$K_2$-regularity and normality

TL;DR

The paper establishes that $K_2$-regularity implies normality for noetherian algebras of finite normalization over fields of characteristic zero, extending Vorst's ideas via Milnor squares and $K$-theory excision. It then connects higher du Bois singularities to $K$-theory, showing that $K_{p+1}$-regularity implies $p$-du Bois properties and hence regularity in codimension $p$, with sharper conclusions for local complete intersections: $K_{p+1}$-regularity yields normality and regularity in codimension $2p$. By leveraging Mustaţă–Popa results, the authors obtain stronger bounds in the lci case, notably that a local complete intersection affine scheme that is $K_{p+1}$-regular is normal and regular in codimension $2p$, and in the surface case, $K_2$-regularity forces full regularity. Overall, the work deepens the link between algebraic $K$-theory and singularity theory, providing sharper criteria and revealing the role of $p$-du Bois properties in controlling regularity.

Abstract

We take a fresh look at the relationship between $K$-regularity and regularity of schemes, proving two results in this direction. First, we show that $K_2$-regular affine algebras over fields of characteristic zero are normal. Second, we improve on Vorst's $K$-regularity bound in the case of local complete intersections; this is related to recent work on higher du Bois singularities.

Theorems & Definitions (38)

• Theorem 1
• Remark 2
• Theorem 3
• Example 4
• Remark 5
• Lemma 1.3
• proof
• Corollary 1.4
• proof
• Lemma 1.5