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Prime ideals in two-dimensional Noetherian domains and fiber products and connected sums
This thesis concerns three topics in commutative algebra: 1) The projective line over the integers (Chapter 2), 2) Prime ideals in two-dimensional quotients of mixed power series-polynomial rings (Chapter 3), 3) Fiber products and connected sums of local rings (Chapter 4). ^ In the first chapter we introduce basic terminology used in this thesis for all three topics. ^ In the second chapter we consider the partially ordered set (poset) of prime ideals of the projective line Proj( Z [h, k]) over the integers Z , and we interpret this poset as Spec( Z [x]) ∪ Spec( Z1x ) with an appropriate identification. ^ We have some new results that support Aihua Li and Sylvia Wiegand's conjecture regarding the characterization of Proj( Z [h, k]). In particular we show that a possible axiom for Proj( Z [h, k]) proposed by Arnavut, Li and Wiegand holds for some previously unknown cases. ^ We study the sets of prime ideals of polynomial rings, power series rings and mixed power series-polynomial rings in Chapter 3. Let R be a one-dimensional Noetherian domain and let x and y be indeterminates. We describe the prime spectra of certain two-dimensional quotients of mixed power series/polynomial rings over R, that is, Spec( Rx yQ ) and Spec( Ry xQ′ ), where Q and Q' are certain height-one prime ideals of R[[x]][y] and R[y][[x]] respectively. ^ In the last chapter we describe some ring-theoretic and homological properties of fiber products and connected sums of local rings. For Gorenstein Artin k-algebras R and S where k is a field, the connected sum, R#kS, is a quotient of the classical fiber product. We give basic properties of connected sums over a field and show that certain Gorenstein local k-algebras decompose as connected sums. We generalize structure theorems given by Sally, Elias and Rossi that show two types of Gorenstein local k-algebras are connected sums.^
Applied Mathematics|Mathematics|Theoretical Mathematics
Celikbas, Ela, "Prime ideals in two-dimensional Noetherian domains and fiber products and connected sums" (2012). ETD collection for University of Nebraska - Lincoln. AAI3523374.