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Let 0 → G < sub > n </ sub > → … → G < sub > 0 </ sub > → 0 denote a finite complex of free R-modules such that ⊕< sub > i </ sub > H < sub > i </ sub >( G < sub >•</ sub >) has finite length but is not 0.

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Let 0 → G < sub > n </ sub > → … → G < sub > 0 </ sub > → 0 denote a finite complex of free R-modules such that H < sub > i </ sub >( G < sub >•</ sub >) has finite length for i > 0 and H < sub > 0 </ sub >( G < sub >•</ sub >) has a minimal generator that is killed by a power of the maximal ideal of R. Then dim R ≤ n.

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