Invariant chains in algebra and discrete geometry

TL;DR

This paper explores the relationship between local and global finite generation in algebraic structures like cones, monoids, and ideals, introducing new frameworks and proofs for stabilization of invariant chains.

Contribution

It establishes local-global correspondences for finite generation, extends the theory to new classes of algebraic objects, and provides a novel proof of stabilization for Inc-invariant chains.

Findings

Finite generation of cones and monoids relates to equivariant finite generation.

Examples show no direct Noetherian analogy for cones and monoids.

New proof confirms stabilization of Inc-invariant chains of ideals.

Abstract

We relate finite generation of cones, monoids, and ideals in increasing chains (the local situation) to equivariant finite generation of the corresponding limit objects (the global situation). For cones and monoids there is no analog of Noetherianity as in the case of ideals and we demonstrate this in examples. As a remedy, we find local-global correspondences for finite generation. These results are derived from a more general framework that relates finite generation under closure operations to equivariant finite generation under general families of maps. We also give a new proof that non-saturated Inc-invariant chains of ideals stabilize, closing a gap in the literature.