Let $X$ be a scheme over a field $k$. (Feel free to assume that $X$ is an algebraic variety, if needed.) Also, let $M^\bullet$ be a complex in the derived category of quasi-coherent sheaves $\mathsf{D}_\text{qc}(\mathcal{O}_X)$. (Feel free to assume $M^\bullet$ bounded if needed. But I would rather not suppose it coherent.)
I want to understand what kind of information we can get from $M^\bullet$ if we know its derived fibers $\mathsf{L}x^\ast M^\bullet$ for every $x\in X(k)$. Let me be concrete and give two precise questions:
- Is it true that $M^\bullet=0$ if $\mathsf{L}x^\ast M^\bullet=0$ for every $x\in X(k)$? (This is the content of this answer, if we allow $x$ to be every point of $X$, not only the $k$-points.)
- Let $\mathsf{D}^b_\text{lf}(\mathcal{O}_X)$ be the full subcategory of $\mathsf{D}_\text{qc}(\mathcal{O}_X)$ whose objects $M^\bullet$ satisfy that $\mathscr{H}^i(M^\bullet)$ is locally free and zero for $i$ large enough. Then, is it true that $M^\bullet\in \mathsf{D}^b_\text{lf}(\mathcal{O}_X)$ if, for every $x\in X(k)$, $\mathsf{L}_i x^\ast M^\bullet$ is finite-dimensional and zero for $i$ large enough?
Best Answer
No to both questions:
Take $M = K(X)$, the field of rational functions on $X$.
Take $M$ to be the structure sheaf of any regular point on $X$ (assuming $\dim(X) > 0$).