Math 2.0

Category theory; homological algebra Definition

Exact functor

Builds on Abelian categoryAdditive functorShort exact sequence

Definition

Let ๐’œ\mathcal{A} and โ„ฌ\mathcal{B} be abelian categories and let Fโ€‰โฃ:๐’œโ†’โ„ฌF \colon \mathcal{A} \to \mathcal{B} be an additive functor. Let 0โ†’Aโ€ฒโ†’Aโ†’Aโ€ฒโ€ฒโ†’00 \to A' \to A \to A'' \to 0 denote an arbitrary short exact sequence in ๐’œ\mathcal{A}.

  1. FF is left exact if 0โ†’F(Aโ€ฒ)โ†’F(A)โ†’F(Aโ€ฒโ€ฒ)0 \to F(A') \to F(A) \to F(A'') is exact for every such sequence.
  2. FF is right exact if F(Aโ€ฒ)โ†’F(A)โ†’F(Aโ€ฒโ€ฒ)โ†’0F(A') \to F(A) \to F(A'') \to 0 is exact for every such sequence.
  3. FF is exact if 0โ†’F(Aโ€ฒ)โ†’F(A)โ†’F(Aโ€ฒโ€ฒ)โ†’00 \to F(A') \to F(A) \to F(A'') \to 0 is exact for every such sequence, that is, if FF is both left exact and right exact.

Remarks

An exact functor preserves all exact sequences, not only the short ones. A functor is left exact if and only if it preserves kernels, and right exact if and only if it preserves cokernels. For a contravariant functor the same words are used after replacing ๐’œ\mathcal{A} by its opposite category.

Example: for a ring RR and an RR-module MM, the functor HomโกR(M,โˆ’)\Hom_R(M, -) is left exact, and MโŠ—Rโˆ’M \otimes_R - is right exact. The functor โ„šโŠ—โ„คโˆ’\Q \otimes_{\Z} - on abelian groups is exact, because โ„š\Q is a flat โ„ค\Z-module.

Non-example: F=โ„ค/2โ„คโŠ—โ„คโˆ’F = \Z/2\Z \otimes_{\Z} - is right exact but not exact. Apply it to 0โ†’โ„คโ†’2โ„คโ†’โ„ค/2โ„คโ†’00 \to \Z \xrightarrow{2} \Z \to \Z/2\Z \to 0. The first map becomes multiplication by 22 on โ„ค/2โ„ค\Z/2\Z, which is zero and so not injective.

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Abelian category, concept, no evidence levelAbelian categoryconceptAdditive functor, concept, no evidence levelAdditive functorconceptShort exact sequence, concept, no evidence levelShort exact sequenceconceptExact functor, definition, no evidence levelExact functordefinition
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  1. Abelian category concept
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  4. Exact functor definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Exact_functor
About this page
TagAK2XQC04PC
URIm2://AK2XQC04PC
Kinddefinition
WritingW0 stub
MSC18E10, 18G10
DefinesExact functor
UsesAbelian category, Additive functor, Short exact sequence

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