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Let $R$ be a local Cohen-Macaulay ring with canonical module $\omega_{R}$. We investigate the following question of Huneke: If the sequence of Betti numbers $\{\beta^{R}_{i}(\omega_{R})\}$ has polynomial growth, must $R$ be Gorenstein? This question is well-understood when $R$ has minimal multiplicity. We investigate this question for a more general class of rings which we say are homologically of minimal multiplicity. We provide several characterizations of the rings in this class and establish a general ascent and descent result.
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