  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.
