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Cohen Structure Theorem
In mathematics, the Cohen structure theorem, introduced by , describes the structure of complete Noetherian local rings. Some consequences of Cohen's structure theorem include three conjectures of Krull Krull is a surname originating from Prussian nobility. People *Alexander Krull (born 1970), German singer *Annie Krull (1876–1947), German operatic soprano *Germaine Krull (1897–1985), photographer * Hasso Krull (born 1964), Estonian po ...: *Any complete regular equicharacteristic Noetherian local ring is a ring of formal power series over a field. (Equicharacteristic means that the local ring and its residue field have the same characteristic, and is equivalent to the local ring containing a field.) *Any complete regular Noetherian local ring that is not equicharacteristic but is unramified is uniquely determined by its residue field and its dimension. *Any complete Noetherian local ring is the image of a complete regular Noetherian local ring. Statement The mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completion (ring Theory)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions ''R'' on a space ''X'' concentrates on a formal neighborhood of a point of ''X'': heuristically, this is a neighborhood so small that ''all'' Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in the case when ''R'' has a metric given by a non-Archimedean absolute value. General construction Suppose that ''E'' is an abelian group with a descending filtration : E = F^0 E \supset F^1 E \supset F^2 E \supset \cdots \, of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |