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- W2953503145 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 $A$ are all isomorphic if and only if $A$ is noetherian, if and only if the center $Z$ of $A$ is noetherian, if and only if $A$ is a finitely generated $Z$-module. Furthermore, we show that $Z$ 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." @default.
- W2953503145 created "2019-07-12" @default.
- W2953503145 creator A5005824086 @default.
- W2953503145 date "2011-09-21" @default.
- W2953503145 modified "2023-09-27" @default.
- W2953503145 title "Nonnoetherian geometry and toric superpotential algebras" @default.
- W2953503145 hasPublicationYear "2011" @default.
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