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• W190115663 abstract "We study (local) properties of complex analytic mappings (and modules over them) by analyzing their underlying family of fibres. Two important properties of such mappings, namely openness and flatness, are our main object of interest. Failure of either of these properties manifests itself as some sort of degeneracy in the family of fibres. The first goal of this thesis is to develop criteria that allow one to effectively (i.e., computationally) detect these degeneracies, and in addition, that can be applied to the case of mappings with singular targets. This is the subject of Chapters 2 and 3. Particularly regarding flatness, no such algorithms that work in the general setting of singular targets were known before. We prove that a mapping germ φ : X → Y (under some assumptions) is flat (resp. open) if and only if after pulling φ back by the blowing-up of the origin in Y , the special fibre does not contain an irreducible component (resp. an isolated irreducible component). Algebraically, this criterion is equivalent to a test for a specific zero-divisor in the local ring (resp. reduced local ring) of the pullback. Also, as a generalization of previous flatness criteria of Auslander’s type to the case of singular bases, we prove that an analytic Rmodule F is flat if and only if the analytic tensor product F ⊗R · · · ⊗RF } {{ } n times ⊗RS has no vertical components, where R is a complex analytic algebra which is an integral domain of dimension n, and S is the local ring of a desingularization of Specan(R). Our second goal is to characterize different modes of the above mentioned degeneracies. This is the topic of Chapter 4. We study the verticality index of a mapping, defined as the highest fibred power in which no verical components emerge. This is a gauge which measures the level of non-openness" @default.
• W190115663 created "2016-06-24" @default.
• W190115663 creator A5060493177 @default.
• W190115663 date "2013-01-01" @default.
• W190115663 modified "2023-09-27" @default.
• W190115663 title "On degeneracies in the family of fibres of a complex analytic mapping" @default.
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