Projective Algebraic Geometry II: Regular Functions, Local Rings, Morphisms

1999 ◽  
pp. 129-141
Author(s):  
Peter Falb
Keyword(s):  
Algebraic Geometry ◽  
Local Rings ◽  
Regular Functions

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 On Regular Sequences

1966 ◽  
Vol 27 (1) ◽  
pp. 355-356 ◽  
Author(s):  
J. Dieudonné
Keyword(s):  
Algebraic Geometry ◽  
Noetherian Ring ◽  
Regular Sequence ◽  
Local Rings ◽  
Regular Sequences

The concept of regular sequence of elements of a ring A (first introduced by Serre under the name of A-sequence [2]), has far-reaching uses in the theory of local rings and in algebraic geometry. It seems, however, that it loses much of its importance when A is not a noetherian ring, and in that case, it probably should be superseded by the concept of quasi-regular sequence [1].

Examples of Factorial Rings in Algebraic Geometry

1987 ◽  
Vol 30 (1) ◽  
pp. 75-79
Author(s):  
Jacek Bochnak ◽  
Wojciech Kucharz
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Variety ◽  
Cohomology Group ◽  
Connected Components ◽  
Real Algebraic Variety ◽  
Regular Functions ◽  
Complex Valued

AbstractWe show that the ring of complex-valued regular functions on an affine irreducible nonsingular real algebraic variety X is factorial if dim X = 1 or dim X = 2 and X has no compact connected components or X is compact and the second cohomology group of X with integral coefficients vanishes.

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