In the category of groups, there are lots of "exact sequences", e.g. 4 → H → 2, that neither split nor cosplit, where H is the eight-element group of quaternions, and lots of sequences like 4 → D → 2 that split but do not cosplit, where D is the eight-element dihedral group. By "2" and "4" I mean the cyclic groups of those orders. By "exact sequence" A → B → C, I mean that A is the kernel of the quotient B → C (equivalently C is the cokernel of the subobject A → B). A sequence A → B → C "splits" if there is a map C → B so that the compotision C → B → C is the identity; cosplitting is on the other side.


So in groups, a split exact sequence does not necessarily cosplit. (In fact, I have a hard time thinking of any cosplit sequences.) On the other hand, my friends who do ring theory state definitions like "A ring is semisimple if any short exact sequence of modules splits". Why don't they ask for the sequence to cosplit? Does that come for free? (Or am I misremembering the definition?)


More generally, what conditions does one have to place on a category so that "splits" implies "cosplits"?



When does "splits" imply "cosplits"?
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 (1B-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 (1B-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](http://en.wikipedia.org/wiki/Mitchell%27s_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?


