Split

Split(s) or The Split may refer to:

Places

Music

Games

Sports and dance

Splitting lemma

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for a short exact sequence are equivalent.

Given a short exact sequence with maps q and r:

one writes the additional arrows t and u for maps that may not exist:

Then the following statements are equivalent:

The short exact sequence is called split if any of the above statements hold.

(The word "map" refers to morphisms in the abelian category we are working in, not mappings between sets.)

It allows one to refine the first isomorphism theorem:

the first isomorphism theorem states that in the above short exact sequence, (i.e. "C" isomorphic to the coimage of "r" or cokernel of "q")

if the sequence splits, then , and the first isomorphism theorem is just the projection onto C.

It is a categorical generalization of the rank–nullity theorem (in the form ) in linear algebra.

Proof

First, to show that (3) implies both (1) and (2), we assume (3) and take as t the natural projection of the direct sum onto A, and take as u the natural injection of C into the direct sum.

Split (Lush album)

Split is the second album by British shoegazing act Lush, released on 4AD on June 13, 1994 in the UK and a day later in the US. Two singles were released from the album: "Desire Lines" and "Hypocrite," both released on May 30, 1994. Split was reissued by 4AD on CD in July 2001.

Reception

Select gave the album a negative review of two out of five. The review described the album as "mid-paced stuff, fitting between melancholy and listlessness." "There's nothing wrong with a dose of heavyweight introspection per se. But a pretty deft touch is needed to translate it movingly to the recording studio."

Andy Kellman, writing for Allmusic, was far more positive, giving the album 4.5 stars out of five: "Split touches on most forms of emotional turbulence. A legitimizing stunner, the record prevented the band from being lost amidst the bunker of form-over-function dream pop bands. Split shattered every negative aspect of those failed acts with flying colors. A fantastic record within any realm."