Proof.
A regular local ring is Cohen-Macaulay, see Lemma 10.106.3. Hence the equivalences (1) \Leftrightarrow (2) and (3) \Leftrightarrow (4), see Proposition 10.103.4. By Lemma 10.106.4 the ideal \mathop{\mathrm{Ker}}(A \to B) can be generated by \dim (A) - \dim (B) elements. Hence we see that (4) implies (2).
It remains to show that (1) implies (4). We do this by induction on \dim (A) - \dim (B). The case \dim (A) - \dim (B) = 0 is trivial. Assume \dim (A) > \dim (B). Write I = \mathop{\mathrm{Ker}}(A \to C) and J = \mathop{\mathrm{Ker}}(A \to B). Note that J \subset I. Our assumption is that the minimal number of generators of I is \dim (A) - \dim (C). Let \mathfrak m \subset A be the maximal ideal. Consider the maps
J/ \mathfrak m J \to I / \mathfrak m I \to \mathfrak m /\mathfrak m^2
By Lemma 10.106.4 and its proof the composition is injective. Take any element x \in J which is not zero in J /\mathfrak mJ. By the above and Nakayama's lemma x is an element of a minimal set of generators of I. Hence we may replace A by A/xA and I by I/xA which decreases both \dim (A) and the minimal number of generators of I by 1. Thus we win.
\square
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