exact functor ExactFunctor 2002-01-05 15:35:35 2003-09-20 21:56:37 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A covariant functor $F$ is said to be {\it left exact} if whenever $$0 \to A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C$$ is an exact sequence, then $$0 \to FA \buildrel F\alpha \over \longrightarrow FB \buildrel F\beta \over \longrightarrow FC$$ is also an exact sequence. A covariant functor $F$ is said to be {\it right exact} if whenever $$A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C \to 0$$ is an exact sequence, then $$FA \buildrel F\alpha \over \longrightarrow FB \buildrel F\beta \over \longrightarrow FC \to 0$$ is also an exact sequence. A contravariant functor $F$ is said to be {\it left exact} if whenever $$A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C \to 0$$ is an exact sequence, then $$0 \to FC \buildrel F\beta \over \longrightarrow FB \buildrel F\alpha \over \longrightarrow FA$$ is also an exact sequence. A contravariant functor $F$ is said to be {\it right exact} if whenever $$0 \to A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C$$ is an exact sequence, then $$FC \buildrel F\beta \over \longrightarrow FB \buildrel F\alpha \over \longrightarrow FA \to 0$$ is also an exact sequence. A (covariant or contravariant) functor is said to be {\it exact} if it is both left exact and right exact.