Dimension and singularity theory for local rings of finite embedding dimension

TL;DR

This paper develops an algebraic framework for local rings of finite embedding dimension, extending dimension concepts, generalizing singularity notions, and applying these to derive bounds and characterizations of ring properties.

Contribution

It introduces new extensions of dimension, generalizes singularity concepts, and applies homological theorems to local rings, especially in mixed characteristic.

Findings

Bounds on Betti and Bass numbers in Cohen-Macaulay rings

Characterizations of ring properties via uniform arithmetic behavior

Validation of the Improved New Intersection Theorem in certain mixed characteristic cases

Abstract

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally, variants of the homological theorems are shown to hold in equal characteristic. This theory is then applied to Noetherian local rings in order to get: (i) over a Cohen-Macaulay local ring, uniform bounds on the Betti numbers of a Cohen-Macaulay module in terms of dimension and multiplicity, and similar bounds for the Bass numbers of a finitely generated module; (ii) a characterization for being respectively analytically unramified, analytically irreducible, unmixed, quasi-unmixed, normal, Cohen-Macaulay, pseudo-rational, or weakly F-regular in terms of certain uniform arithmetic behavior; (iii) in mixed characteristic, the Improved New Intersection Theorem when the residual characteristic or ramification index is large with respect to dimension (and some other numerical invariants).