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The Snake lemma associates to a commutative diagram with exact rows an exact sequenceFor $K=L_1\cup L_2$, apply it degree by degree to the short exact sequence of chain complexesThe resulting connecting maps splice the kernels modulo boundaries into the Mayer--Vietoris sequence

Now let $K$ satisfy the stated simplicial pseudomanifold conditions. In an $n$-cycle, cancellation at a common $(n-1)$-face determines the coefficient of either incident $n$-simplex from the other, up to the orientation sign. Connectivity through such faces therefore determines every top-simplex coefficient from one integer. If the signs are globally compatible, their oriented sum is a cycle and generates $H_n(K)\cong \mathbb{Z}$; if they are inconsistent, that integer must be zero and $H_n(K)=0$.

Let $x$ be the resulting fundamental class, represented by $z$. Since the intersection has dimension below $n$, split uniquely
$z=z_1+z_2$ into top chains in $L_1$ and $L_2$. Thenand the Mayer--Vietoris boundary isIt is nonzero exactly when both $L_1$ and $L_2$ contain top-dimensional simplices; if one side contains none, the fundamental cycle already lies in the other side, while if both do, connectedness forces a nonempty interface and its oriented boundary represents a nonzero class.

Finally take $K\cong S^3$, $L_1\cong S^1\times D^2$, and
$L_1\cap L_2\cong T^2$. The preceding boundaryis a nonzero map between copies of $\mathbb{Z}$ and sends the fundamental class to the oriented boundary torus, hence is an isomorphism. Exactness then givesThe next part of the sequence isSince $H_1(L_1)\cong \mathbb{Z}$, it follows that
$H_1(L_2)\cong \mathbb{Z}$. The degree-zero sequence makes $L_2$ connected. Therefore

A complex in an abelian category is a sequence $(A_n,d_n)$ with $d_n{d}_{n+1}=0$. A sequence is exact when the image of every incoming map equals the kernel of the outgoing map.

The Five lemma says that in a morphism between exact five-term sequences, suitable epimorphism assumptions on the left and monomorphism assumptions on the right, together with isomorphisms in the four surrounding positions, force the middle map to be an isomorphism.

The Snake lemma associates to a morphism of short exact sequences the exact sequence

Apply the Snake lemma degree by degree to a short exact sequence of complexesIf $[c]\in H_n(C)$, lift a cycle $c$ to $b\in B_n$. Its boundary maps to zero in $C_{n-1}$, so it comes from a cycle $a\in A_{n-1}$; define $\partial [c]=[a]$. The Snake-lemma exactness and independence checks yieldThis is the algebraic Mayer–Vietoris theorem for homology objects.

An abelian category is an additive category with all kernels and cokernels in which every monomorphism is a kernel and every epimorphism is a cokernel. If $m:A\rightarrowtail B$ is monic and $c=\mathrm{coker}m$, then $m$ factors through $k=\ker c$. The canonical coimage-to-image morphism is an isomorphism in an abelian category, so this factorization identifies $A$ with $\ker c$; hence $m$ is the kernel of its own cokernel. Dually, every epimorphism is the cokernel of its own kernel. The assignments $m\mapsto \mathrm{coker}m$ and $q\mapsto \ker q$ therefore give inverse bijections between subobjects and quotient objects of a fixed object. Well-poweredness is consequently equivalent to well-copoweredness.

The category ${\mathcal{C}}^{\mathcal{A}}$ of complexes has chain complexes as objects and chain maps as morphisms. Write $Z_n=\ker d_n$ and $Q_n=\mathrm{coker}{d}_{n+1}$. The equation $d_n{d}_{n+1}=0$ gives both an induced map $\mathrm{im}{d}_{n+1}\to Z_n$ and a map $Q_n\to \mathrm{im}{d}_n$. The homology object has the two canonically isomorphic descriptionsPassing to the opposite category exchanges these descriptions, proving self-duality.

Let $S_n(A)=A[n]$. A chain map $A[n]\to C_{\bullet }$ is exactly a map $A\to Z_n(C)$, while a chain map $C_{\bullet }\to A[n]$ is exactly a map $Q_n(C)\to A$. Therefore

The Snake lemma states that a commutative diagram with exact rows yields the exact sequence of the three kernels, followed by its connecting morphism and the three cokernels. Apply it degree by degree to a short exact sequence of complexes $0\to A_{\bullet }\to B_{\bullet }\to C_{\bullet }\to 0$, using the diagrams of cycles, boundaries, and degree objects. The connecting map sends a cycle of $C_n$ to a lift in $B_n$, takes its boundary in $B_{n-1}$, and identifies that boundary with a class in $H_{n-1}(A)$. The Snake lemma gives exactness and producesthe algebraic Mayer–Vietoris theorem.