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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On derived functors of limit


Author: Dana May Latch
Journal: Trans. Amer. Math. Soc. 181 (1973), 155-163
MSC: Primary 18E25
MathSciNet review: 0323866
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Abstract: If $ \mathcal{A}$ is a cocomplete category with enough projectives and $ {\mathbf{C}}$ is a $ \downarrow $-finite small category, then there is a spectral sequence which shows that the cardinality of $ {\mathbf{C}}$ and colimits over finite initial subcategories $ {\mathbf{C'}}$ of $ {\mathbf{C}}$ are determining factors for computation of derived functors of colimit. Applying a recent result of Mitchell to this spectral sequence we show that if the cardinality of $ {\mathbf{C}}$ is at most $ \aleph _{n}$, and the flat dimension of $ {\Delta ^ \ast }Z$ (constant diagram of type $ {{\mathbf{C}}^{{\text{op}}}}$ with value $ Z$) is $ k$, then the derived functors of $ {\lim _{\mathbf{C}}}:\mathcal{A}{b^{\mathbf{C}}} \to \mathcal{A}b$ vanish above dimension $ n + 1 + k$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0323866-0
PII: S 0002-9947(1973)0323866-0
Keywords: Derived functors of limit, downward finite small category, directed completion of a category $ \mathcal{A}$, homological dimension, flat dimension
Article copyright: © Copyright 1973 American Mathematical Society