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Full-Text Articles in Algebra
Equisingular Approximation Of Analytic Germs, Aftab Yusuf Patel
Equisingular Approximation Of Analytic Germs, Aftab Yusuf Patel
Electronic Thesis and Dissertation Repository
This thesis deals with the problem of approximating germs of real or complex analytic spaces by Nash or algebraic germs. In particular, we investigate the problem of approximating analytic germs in various ways while preserving the Hilbert-Samuel function, which is of importance in the resolution of singularities. We first show that analytic germs that are complete intersections can be arbitrarily closely approximated by algebraic germs which are complete intersections with the same Hilbert-Samuel function. We then show that analytic germs whose local rings are Cohen-Macaulay can be arbitrarily closely approximated by Nash germs whose local rings are Cohen- Macaulay and …
Commutative Rings Graded By Abelian Groups, Brian P. Johnson
Commutative Rings Graded By Abelian Groups, Brian P. Johnson
Department of Mathematics: Dissertations, Theses, and Student Research
Rings graded by Z and Zd play a central role in algebraic geometry and commutative algebra, and the purpose of this thesis is to consider rings graded by any abelian group. A commutative ring is graded by an abelian group if the ring has a direct sum decomposition by additive subgroups of the ring indexed over the group, with the additional condition that multiplication in the ring is compatible with the group operation. In this thesis, we develop a theory of graded rings by defining analogues of familiar properties---such as chain conditions, dimension, and Cohen-Macaulayness. We then study the …