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Concept
Exact functor
Summary
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled.
Let P and Q be abelian categories, and let F: P→Q be a covariant additive functor (so that, in particular, F(0) = 0). We say that F is an exact functor if whenever
is a short exact sequence in P then
is a short exact sequence in Q. (The maps are often omitted and implied, and one says: "if 0→A→B→C→0 is exact, then 0→F(A)→F(B)→F(C)→0 is also exact".)
Further, we say that F is
left-exact if whenever 0→A→B→C→0 is exact then 0→F(A)→F(B)→F(C) is exact;
right-exact if whenever 0→A→B→C→0 is exact then F(A)→F(B)→F(C)→0 is exact;
half-exact if whenever 0→A→B→C→0 is exact then F(A)→F(B)→F(C) is exact. This is distinct from the notion of a topological half-exact functor.
If G is a contravariant additive functor from P to Q, we similarly define G to be
exact if whenever 0→A→B→C→0 is exact then 0→G(C)→G(B)→G(A)→0 is exact;
left-exact if whenever 0→A→B→C→0 is exact then 0→G(C)→G(B)→G(A) is exact;
right-exact if whenever 0→A→B→C→0 is exact then G(C)→G(B)→G(A)→0 is exact;
half-exact if whenever 0→A→B→C→0 is exact then G(C)→G(B)→G(A) is exact.
It is not always necessary to start with an entire short exact sequence 0→A→B→C→0 to have some exactness preserved. The following definitions are equivalent to the ones given above:
F is exact if and only if A→B→C exact implies F(A)→F(B)→F(C) exact;
F is left-exact if and only if 0→A→B→C exact implies 0→F(A)→F(B)→F(C) exact (i.e. if "F turns kernels into kernels");
F is right-exact if and only if A→B→C→0 exact implies F(A)→F(B)→F(C)→0 exact (i.e. if "F turns cokernels into cokernels");
G is left-exact if and only if A→B→C→0 exact implies 0→G(C)→G(B)→G(A) exact (i.e.
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