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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.
Horseshoe lemma

Let RR be a ring and

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

a short exact sequence of RR-modules (see ).

Statement (projective version)

Suppose we are given projective resolutions

⋯→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

with each PnA,PnCP_n^A, P_n^C . Then there exists a projective resolution of BB,

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

and chain maps fitting into a short exact sequence of chain complexes

0→P∙A→P∙B→P∙C→0 0 \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 for all nn, and the augmentation maps recover 0→A→B→C→00\to A\to B\to C\to 0 in degree 00.

Equivalently: you can “splice” the two resolutions into one, degreewise as a direct sum, with differentials chosen so that exactness holds.

Statement (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 ).

Examples

Example 1 (building a resolution of Z/6\mathbb Z/6 from Z/2\mathbb Z/2 and Z/3\mathbb Z/3)

In Ab\mathbf{Ab} (i.e. R=ZR=\mathbb Z), there is a short exact sequence

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

Use 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.

Horseshoe 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,\quad P_0^B \cong \mathbb Z\oplus \mathbb Z,

and an exact sequence of complexes 0→P∙Z/2→P∙Z/6→P∙Z/3→00\to P_\bullet^{\mathbb Z/2}\to P_\bullet^{\mathbb Z/6}\to P_\bullet^{\mathbb Z/3}\to 0. This is not minimal, but it is explicit and sufficient to compute and groups.

Example 2 (computing Tor⁡1\operatorname{Tor}_1 from a short exact sequence)

Let 0→A→B→C→00\to A\to B\to C\to 0 be short exact and fix an RR-module NN. Take a projective resolution P∙BP_\bullet^B of BB. By horseshoe, you can choose compatible resolutions so that 0→P∙A→P∙B→P∙C→00\to P_\bullet^A\to P_\bullet^B\to P_\bullet^C\to 0 is exact. Tensoring degreewise with NN gives a short exact sequence of chain complexes (tensor is right exact; see ), and the resulting long exact homology sequence identifies the connecting map as the boundary in .

Example 3 (injective horseshoe and Ext⁡\operatorname{Ext})

If you want Ext⁡Rn(−,I)\operatorname{Ext}^n_R(-,I) computations, an injective horseshoe lets you choose compatible injective resolutions to compute

Ext⁡Rn(C,M)→Ext⁡Rn(B,M)→Ext⁡Rn(A,M) \operatorname{Ext}^n_R(C,M)\to \operatorname{Ext}^n_R(B,M)\to \operatorname{Ext}^n_R(A,M)

via a short exact sequence of cochain complexes and the induced long exact cohomology sequence (see ).