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arXiv · 2606.11443

Regularity is bounded on a quasi-excellent Noetherian scheme

Alessandro De Stefani·Jack Jeffries·Nawaj KC·Luis Núñez-Betancourt
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Abstract

A point of a scheme has an associated tangent cone, the spectrum of a standard graded algebra encoding the local singularity. Its homological complexity can be measured by its graded Betti table: a matrix that records a part of the structure of its graded, minimal free resolution over a polynomial ring. A natural question is whether the homological complexity of the tangent cones varies arbitrarily across a scheme. In this paper, we show that this is not the case for a quasi-excellent Noetherian scheme; over such schemes, only finitely many graded Betti tables can occur. More generally, we show that a coherent sheaf over a quasi-excellent Noetherian scheme admits finitely many graded Betti tables, and that the constancy loci for the graded Betti table are constructible. As an immediate consequence, regularity is bounded on a quasi-excellent Noetherian scheme.

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Alessandro De Stefani, Jack Jeffries, Nawaj KC, Luis Núñez-Betancourt. 2026-06-09. Regularity is bounded on a quasi-excellent Noetherian scheme. https://arxiv.org/abs/2606.11443

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