Depth drops by one after quotienting by a regular element

Regularity includes $M\mathrm{/}xM\mathrm{\ne }0$. Since $x\in I$ acts injectively on $M$, no nonzero element of $M$ is annihilated by $I$. Thus ${\mathrm{Ext}}_R^0(R\mathrm{/}I,M)=0$.

Apply ${\mathrm{Ext}}_R(R\mathrm{/}I,-)$ to $0\to M\stackrel{x}{\to }M\to M\mathrm{/}xM\to 0$. Multiplication by $x$ on each Ext group is zero because $x$ annihilates $R\mathrm{/}I$, so the long exact sequence yields $0\to E^i(M)\to E^i(M\mathrm{/}xM)\to E^{i+1}(M)\to 0$, where $E^i(T)={\mathrm{Ext}}_R^i(R\mathrm{/}I,T)$. By step 1.1, the first nonzero Ext degree for $M$ is either a positive integer $d$ or $\mathrm{\infty }$. The sequence makes the first degree for $M\mathrm{/}xM$ equal $d-1$ when $d$ is finite and $\mathrm{\infty }$ when all groups vanish. The Ext characterization gives the displayed depth formula with its stated convention.