On derived functors of limit

Latch, Dana May

doi:10.1090/S0002-9947-1973-0323866-0

On derived functors of limit
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by Dana May Latch

Trans. Amer. Math. Soc. 181 (1973), 155-163

DOI: https://doi.org/10.1090/S0002-9947-1973-0323866-0

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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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