I have some trouble visualizing (quasi-)coherent sheaves with finite support (for example a point) or even finite length.
Algebraically, in the affine case over $\text{Spec} A$, one point support should correspond to modules over $A$ whose localization at every prime but some $\mathfrak p$ is zero. What do such $A$-modules look like? Would the additional requirement length 1 imply that the module localized at $\mathfrak p$ is equal to $A_{\mathfrak p}$? Edit: thanks Pavel.
The question arose when coming across various instances where sheaf quotients are taken, for example considering a maximal flag of coherent sub-$\mathcal O_{\overline X}$ given by $\mathcal F=\mathcal F_0\supset\mathcal F_{-1}\supset\cdots\supset\mathcal F_{-t}=\mathcal F(-P)$ such that length$(\mathcal F_{k+1}/\mathcal F_k)=1$, where $P$ is a certain closed point, $\overline X$ a reduced, irreducible complete curve, $\mathcal F$ a torsion-free coherent $\mathcal O_{\overline X}$-Module. (I can provide more context if needed). Essentially I am asking: what is/how to think about $\mathcal F_{k+1}/\mathcal F_k$? If I understand correctly, length 1 is a stronger property than being supported at a point.
Another related example would be Drinfeld's shtukas, where we also have cokernels with "small" support.
An answer that illuminates the geometric point of view (for example on a curve, via sections/functions on the curve etc) and some of the algebra (affine case?) would be greatly appreciated. Thanks.