Ext concentration for a Cohen–Macaulay quotient of a regular local ring

The quotient's depth over $R$ equals its depth over $B$, since multiplication by a sequence of elements and regularity are unchanged after passing to their images in $B$. Thus [F1] gives ${\mathrm{pd}}_{R}B=N-d=c$, proving vanishing for $q>c$. By [F3], $\mathrm{ht}I=c$: the dimension of $R\mathrm{/}I$ is the maximum of the dimensions $R\mathrm{/}\mathfrak{p}$ for minimal primes $\mathfrak{p}$ over $I$.

Inductively maintain a regular tuple $f_1,\dots ,f_j\in I$ with $Q=R\mathrm{/}(f_1,\dots ,f_j)$ CM of dimension $N-j$; the case $j=0$ is [F1]. For $j<c$, every associated prime of $Q$, viewed in $R$, has quotient dimension $N-j$, hence height $j$ by [F3]. None contains $I$, whose height is $c>j$. Prime avoidance [F2] gives $f_{j+1}\in I$ regular on $Q$. It avoids every minimal prime of $Q$ by [F5], so every prime containing $(f_1,\dots ,f_{j+1})$ strictly contains a height-$j$ minimal prime and has height at least $j+1$. By [F3], $e=\dim (Q\mathrm{/}{f}_{j+1}Q)\le N-j-1$. A parameter tuple of length $e$ on this nonzero quotient, supplied by [F5], lifts together with $f_{j+1}$ to a tuple with finite-length quotient on $Q$. The minimal-length assertion of [F5] gives $N-j\le e+1$, hence $e=N-j-1$ and this lifted tuple is a system of parameters for $Q$. Thus $f_{j+1}$ is a regular parameter element, and [F2] makes the next quotient CM, completing the induction. All quotients are nonzero since the ideals lie in the maximal ideal. This constructs the length-$c$ regular sequence, including the empty tuple if $c=0$.

If $f$ is regular in a ring $Q$ and annihilates a $Q$-module $M$, the two-term resolution $[Q\stackrel{f}{\to }Q]$ of $Q\mathrm{/}fQ$ shows $R{\mathrm{Hom}}_Q(Q\mathrm{/}fQ,Q)\cong (Q\mathrm{/}fQ)[-1]$. Derived adjunction [F4] gives ${\mathrm{Ext}}_Q^q(M,Q)={\mathrm{Ext}}_{Q\mathrm{/}fQ}^{q-1}(M,Q\mathrm{/}fQ)$. Apply this successively to the sequence in step 2.1, all of which annihilate $B$. The result is ${\mathrm{Ext}}_R^q(B,R)={\mathrm{Ext}}_{R\mathrm{/}(f_1,\dots ,f_c)}^{q-c}(B,R\mathrm{/}(f_1,\dots ,f_c))=0$ for $q<c$, since negative Ext between modules vanishes. Along with step 1.1 this proves concentration. An invertible module over a local ring is free of rank one, so the same assertion holds for $L$. AC is inherited from the cited suppliers, including the derived adjunction [F4]; no complete-intersection assumption on $B$ was introduced.