Integrally closed $\mathfrak{m}$-primary ideals have extremal resolutions

TL;DR

This paper proves that integrally closed $rak{m}$-primary ideals in local rings have maximal complexity and curvature, and characterizes complete intersection rings based on these invariants, providing simplified proofs of known results.

Contribution

It establishes that such ideals always have maximal complexity and curvature, and characterizes complete intersection rings through these invariants, with concise proofs.

Findings

Integrally closed $rak{m}$-primary ideals have maximal complexity and curvature.

Complete intersection rings are characterized by the complexity and curvature of these ideals.

Provides short proofs of known results on projective, injective, and Gorenstein dimensions.

Abstract

We show that every integrally closed $\mathfrak{m}$-primary ideal $I$ in a commutative Noetherian local ring $(R,\mathfrak{m},k)$ has maximal complexity and curvature, i.e., $ {\rm cx}_R(I) = {\rm cx}_R(k) $ and $ {\rm curv}_R(I) = {\rm curv}_R(k) $. As a consequence, we characterize complete intersection local rings in terms of complexity, curvature and complete intersection dimension of such ideals. The analogous results on projective, injective and Gorenstein dimensions are known. However, we provide short proofs of these results as well.