Nine lemma

The Neunerlemma, named because of the structure of the diagram shown below, also 3x3 lemma is a mathematical statement of commuting diagrams and exact sequences which is valid for any abelian category and the category of groups.

Statement

If ( in an abelian category, or the category of groups) the diagram

Commutative and all columns, and the bottom two rows are exact, so the top row is exact. Likewise, the following applies: If all columns, and the top two lines exactly, so the bottom row is exact.

Evidence

The proof is by diagram chase, first under the assumption that the graph relates to the category of groups. For simplicity, all horizontal pictures with h, all vertical are denoted by v. The neutral element of the group name, respectively. The evidence shows the typical characteristic of diagram hunts that although the documentary evidence consists entirely of trivial steps to be taken together, however confusing or unmotivated act - only if you tracked the steps in the diagram, the Zusammanhänge be obvious.

Be first all columns, and the bottom two lines exactly.

  • It has to do so. It follows from the injectivity of the well and of finally.
  • If so, is so.
  • It has, so, so for a. It follows also, that for a. Then, from which already follows.
  • If so there is a with. Because there is a with. Next there is a with, ie. Thus differ and around for a suitable, ie it applies. Then and finally.

All points together show the accuracy of the first row.

Be now all the columns and the top two lines exactly.

  • If so for one and then for one, each by surjectivity of or. Then is.
  • If, as a. Then.
  • It has and we choose one with, so, so for a. Continue for one. Then, they are for a. Finally.
  • It has and we choose to work, and hence for a. It is, therefore, already. Consequently, for a. For reasons already and thus follows.

Together, this results in turn the accuracy of the last line.

The first carried out for groups of proof applies (possibly in additive notation translated) as well as for abelian groups or even for modules over a ring. By the embedding theorem of Mitchell but this is already sufficient to prove the Neunerlemma for all abelian categories.

Mathematics Commutative diagram Mitchell's embedding theorem Snake lemma
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