5.3. Split-exact sequences
We will be particularly interested in exact sequences which are split in
the following sense.
0NM8
Definition 5.18 . ΞΊ \kappa β \infty ΞΊ \kappa
π \textstyle{{\mathcal{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces} f \scriptstyle{f} β¬ \textstyle{{\mathcal{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces} g \scriptstyle{g} π \textstyle{\mathcal{C}}
is called split-exact if there exist exact functors i : β¬ β π i\colon{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}} j : π β β¬ j\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}} f f g g i β f β Id i\circ f\simeq\Id g β j β Id g\circ j\simeq\Id
We will also be interested in (split-) exact sequences of spectral categories.
0NM9
Definition 5.19 . π β β¬ β π {\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}} β \infty
N β‘ ( Ξ© β β Mod β ( π ) cf ) βΆ N β‘ ( Ξ© β β Mod β ( β¬ ) cf ) βΆ N β‘ ( Ξ© β β Mod β ( π ) cf ) \mathrm{N}(\Omega^{\infty}\Mod({\mathcal{A}})^{\cf})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}(\Omega^{\infty}\Mod({\mathcal{B}})^{\cf})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}(\Omega^{\infty}\Mod({\mathcal{C}})^{\cf})
is exact.
The following characterization is an immediate corollary of propositionΒ 5.15
0NMA
Proposition 5.20 . A sequence π β β¬ β π {\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}
π β‘ ( π ) βΆ π β‘ ( β¬ ) βΆ π β‘ ( π ) {\mathcal{D}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}({\mathcal{B}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}({\mathcal{C}})
is exact.
Next, observe that we can relate these notions as follows (the proof
of which is immediate):
0NMB
Proposition 5.21 . Let π β β¬ β π {\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}} Ξ¨ perf β ( π ) β Ξ¨ perf β ( β¬ ) β Ξ¨ perf β ( π ) \Psi_{\perf}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\perf}({\mathcal{B}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\perf}({\mathcal{C}}) β \infty
We also have an essential converse statement.
0NMC
Proposition 5.22 . Let π β β¬ β π {\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}} β \infty
π ~ βΆ β¬ ~ βΆ π ~ \widetilde{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathcal{B}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathcal{C}}}
such that
Ξ¨ perf β ( π ~ β β¬ ~ β π ~ ) \Psi_{\perf}(\widetilde{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathcal{B}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathcal{C}}}) π β β¬ β π {\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}
0NMD
Proof. This follows from propositionΒ 4.28