > 0, is the integral closure of R in the algebraic closure of its quotient field Cohen-Macaulay? This leads to the following splitting problem: Given R = k{X(,1),X(,2),..,X(,n)}/(f) with char k = p such that f is homogenous with an isolated singularity at the origin and 0 (--->) R (--->) S where S is a module finite ring extension of R, when is R a direct summ and as an R-module of S? In Chapter 1 we have the following results: PROPOSITION: Let (R,m) be a normal domain containing a field with positive characteristic such that Gr(,m)R is a Cohen-Macaulay domain. Let x(,1),x(,2),..,x(,n) be in R such that {x(,1)}(,k), {x(,2)}(,k), .., {x(,n)}(,k) form a system of parameters in Gr(,m)R. Let w (ELEM) R such that ord(,m)(w) = k + d where d (LESSTHEQ) k. Let S be an integral extension of R such that lies S in a finite Galois extension of the quotient field of R and suppose w (NOT ELEM) (x(,1),...,x(,n))R. Assume m('n) are integrally closed ideals. Then (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) PROPOSITION: If R = k{X(,1),X(,2),..,X(,n)}/(f) where f is a homogeneous polynomial of degree n with an isolated singularity at the origin. Then there is an S, module-finite over R such that 0 (--->) R (--->) S does not split. The existence of the small Cohen-Macaulay modules (i.e. finitely generated ones) over complete local rings is known for dim R (LESSTHEQ) 2 and dim R = 3 where R is the completion of a finitely generated non-negatively graded algebra over a field of char p > 0. We show the following: PROPOSITION: Let k be an algebraically closed field and C(,1),..,C(,n) be projective smooth curves over k. Then there is a small Cohen-Macaulay module for the coordinate ring of C(,1) x C(,2) x .. x C(,n). In Chapter 3 we will construct big Cohen-Macaulay algebras with unit for certain rings by trivializing relations on a system of parameters. |
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