Nonnoetherian geometry

Abstract

Communicated by P. SmithWe 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.

Nonnoetherian coordinate rings with unique maximal depictions

“…Indeed, (i) holds since by Proposition 3.4, [B1,Theorem 3.13.2]: 3 if R is a nonnoetherian subalgebra of a finitely generated k-algebra S, and there is some m ∈ ι(U c S/R ) satisfying Proposition 3.8. Suppose each I i is a radical ideal of S.…”

“…Recall that a depiction of a nonnoetherian domain R is a finitely generated kalgebra S that is as close to R as possible, in a suitable geometric sense (Definition 2.1). For example, if R is depicted by S, then R and S have equal Krull dimension, and their maximal spectra are birationally equivalent [B1,Theorem 2.5]. Set…”

“…In the published version of[B1, Theorem 3.13.2], S is assumed to be a depiction of R, but this is not used in the proof of the theorem.…”

“…This proposition is erroneously claimed as a corollary to[B1, Theorem 3.13, published version] [B1,. Theorem 3.13] assumes that S is a depiction of R, but if R is noetherian, then S will not be a depiction of R. Indeed, in this case the only depiction of R will be itself[B1, Theorem 3.12], and R = S if I is a non-maximal ideal of S.…”

The Bell states in noncommutative algebraic geometry

“…Proof. I is clearly a maximal ideal of R, and so I is a closed point of Spec R. The claim that U = Max S \ Z(I) follows from [B,Proposition 2.8], and birationality follows from [B,Theorem 2.5.3].…”