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.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
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
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Abelian category concept
  2. Additive functor concept
  3. Short exact sequence concept
  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
Changed2 November 2026

Markdown ยท JSON ยท Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.