Horseshoe lemma explained
In homological algebra, the horseshoe lemma, also called the simultaneous resolution theorem, is a statement relating resolutions of two objects
and
to resolutions ofextensions of
by
. It says that if an object
is an extension of
by
, then a resolution of
can be built up
inductively with the
nth item in the resolution equal to the
coproduct of the
nth items in the resolutions of
and
. The name of the lemma comes from the shape of the diagram illustrating the lemma's hypothesis.
Formal statement
Let
be an
abelian category with
enough projectives. If
is a diagram in
such that the column is
exact and therows are projective resolutions of
and
respectively, thenit can be completed to a commutative diagram
where all columns are exact, the middle row is a projective resolutionof
, and
for all
n. If
is anabelian category with
enough injectives, the
dual statement also holds.
The lemma can be proved inductively. At each stage of the induction, the properties of projective objects are used to define maps in a projective resolution of
. Then the
snake lemma is invoked to show that the simultaneous resolution constructed so far has exact rows.
See also
References
- Book: Henri . Cartan . Henri Cartan . Samuel . Eilenberg . Samuel Eilenberg . [{{GBurl|0268b52ghcsC|pg=PR11}} Homological algebra ]. Princeton University Press . 1956 . 1999 . 978-0-691-04991-5 .
- Book: Osborne, M. Scott . [{{GBurl|Zdk7w3Jl64oC|pg=PP5}} Basic homological algebra ]. Springer . 2000 . 978-0-387-98934-1 .
