On derived functors of limit

Abstract

If A \mathcal {A} is a cocomplete category with enough projectives and C {\mathbf {C}} is a ↓ \downarrow -finite small category, then there is a spectral sequence which shows that the cardinality of C {\mathbf {C}} and colimits over finite initial subcategories C ′ {\mathbf {C’}} of C {\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 C {\mathbf {C}} is at most ℵ n \aleph _{n} , and the flat dimension of Δ ∗ Z {\Delta ^ \ast }Z (constant diagram of type C op {{\mathbf {C}}^{{\text {op}}}} with value Z Z ) is k k , then the derived functors of lim C : A b C → A b {\lim _{\mathbf {C}}}:\mathcal {A}{b^{\mathbf {C}}} \to \mathcal {A}b vanish above dimension n + 1 + k n + 1 + k .