# Snake Lemma on a Square

On a paper I'm reading, the author (Fulton) asks to apply the Snake Lemma to a square diagram:

how is that possible? I tried to construct a standard snake lemma diagram starting from this one, but didn't succeed. In this contest $R$ is a discrete valuation ring, and $A,B,C$ are matrices with entries in $R$ with determinant not $0$ which of course satisfy $C=AB$ (from the diagram).

• What is the paper? Does the author indicate the result you are supposed to prove by applying the Snake Lemma? – André 3000 May 19 '17 at 3:31
• @Quasicoherent sure, we are supposed to prove that $0 \to coker B \to coker C \to coker A \to 0$ is exact. (The paper is "Eigenvalues, Invariant Factors, Highest Weights and Schubert Calculus"). – Maffred May 19 '17 at 3:41

Now the snake lemma is applicable, and $\ker(\operatorname{coker}B\to 0) = \operatorname{coker}B$ and $\operatorname{coker}(\operatorname{coker}B\to 0) = 0$. This gives you the exact sequence $$\ker C\to\ker A\to\operatorname{coker}B\to\operatorname{coker}C\to\operatorname{coker}A\to0,$$ and as $R$ is a DVR and $A$ has nonzero determinant, $\det A$ is a (nonzero) non-zero-divisor in $R$, and hence $\ker A = 0$.