Regularity of Analytic Algebra of Prime Characteristic

The recent progress of modern algebra in analysing, from the algebraic standpoint, the foundations of algebraic geometry, has been marked by the rapid development of what may be called ‘analytic algebra’. By this we mean the topological theories of Noetherian rings that arise when one uses ideals to define neighbourhoods; this includes, for instance, the theory of power-series rings and of local rings. In the present paper some applications are made of this kind of algebra to some problems connected with the notion of a branch of a variety at a point.

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Frobenius closure and prime characteristic singularities

10.32469/10355/78133 ◽

2020 ◽

Author(s):

◽

Kyle Logan Maddox

Keyword(s):

Local Ring ◽

Local Cohomology ◽

Sufficient Condition ◽

Zariski Topology ◽

Prime Characteristic ◽

Joint Work ◽

University Of Missouri ◽

Mild Conditions ◽

Flat Maps ◽

The University

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] This dissertation outlines several results about prime characteristic singularities for which the nilpotent part under the induced Frobenius action on local cohomology is either finite colength or the entire module, collectively referred to here as nilpotent singularities. First, we establish a sufficient condition for the finiteness of the Frobenius test exponent for a local ring and apply it to conclude that nilpotent singularities have finite Frobenius test exponent. In joint work with Jennifer Kenkel, Thomas Polstra, and Austyn Simpson, we show that under mild conditions nilpotent singularities descend and ascend along faithfully flat maps. Consequently, we then prove that the loci of primes which are weakly F-nilpotent and F-nilpotent are open in the Zariski topology for rings which are either F-finite or essentially of fiiite type over an excellent local ring.

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Coordinates for analytic operator algebras

Glasgow Mathematical Journal ◽

10.1017/s0017089500007527 ◽

1989 ◽

Vol 31 (1) ◽

pp. 31-47

Author(s):

Baruch Solel

Keyword(s):

Abelian Group ◽

Von Neumann Algebra ◽

Operator Algebras ◽

Dual Group ◽

Positive Semigroup ◽

Analytic Operator ◽

Von Neumann ◽

Finite Von Neumann Algebra ◽

Analytic Algebra ◽

Image Position

Let M be a σ-finite von Neumann algebra and α = {αt}t∈A be a representation of a compact abelian group A as *-automorphisms of M. Let Γ be the dual group of A and suppose that Γ is totally ordered with a positive semigroup Σ⊆Γ. The analytic algebra associated with α and Σ iswhere spα(a) is Arveson's spectrum. These algebras were studied (also for A not necessarily compact) by several authors starting with Loebl and Muhly [10].

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