On the stack of 0-dimensional coherent sheaves: structural aspects

Let $X$ be a quasiprojective scheme. In this expository note we collect a series of useful structural results on the stack $\mathcal{C}o{h}^n(X)$ parametrising $0$-dimensional coherent sheaves of length $n$ over $X$. For instance, we discuss its functoriality (in particular its behaviour along étale maps), the support morphism to ${\mathrm{S}\mathrm{y}\mathrm{m}}^n(X)$, and its relationship with the Quot scheme of points ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_X(\mathcal{E},n)$ for fixed $\mathcal{E}\in \mathrm{C}\mathrm{o}\mathrm{h}(X)$.