1. Nonnoetherian geometry.
- Author
-
Beil, Charlie
- Subjects
- *
COMMUTATIVE algebra , *NOETHERIAN rings , *KRULL rings , *DIMENSION theory (Algebra) , *ALGEBRAIC varieties - Abstract
We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive-dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of are all isomorphic if and only if is noetherian, if and only if the center of is noetherian, if and only if is a finitely generated -module. Furthermore, we show that is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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