Proposition 90.13.4. Let $A \to B$ be a local complete intersection map. Then $L_{B/A}$ is a perfect complex with tor amplitude in $[-1, 0]$.

**Proof.**
Choose a surjection $P = A[x_1, \ldots , x_ n] \to B$ with kernel $J$. By Lemma 90.10.3 we see that $J/J^2 \to \bigoplus B\text{d}x_ i$ is quasi-isomorphic to $\tau _{\geq -1}L_{B/A}$. Note that $J/J^2$ is finite projective (More on Algebra, Lemma 15.31.3), hence $\tau _{\geq -1}L_{B/A}$ is a perfect complex with tor amplitude in $[-1, 0]$. Thus it suffices to show that $H^ i(L_{B/A}) = 0$ for $i \not\in [-1, 0]$. This follows from (90.7.0.1)

and Lemma 90.13.3 to see that $H^ i(L_{B/P})$ is zero unless $i \in \{ -1, 0\} $. (We also use Lemma 90.4.7 for the term on the left.) $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)