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arXiv · 2610.29309

On some problems concerning the growth of Betti numbers

Kaito Kimura
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Abstract

We investigate several problems concerning the growth of Betti numbers over commutative noetherian rings. First, we establish uniform positive lower and finite upper bounds for the ratios of consecutive Betti numbers of finitely generated modules of infinite projective dimension over an arbitrary local ring, giving an affirmative answer to a question of Avramov. We also prove the upper semicontinuity of the curvature of residue fields of localizations over excellent rings. Next, using zeros of Hilbert series, we obtain criteria for exponential growth of Betti and Bass numbers of graded modules. As applications, we establish exponential growth of the Betti numbers of canonical modules for non-Gorenstein positively graded Cohen-Macaulay domains. Furthermore, we show that the first Betti number of the canonical module of a non-Gorenstein generically Gorenstein Cohen-Macaulay local ring is strictly larger than the zeroth. In particular, both questions of Jorgensen and Leuschke have affirmative answers for Cohen-Macaulay quotients of Gorenstein local rings by monomial ideals in regular sequences.

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Kaito Kimura. 2026-10-21. On some problems concerning the growth of Betti numbers. https://arxiv.org/abs/2610.29309

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