scholarly journals On the canonical topology of an analytic algebra and of an analytic module

1965 ◽  
Vol 79 ◽  
pp. 129-153 ◽  
Author(s):  
M. Jurchescu
Keyword(s):  
Analytic Algebra

Some results concerning the local analytic branches of an algebraic variety

1953 ◽  
Vol 49 (3) ◽  
pp. 386-396 ◽  
Author(s):  
D. G. Northcott
Keyword(s):  
Algebraic Geometry ◽  
Power Series ◽  
Algebraic Variety ◽  
Recent Progress ◽  
Rapid Development ◽  
Noetherian Rings ◽  
Local Rings ◽  
Power Series Rings ◽  
Analytic Algebra ◽  
Topological Theories

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.

scholarly journals Coordinates for analytic operator algebras

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].

The analytic algebra associated to a quasi-analytic local algebra

1969 ◽  
Vol 20 (6) ◽  
pp. 612-615
Author(s):  
C. Banica ◽  
O. Stanasila
Keyword(s):  
Local Algebra ◽  
Analytic Algebra

Applications of the asymptotic range to analytic subalgebras of groupoid C*-algebras

1992 ◽  
Vol 12 (2) ◽  
pp. 341-358 ◽  
Author(s):  
Baruch Solel
Keyword(s):  
Total Order ◽  
C Algebras ◽  
Maximality Property ◽  
Unit Space ◽  
Analytic Algebra

AbstractFor a 1-cocycle c on a principal r-discrete groupoid G, that vanishes only on the unit space of G, we show that the asymptotic range of c, , is an invariant for the total order c−1([0, ∞]). It follows that is also an invariant (with respect to isometric isomorphisms) of the triangular analytic algebra supported on c−1([0, ∞]). We also prove that if and only if the analytic algebra has a certain maximality property.

scholarly journals Bad extensions of analytic algebra

1987 ◽  
Vol 111 (1) ◽  
pp. 206-209
Author(s):  
H.W Schuster ◽  
A Vogt
Keyword(s):  
Analytic Algebra

Regularity of Analytic Algebra of Prime Characteristic

2013 ◽  
Vol 41 (8) ◽  
pp. 3130-3146 ◽  
Author(s):  
Mamoru Furuya ◽  
Hiroshi Niitsuma
Keyword(s):  
Prime Characteristic ◽  
Analytic Algebra

scholarly journals A generalization of Milnor’s formula

2021 ◽  
Author(s):  
Matthias Zach
Keyword(s):  
Holomorphic Function ◽  
Euler Characteristic ◽  
Critical Points ◽  
Betti Number ◽  
Milnor Number ◽  
Analytic Space ◽  
Isolated Singularity ◽  
Milnor Fiber ◽  
Analytic Algebra

AbstractThe Milnor number $$\mu _f$$ μ f of a holomorphic function $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$ f : ( C n , 0 ) → ( C , 0 ) with an isolated singularity has several different characterizations as, for example: 1) the number of critical points in a morsification of f, 2) the middle Betti number of its Milnor fiber $$M_f$$ M f , 3) the degree of the differential $${\text {d}}f$$ d f at the origin, and 4) the length of an analytic algebra due to Milnor’s formula $$\mu _f = \dim _{\mathbb {C}}{\mathcal {O}}_n/{\text {Jac}}(f)$$ μ f = dim C O n / Jac ( f ) . Let $$(X,0) \subset ({\mathbb {C}}^n,0)$$ ( X , 0 ) ⊂ ( C n , 0 ) be an arbitrarily singular reduced analytic space, endowed with its canonical Whitney stratification and let $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$ f : ( C n , 0 ) → ( C , 0 ) be a holomorphic function whose restriction f|(X, 0) has an isolated singularity in the stratified sense. For each stratum $${\mathscr {S}}_\alpha $$ S α let $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) be the number of critical points on $${\mathscr {S}}_\alpha $$ S α in a morsification of f|(X, 0). We show that the numbers $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) generalize the classical Milnor number in all of the four characterizations above. To this end, we describe a homology decomposition of the Milnor fiber $$M_{f|(X,0)}$$ M f | ( X , 0 ) in terms of the $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) and introduce a new homological index which computes these numbers directly as a holomorphic Euler characteristic. We furthermore give an algorithm for this computation when the closure of the stratum is a hypersurface.

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