The Stacks project

Example 98.24.1. Let $\Lambda , S, W, \mathcal{F}$ be as in Example 98.22.3. Assume that $W \to S$ is proper and $\mathcal{F}$ coherent. By Cohomology of Schemes, Remark 30.22.2 there exists a finite complex of finite projective $\Lambda $-modules $N^\bullet $ which universally computes the cohomology of $\mathcal{F}$. In particular the obstruction spaces from Example 98.22.3 are $\mathcal{O}_ x(M) = H^1(N^\bullet \otimes _\Lambda M)$. Hence with $K^\bullet = N^\bullet \otimes _\Lambda A[-1]$ we see that $\mathcal{O}_ x(M) = H^2(K^\bullet \otimes _ A^\mathbf {L} M)$.