Math 2.0

Category theory; homological algebra Homological algebra: complexes and exact sequences Lemma

Nine lemma

Builds on Abelian categoryShort exact sequenceCommutative diagramSnake lemma

Statement

Let ๐’œ\mathcal{A} be an abelian category, such as the category of modules over a ring. Consider a commutative diagram 000โ†“โ†“โ†“0โ†’A1โ†’B1โ†’C1โ†’0โ†“โ†“โ†“0โ†’A2โ†’B2โ†’C2โ†’0โ†“โ†“โ†“0โ†’A3โ†’B3โ†’C3โ†’0โ†“โ†“โ†“000\begin{array}{ccccccccc} & & 0 & & 0 & & 0 & & \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & A_1 & \to & B_1 & \to & C_1 & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & A_2 & \to & B_2 & \to & C_2 & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & A_3 & \to & B_3 & \to & C_3 & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ & & 0 & & 0 & & 0 & & \end{array} in ๐’œ\mathcal{A} in which the three columns are short exact sequences.

  1. If the second and third rows are short exact, then the first row is short exact.
  2. If the first and second rows are short exact, then the third row is short exact.

Remarks

For part 1, apply the snake lemma to the map from the second row to the third row given by the vertical morphisms. Since the columns are exact, the kernels of these vertical morphisms are A1,B1,C1A_1, B_1, C_1 and their cokernels are 00. The exact sequence of the snake lemma becomes 0โ†’A1โ†’B1โ†’C1โ†’00 \to A_1 \to B_1 \to C_1 \to 0, which is the first row. Part 2 is the dual statement and follows in the same way from the map from the first row to the second row, whose cokernels are A3,B3,C3A_3, B_3, C_3. Example: let AโŠ†BโŠ†CA \subseteq B \subseteq C be modules. Take the rows 0โ†’Aโ†’Aโ†’0โ†’00 \to A \to A \to 0 \to 0, then 0โ†’Bโ†’Cโ†’C/Bโ†’00 \to B \to C \to C/B \to 0, then 0โ†’B/Aโ†’C/Aโ†’C/Bโ†’00 \to B/A \to C/A \to C/B \to 0, with the evident maps. The columns 0โ†’Aโ†’Bโ†’B/Aโ†’00 \to A \to B \to B/A \to 0, 0โ†’Aโ†’Cโ†’C/Aโ†’00 \to A \to C \to C/A \to 0 and 0โ†’0โ†’C/Bโ†’C/Bโ†’00 \to 0 \to C/B \to C/B \to 0 are short exact, and so are the first two rows. Part 2 gives that the third row is short exact, which is the third isomorphism theorem (C/A)/(B/A)โ‰…C/B(C/A)/(B/A) \cong C/B.

Going deeper

For a student

The lemma is also called the 3ร—33 \times 3 lemma. It is a quick consequence of the snake lemma, and it is used to compare filtrations and to build short exact sequences of quotients. A typical use is the third isomorphism theorem in an abelian category. It also appears when one shows that constructions such as the mapping cone or the total complex respect short exact sequences.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Abelian category, concept, no evidence levelAbelian categoryconceptCokernel, concept, no evidence levelCokernelconceptCommutative diagram, concept, no evidence levelCommutative diagramconceptEpimorphism, concept, no evidence levelEpimorphismconceptExact sequence, concept, no evidence levelExact sequenceconceptKernel, concept, no evidence levelKernelconceptMonomorphism, concept, no evidence levelMonomorphismconceptPseudo-element, concept, no evidence levelPseudo-elementconceptShort exact sequence, concept, no evidence levelShort exact sequenceconceptSnake lemma, statement, no evidence levelSnake lemmastatementNine lemma, statement, no evidence levelNine lemmastatement
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. Cokernel concept
  3. Commutative diagram concept
  4. Epimorphism concept
  5. Exact sequence concept
  6. Kernel concept
  7. Monomorphism concept
  8. Pseudo-element concept
  9. Short exact sequence concept
  10. Snake lemma statement
  11. Nine lemma statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Nine_lemma
About this page
Tag6ZRRH96VZT
URIm2://6ZRRH96VZT
Kindstatement, lemma
WritingW0 stub
MSC18G35, 18E10
StatusThis copy of the site has no build data. Run m2 status 6ZRRH96VZT.
AboutShort exact sequence
UsesAbelian category, Short exact sequence, Commutative diagram, Snake lemma
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Changed2 November 2026

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