The Stacks project

Lemma 15.78.4. Let $A \to B$ be a local complete intersection as in Definition 15.32.2. Then $\mathop{N\! L}\nolimits _{B/A}$ is a perfect object of $D(B)$ with tor amplitude in $[-1, 0]$.

Proof. Write $B = A[x_1, \ldots , x_ n]/I$. Then $\mathop{N\! L}\nolimits _{B/A}$ is represented by the complex

\[ I/I^2 \longrightarrow \bigoplus B \text{d}x_ i \]

of $B$-modules with $I/I^2$ placed in degree $-1$. Since the term in degree $0$ is finite free, this complex has tor-amplitude in $[-1, 0]$ if and only if $I/I^2$ is a flat $B$-module, see Lemma 15.63.2. By definition $I$ is a Koszul regular ideal and hence a quasi-regular ideal, see Section 15.31. Thus $I/I^2$ is a finite projective $B$-module (Lemma 15.31.3) and we conclude both that $\mathop{N\! L}\nolimits _{B/A}$ is perfect and that it has tor amplitude in $[-1, 0]$. $\square$


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