Horseshoe lemma

The horseshoe lemma is one of the foundations of homological algebra . It says that the three modules can be resolved in a short exact sequence ( projective or injective ) in such a way that a short exact sequence of resolutions is created.

The result appears - albeit without a name - in the book by Cartan and Eilenberg in 1956 .

The lemma

Be a short exact sequence of modules, or more generally of objects in an Abelian category . Be and projective resolutions. Then there is a projective resolution and chain homomorphisms such that ${\displaystyle 0\rightarrow L\,{\xrightarrow {f}}\,M\,{\xrightarrow {g}}N\rightarrow 0}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle P_{*}^{L}\,{\xrightarrow {\varepsilon ^{L}}}\,L}$ ${\displaystyle P_{*}^{N}\,{\xrightarrow {\varepsilon ^{N}}}\,N}$ ${\displaystyle P_{*}^{M}\,{\xrightarrow {\varepsilon ^{M}}}\,M}$ ${\displaystyle P_{*}^{L}\,{\xrightarrow {f_{*}}}\,P_{*}^{M}\,{\xrightarrow {g_{*}}}\,P_{*}^{N}}$

1. ${\displaystyle 0\rightarrow P_{*}^{L}\,{\xrightarrow {f_{*}}}\,P_{*}^{M}\,{\xrightarrow {g_{*}}}\,P_{*}^{N}\rightarrow 0}$is a short exact sequence of chain complexes . That is, in every degree there is a short exact sequence - which is necessarily disintegrating due to the projectivity of . ${\displaystyle 0\rightarrow P_{n}^{L}\,{\xrightarrow {f_{n}}}\,P_{n}^{M}\,{\xrightarrow {g_{n}}}\,P_{n}^{N}\rightarrow 0}$ ${\displaystyle P_{n}^{N}}$
2. The resulting diagram
   
   
   
   commutes. That is, it is and . ${\displaystyle f\circ \varepsilon ^{L}=\varepsilon ^{M}\circ f_{0}}$ ${\displaystyle g\circ \varepsilon ^{M}=\varepsilon ^{N}\circ g_{0}}$

The corresponding statement for injective resolutions also applies.

About the name

The "horseshoe" (injective case)

The input data resembles a horseshoe, the lemma fills the horseshoe.

Applications

• There are two ways to define the term derived functor . The proof that these two paths are equivalent uses the horseshoe lemma and the snake lemma . The two ways:
  1. Construction via a projective or injective resolution.
  2. Characterization as a universal δ-functor .
• The horseshoe lemma also allows the construction of Cartan – Eilenberg resolutions .

literature

• Joseph J. Rotman : An introduction to homological algebra . 2nd Edition. Springer Verlag, New York 2009, ISBN 978-0-387-24527-0 , pp. 349-350 .
• Charles A. Weibel : An introduction to homological algebra (=  Cambridge studies in advanced mathematics . No. 38 ). Cambridge University Press , 1994, ISBN 0-521-43500-5 , p. 37 .

Individual evidence

1. ^ Henri Cartan , Samuel Eilenberg : Homological Algebra (=  Princeton Mathematical Series . No. 19 ). Princeton University Press , 1956, LCCN  53-010148 , p. 80, Proposition V.2.2 .