Vol. 57, No. 2, 1975

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Basically bounded functors and flat sheaves

William Charles Waterhouse

Vol. 57 (1975), No. 2, 597–610

A functor is here called basically bounded if, roughly speaking, it is determined by its values on objects of some bounded cardinality. For functors on R-algebras, it is shown that common constructions involving basically bounded functors can again be computed on algebras of bounded size, and hence are uniquely defined irrespective of any special set-theoretic assumptions. Even operations which seem to require arbitrarily large algebras—computing Čech cohomology and sheafifications in the flat topology, forming Ext groups and sheaves—turn out to be basically bounded. The proofs use homological algebra and a notion of approximation by small coverings.

Mathematical Subject Classification 2000
Primary: 14F20
Received: 24 April 1973
Published: 1 April 1975
William Charles Waterhouse