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https://hdl.handle.net/2142/72542
Description
Title
Weak Purity for Gorenstein Rings
Author(s)
Borek, Adam Richard
Issue Date
1993
Doctoral Committee Chair(s)
Griffith, Phillip A.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
In the language of local algebra, the classical purity of branch locus theorem states that a module finite ring extension of local normal domains $B\to A$ which is unramified in codimension one, with B a regular local ring, is unramified (and in this setting, etale). When the ring B is merely Gorenstein, there are numerous examples which show that A need not be Cohen-Macaulay, hence that the extension need not be unramified. In a related context, the main portion of the thesis addresses the purity of the extension $B\to A.$ The setting is as follows: $B\to A$ is a module finite extension of normal rings, unramified in codimension one with B an excellent local equicharacteristic Gorenstein domain of dimension at least five (under additional conditions, the mixed characteristic case is considered). In this context, a weak purity holds: if B is "regular" enough (that is, satisfies $(R\sb{k})),$ then A inherits a certain amount of depth (that is, satisfies $(S\sb{k-1}))$ where $k\ge 4.$ This purity is weak in that A acquires "good" properties from B, yet the extension itself need not be "good" (that is, unramified), as is illustrated by an example. Moreover, the method of proof is used to recover Grothendieck's purity theorem for complete intersections in the special case of a hypersurface ring B. Applications are considered: in extensions $B\to A$ similar to those above which are normal (with Galois group G), a relationship between codimension two primes of A which are fixed under the action of G and small MCM A-modules (via "Bourbaki"-exact sequences) is examined; depth properties of divisorial B-ideals of finite order in Cl(B) are investigated; and related ideas are studied.
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