Lemma 97.24.4. In Situation 97.24.2 assume that (iv) of Lemma 97.24.3 holds and that $K^\bullet $ is a perfect object of $D(A)$. In this case, if $x$ is versal at a closed point $u_0 \in U$ then there exists an open neighbourhood $u_0 \in U' \subset U$ such that $x$ is versal at every finite type point of $U'$.

**Proof.**
We may assume that $K^\bullet $ is a finite complex of finite projective $A$-modules. Thus the derived tensor product with $K^\bullet $ is the same as simply tensoring with $K^\bullet $. Let $E^\bullet $ be the dual perfect complex to $K^\bullet $, see More on Algebra, Lemma 15.74.15. (So $E^ n = \mathop{\mathrm{Hom}}\nolimits _ A(K^{-n}, A)$ with differentials the transpose of the differentials of $K^\bullet $.) Let $E \in D^{-}(A)$ denote the object represented by the complex $E^\bullet [-1]$. Let $\xi \in H^1(\text{Tot}(K^\bullet \otimes _ A \mathop{N\! L}\nolimits _{A/\Lambda }))$ be the element constructed in Lemma 97.24.3 and denote $\xi : E = E^\bullet [-1] \to \mathop{N\! L}\nolimits _{A/\Lambda }$ the corresponding map (loc.cit.). We claim that the pair $(E, \xi )$ satisfies all the assumptions of Lemma 97.23.4 which finishes the proof.

Namely, assumption (i) of Lemma 97.23.4 follows from conclusion (1) of Lemma 97.24.3 and the fact that $H^2(K^\bullet \otimes _ A^\mathbf {L} -) = \mathop{\mathrm{Ext}}\nolimits ^1(E, -)$ by loc.cit. Assumption (ii) of Lemma 97.23.4 follows from conclusion (2) of Lemma 97.24.3 and the fact that $H^1(K^\bullet \otimes _ A^\mathbf {L} -) = \mathop{\mathrm{Ext}}\nolimits ^0(E, -)$ by loc.cit. Assumption (iii) of Lemma 97.23.4 is clear. $\square$

## 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Your email address will not be published.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)