The \textit{quot} functor of a quasi-coherent sheaf (Q1741599)

The author studies the quot functor of a quasi-coherent sheaf over a projective scheme. Let $X$ be a projective scheme over an algebraically closed field $\mathbf{k}$, $\mathcal{E}$ be a quasi-coherent sheaf over $X$, and $h$ be a Hilbert polynomial. Let $\mathrm{quot}_h^X(\mathcal{E}_{(-)})$ be the contravariant functor parametrizing isomorphism classes of coherent quotients $Q_T$ of $\mathcal{E}_T = \mathcal{E}\otimes_{\mathbf{k}}\mathcal{O}_T$, for every scheme $T$ over $\mathbf{k}$, such that $Q_T$ is flat over $T$ and has Hilbert polynomial $h$. Grothendieck's classical result states that if $\mathcal{E}$ is coherent, then the above functor is representable. The main theorem of the paper asserts that if $\mathcal{E}$ is a quasi-coherent $\mathcal{O}_X$-module, then there is a scheme $\text{Quot}^X_h(\mathcal{E})$ representing the functor $\mathrm{quot}_h^X(\mathcal{E}_{(-)})$. The main idea in the author's construction of $\text{Quot}^X_h(\mathcal{E})$ is a version of Grothendieck's Grassmannian embedding combined with a result of Deligne, realizing quasi-coherent sheaves as ind-objects in the category of quasi-coherent sheaves of finite presentation. \par Section 2 collects some background materials such as limits and quasi-compact schemes, ind-objects and the above theorem of Deligne, representable functors, and Castelnuovo-Mumford regularity. Section 3 is devoted to the filtering construction of the schematic Grassmannian. The author proves the main theorem in Section 4.