Euler class group of certain overrings of a polynomial ring

2017 ◽  
Vol 9 (3) ◽  
pp. 341-365
Author(s):  
Alpesh M. Dhorajia
Keyword(s):  
Polynomial Ring ◽  
Class Group ◽  
Euler Class ◽  
Euler Class Group

scholarly journals Revisiting Nori's question and homotopy invariance of Euler class groups

2010 ◽  
Vol 8 (3) ◽  
pp. 451-480 ◽  
Author(s):  
Mrinal Kanti Das
Keyword(s):  
Class Group ◽  
Euler Class ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Class Groups ◽  
Homotopy Invariance ◽  
Smoothness Assumption ◽  
Polynomial Extension ◽  
Euler Class Group ◽  
Smooth Affine

AbstractThis paper examines the relation between the Euler class group of a Noetherian ring and the Euler class group of its polynomial extension. When the ring is a smooth affine domain, the two groups are canonically isomorphic. This is a consequence of a theorem of Bhatwadekar-Sridharan, which they proved in order to answer a question of Nori on sections of projective modules over such rings. If the smoothness assumption is removed, the result of Bhatwadekar-Sridharan is no longer valid and also the Euler class groups above are not in general isomorphic. In this paper we investigate a variant of Nori's question for arbitrary Noetherian rings and derive several consequences to understand the relation between various groups in the theory of Euler classes.

On Central Ω-Krull Rings and their Class Groups

1984 ◽  
Vol 36 (2) ◽  
pp. 206-239 ◽  
Author(s):  
E. Jespers ◽  
P. Wauters
Keyword(s):  
Polynomial Ring ◽  
Class Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Semi Group ◽  
Class Groups ◽  
Krull Domain ◽  
Direct Limits ◽  
Commutative Theory ◽  
Necessary And Sufficient

The aim of this note is to study the class group of a central Ω-Krull ring and to determine in some cases whether a twisted (semi) group ring is a central Ω-Krull ring. In [8] we defined an Ω-Krull ring as a generalization of a commutative Krull domain. In the commutative theory, the class group plays an important role. In the second and third section, we generalize some results to the noncommutative case, in particular the relation between the class group of a central Ω-Krull ring and the class group of a localization. Some results are obtained in case the ring is graded. Theorem 3.2 establishes the relation between the class group and the graded class group. In particular, in the P.I. case we obtain that the class group is equal to the graded class group. As a consequence of a result on direct limits of Ω-Krull rings, we are able to derive a necessary and sufficient condition in order that a polynomial ring R[(Xi)i∊I] (I may be infinite) is a central Ω-Krull ring.

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