Return to Answer

Yes, it comes for free. A short exact sequence of abelian groups, or R-modules (R not necessarily commutative), splits iff it cosplits iff the middle term is the sum of the other two terms (sorry if I misread and you already know this!). The key here is that maps of modules and be added/subtracted:

Let the middle maps in 0→A→B→C→0 be f:A→B and g:B→C. Then if there's a splitting q:C→B, then (1_B-qg):B→B is a projection onto A as a submodule of B, i.e. a cosplitting. Together the splitting and cosplitting exhibit B as the direct sum of A and C.

A dual trick shows that a cosplit sequences are split: if p:B→A is a cosplitting, then (1_B-fp):B→B is a projection onto a submodule of itself which is isomorphic to C via g, i.e. a cosplitting, so again B is the sum of A and C via these maps.

More generally, this works in any abelian category. One way to recognize this instantly is via Freyd's Exact Embedding theorem, which roughly implies that you can pretend a diagram in an abelian category is a diagram of R-modules for some R.

... whoah... are you guys related?

Yes, it comes for free. A short exact sequence of abelian groups, or R-modules (R not necessarily commutative), splits iff it cosplits iff the middle term is the sum of the other two terms. The key here is that maps of modules and be added/subtracted:

Let the middle maps in 0→A→B→C→0 be f:A→B and g:B→C. Then if there's a splitting q:C→B, then (1_B-qg):B→B is a projection onto A as a submodule of B, i.e. a cosplitting. Together the splitting and cosplitting exhibit B as the direct sum of A and C.

A dual trick shows that a cosplit sequences are split: if p:B→A is a cosplitting, then (1_B-fp):B→B is a projection onto a submodule of itself which is isomorphic to C via g, i.e. a cosplitting, so again B is the sum of A and C via these maps.

More generally, this same trick works in any abelian category. One way to recognize this instantly is via Freyd's Exact Embedding theorem, which roughly implies that you can pretend a diagram in an abelian category is a diagram of R-modules for some R.

... whoah... are you guys related?

Yes, it comes for free. A short exact sequence of abelian groups, or modules, splits iff it cosplits iff the middle term is the sum of the other two terms (sorry if I misread and you already know this!). The key here is that maps of modules and be added/subtracted:

Let the middle maps in 0→A→B→C→0 be f:A→B and g:B→C. Then if there's a splitting q:C→B, then (1_B-qg):B→B is a projection onto A as a submodule of B, i.e. a cosplitting. Together the splitting and cosplitting exhibit B as the direct sum of A and C.

A dual trick shows that a cosplit sequences are split: if p:B→A is a cosplitting, then (1_B-fp):B→B is a projection onto a submodule of itself which is isomorphic to C via g, i.e. a cosplitting, so again B is the sum of A and C via these maps.

More generally, this works in any abelian category.

Yes, it comes for free. A short exact sequence of abelian groups, or R-modules (R not necessarily commutative), splits iff it cosplits iff the middle term is the sum of the other two terms (sorry if I misread and you already know this!). The key here is that maps of modules and be added/subtracted:

Let the middle maps in 0→A→B→C→0 be f:A→B and g:B→C. Then if there's a splitting q:C→B, then (1_B-qg):B→B is a projection onto A as a submodule of B, i.e. a cosplitting. Together the splitting and cosplitting exhibit B as the direct sum of A and C.

A dual trick shows that a cosplit sequences are split: if p:B→A is a cosplitting, then (1_B-fp):B→B is a projection onto a submodule of itself which is isomorphic to C via g, i.e. a cosplitting, so again B is the sum of A and C via these maps.

More generally, this works in any abelian category. One way to recognize this instantly is via Freyd's Exact Embedding theorem, which roughly implies that you can pretend a diagram in an abelian category is a diagram of R-modules for some R.

... whoah... are you guys related?

Yes, it comes for free. A short exact sequence of abelian groups, or modules, splits iff it cosplits iff the middle term is the sum of the other two terms (sorry if I misread and you already know this!):

Let the middle maps in 0→A→B→C→0 be f:A→B and g:B→C. Then if there's a splitting q:C→B, then (1_B-qg):B→B is a projection onto A as a submodule of B, i.e. a cosplitting. A dual trick shows that a cosplic sequences are split.

Yes, it comes for free. A short exact sequence of abelian groups, or modules, splits iff it cosplits iff the middle term is the sum of the other two terms (sorry if I misread and you already know this!). The key here is that maps of modules and be added/subtracted:

Let the middle maps in 0→A→B→C→0 be f:A→B and g:B→C. Then if there's a splitting q:C→B, then (1_B-qg):B→B is a projection onto A as a submodule of B, i.e. a cosplitting. Together the splitting and cosplitting exhibit B as the direct sum of A and C.

A dual trick shows that a cosplit sequences are split: if p:B→A is a cosplitting, then (1_B-fp):B→B is a projection onto a submodule of itself which is isomorphic to C via g, i.e. a cosplitting, so again B is the sum of A and C via these maps.

More generally, this works in any abelian category.