Splitting lemma

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any , the following statements are equivalent for a short exact sequence If any of these statements holds, the sequence is called a split exact sequence, and the sequence is said to split. In the above short exact sequence, where the sequence splits, it allows one to refine the first isomorphism theorem, which states that: C ≅ B/ker r ≅ B/q(A) (i.e., C isomorphic to the of r or cokernel of q) to: B = q(A) ⊕ u(C) ≅ A ⊕ C where the first isomorphism theorem is then just the projection onto C. It is a generalization of the rank–nullity theorem (in the form V ≅ ker T ⊕ im T) in linear algebra. 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. To prove that 1. implies 3., first note that any member of B is in the set (ker t + q). This follows since for all b in B, b = (b − qt(b)) + qt(b); qt(b) is in im q, and b − qt(b) is in ker t, since t(b − qt(b)) = t(b) − tqt(b) = t(b) − (tq)t(b) = t(b) − t(b) = 0. Next, the intersection of im q and ker t is 0, since if there exists a in A such that q(a) = b, and t(b) = 0, then 0 = tq(a) = a; and therefore, b = 0. This proves that B is the direct sum of im q and ker t. So, for all b in B, b can be uniquely identified by some a in A, k in ker t, such that b = q(a) + k. By exactness ker r = im q. The subsequence B ⟶ C ⟶ 0 implies that r is onto; therefore for any c in C there exists some b = q(a) + k such that c = r(b) = r(q(a) + k) = r(k). Therefore, for any c in C, exists k in ker t such that c = r(k), and r(ker t) = C. If r(k) = 0, then k is in im q; since the intersection of im q and ker t = 0, then k = 0. Therefore, the restriction r: ker t → C is an isomorphism; and ker t is isomorphic to C. Finally, im q is isomorphic to A due to the exactness of 0 ⟶ A ⟶ B; so B is isomorphic to the direct sum of A and C, which proves (3). To show that 2. implies 3.

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Homological algebra

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of .

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In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.