Knowlpedia

Horseshoe lemma

Given a short exact sequence of modules, compatible projective (or injective) resolutions can be spliced to produce a resolution of the middle module.

Let RR be a ring and

0⟶A→iB→pC⟶00 \longrightarrow A \xrightarrow{i} B \xrightarrow{p} C \longrightarrow 0

a of RR-modules. Given projective resolutions P∙A→AP_\bullet^A\to A and P∙C→CP_\bullet^C\to C, the horseshoe lemma provides a projective resolution P∙B→BP_\bullet^B\to B and a short exact sequence of chain complexes

0→P∙A→P∙B→P∙C→00\to P_\bullet^A\to P_\bullet^B\to P_\bullet^C\to 0

such that PnB≅PnA⊕PnCP_n^B\cong P_n^A\oplus P_n^C in every degree nn.

Projective construction

More explicitly, suppose

⋯→P1A→P0A→A→0,⋯→P1C→P0C→C→0\cdots \to P_1^A \to P_0^A \to A \to 0, \qquad \cdots \to P_1^C \to P_0^C \to C \to 0

are resolutions by . The lemma constructs

⋯→P1B→P0B→B→0,\cdots \to P_1^B \to P_0^B \to B \to 0,

whose augmentation maps recover the original short exact sequence. The differentials on the degreewise direct sums are chosen compatibly with the maps ii and pp.

Injective version

Dually, given injective resolutions of AA and CC, one constructs an injective resolution of BB with IBn≅IAn⊕ICnI^n_B\cong I^n_A\oplus I^n_C and a short exact sequence of cochain complexes

0→IA∙→IB∙→IC∙→0.0 \to I^\bullet_A \to I^\bullet_B \to I^\bullet_C \to 0.

See and .

Why it matters

The horseshoe lemma underlies functorial constructions of the long exact sequences in and (see and ), and is a standard way to build resolutions needed to compute derived functors (see ).

Example

In Ab\mathbf{Ab}, the sequence

0→Z/2→Z/6→Z/3→0.0 \to \mathbb Z/2 \to \mathbb Z/6 \to \mathbb Z/3 \to 0.

is exact. Applying the lemma to the standard projective resolutions

0→Z→⋅2Z→Z/2→0,0→Z→⋅3Z→Z/3→0.0\to \mathbb Z \xrightarrow{\cdot 2} \mathbb Z \to \mathbb Z/2 \to 0,\qquad 0\to \mathbb Z \xrightarrow{\cdot 3} \mathbb Z \to \mathbb Z/3 \to 0.

produces a resolution of Z/6\mathbb Z/6 with

P1B≅Z⊕Z,P0B≅Z⊕Z.P_1^B\cong\mathbb Z\oplus\mathbb Z,\qquad P_0^B\cong\mathbb Z\oplus\mathbb Z.

This resolution is generally not minimal, but its compatibility with the original exact sequence is what is useful in homological arguments.