nLab splicing of short exact sequences

Definition

Given a sequence of short exact sequences of the form

$$0 \to X_n \overset{i_n}{\longrightarrow} Y_n \overset{p_n}{\longrightarrow} X_{n+1} \to 0$$

for $n \in \mathbb{Z}$, then their splicing is the horizontal composite sequence

$$\array{ \cdots && \overset{}{\longrightarrow} && Y_{n-1} && \longrightarrow && Y_n && \longrightarrow && Y_{n+1} && \longrightarrow && \cdots \\ && && & {}_{\mathllap{p_{n-1}}}\searrow && \nearrow_{\mathrlap{ i_n } } && {}_{\mathllap{p_n}}\searrow && \nearrow_{\mathrlap{ i_{n+1} }} \\ && && && X_n && && X_{n+1} }$$

which is, evidently, a long exact sequence.

More generally, there is splicing of interlocking systems of long exact sequences. See at Exact couple of a tower of fibrations.

Related entries

• Adams spectral sequence