From a short exact sequence of complexes to a long one in homology
Suppose 0 → A_• → B_• → C_• → 0 is a short exact sequence of chain complexes — exact in each degree. Then homology does not simply split apart; instead the snake lemma, applied degree by degree, knits the homologies into one long exact sequence. The same connecting homomorphism δ we built by hand in guide 2 is what drops you from degree n in C down to degree n−1 in A.
Short exact sequence of complexes: 0 -> A. -> B. -> C. -> 0
Long exact sequence in homology (runs forever both ways):
... -> H_n(A) -> H_n(B) -> H_n(C) --d--> H_{n-1}(A) -> H_{n-1}(B) -> ...
Where d (the connecting map) is the snake's bite applied in each degree.
Derived-functor version (the LES of Ext):
for a short exact sequence 0 -> A -> B -> C -> 0 of modules,
0 -> Hom(C,N) -> Hom(B,N) -> Hom(A,N)
--d--> Ext^1(C,N) -> Ext^1(B,N) -> Ext^1(A,N)
--d--> Ext^2(C,N) -> ...
The connecting map d repairs exactly the surjectivity that Hom(-,N) lost.
Tor has the dual LES, with Tor_n decreasing in degree.Using the long exact sequence
The LES turns into arithmetic. Take 0 → ℤ →(×2) ℤ → ℤ/2 → 0 and apply – ⊗ ℤ/2 to get the Tor long exact sequence. Because ℤ is flat, all its higher Tor vanish, which forces the connecting maps and pins down Tor_1(ℤ/2, ℤ/2) cleanly.
0 -> Z --x2--> Z -> Z/2 -> 0, apply -(x)Z/2. Tor LES:
Tor_1(Z,Z/2) -> Tor_1(Z/2,Z/2) --d--> Z(x)Z/2 --x2--> Z(x)Z/2 -> Z/2(x)Z/2 -> 0
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0 (Z flat) Z/2 --0--> Z/2 Z/2
The map 'x2' on Z(x)Z/2 = Z/2 is multiplication by 2 = 0.
Exactness then gives:
Tor_0(Z/2,Z/2) = Z/2 (x) Z/2 = Z/2,
and Tor_1(Z/2,Z/2) = ker(x2 = 0 map) = Z/2.
Matches gcd(2,2)=2 from guide 4. Z/2 is NOT flat: it has torsion,
and Tor_1 is exactly the witness.The home of the theory, and what comes next
Everything in this track — kernels, cokernels, exact sequences, the snake — needs only a setting where these notions make sense. That setting is an abelian category: a category with a zero object, all kernels and cokernels, and where every monic is a kernel and every epic a cokernel. Modules over a ring form one; so do sheaves, and that generality is exactly why one proof of the snake lemma serves topology, geometry, and number theory at once.
Two doors open from here. Specialize: derived functors of the fixed-point functor over a group give group cohomology, and over a Galois group give Galois cohomology and Hilbert's Theorem 90. Generalize: when a double complex resists direct computation, the spectral sequence organizes its homology into successive approximations — the next serious tool after the long exact sequence, and the natural sequel to this track.