Can we recover the sheaf from the functor?

Let $$k$$ be an algebraically closed field of characteristic zero. Let $$S$$ be a scheme of finite type over $$k$$. Let $$\mathrm{Sch}/S$$ be the category of schemes of finite type over $$S$$. Let $$\mathcal F$$ be a coherent sheaf on $$S$$. Consider the following functor $$\mathbf{Dual}_{\mathcal F}:(\mathrm{Sch}/S)^{\mathrm{op}}\to \mathrm{Set},\quad \big[f:T\to S\big]\mapsto \mathrm{Hom}_{\mathcal O_T}(f^*\mathcal F,\mathcal O_T)$$ where the functor on morphisms is naturally defined.

We can recover the dual $$\mathcal F^\vee:=\mathcal Hom_{\mathcal O_S}(\mathcal F,\mathcal O_S)$$ from $$\mathbf{Dual}_{\mathcal F}$$ by looking at open subsets $$[U\subset S]\in\mathrm{Sch}/S$$. However more is encoded in the functor. For example, if $$S=\mathbb A^1$$ and $$\mathcal F$$ is the skyscraper sheaf at the origin, then $$\mathcal F^\vee=0$$ and $$\mathbf{Dual}_{\mathcal F}(0\hookrightarrow \mathbb A^1)\cong k$$.

I want to ask

To what extent can we recover $$\mathcal F$$ from $$\mathbf{Dual}_{\mathcal F}$$?

A more specific question that I am interested in currently is

Can we characterise $$\mathcal F$$ being locally free in terms of $$\mathbf{Dual}_{\mathcal F}$$?

• That is actually a functor to $\mathbb{A}^1_S$-modules that is representable by an affine morphism over $S$ with an action of $\mathbb{A}^1_S$. The pushforward of the structure sheaf is a quasi-coherent $\mathcal{O}_S$-module with an action of $\mathbb{G}_m$ (inherited from the structure of $\mathbb{A}^1$-module). The first nontrivial $\mathbb{G}_m$-eigensheaf of the pushforward of the structure sheaf equals $\mathcal{F}$. The coherent sheaf $\mathcal{F}$ is locally free if and only if the affine morphism to $S$ is flat. May 4 at 15:21
• @JasonStarr I do not understand you. If I consider $S=\mathrm{Spec}(A)$ and $T=\mathrm{Spec}(B)$ and $f:T\to S$ corresponding to $A\to B$, then the pushforward of the structure sheaf is $B$, as an $A$-module with an action of $\mathbb G_m$? May 4 at 15:27
• Push forward the structure sheaf for the affine morphism that represents the functor. You will get the symmetric algebra on $\mathcal{F}$ with its standard grading corresponding to the action of $\mathbb{G}_m$. The first nontrivial eigensheaf is just the first graded piece of the symmetric algebra, which equals $\mathcal{F}$. May 4 at 15:29
• Isn't the functor you are studying represented by $\mathcal{Spec}_S(\mathrm{Sym}(\mathcal{F}))$? This is the relativised "Spec" construction over $S$ applied to the sheaf of $\mathcal{O}_S$ algebras given by the sum of symmetric powers of $\mathcal{S}$.. OK I just realised that Jason Starr already said this! May 4 at 16:52
• @Kapil Yes, I just realized this after Jason's comments. May 4 at 16:53

I am just posting my final comment as one answer. The set-valued functor $$\textbf{Dual}_{\mathcal{F}}$$ is representable by an affine $$S$$-scheme that is canonically isomorphic to the relative Spec over $$S$$ of the quasi-coherent sheaf of graded $$\mathcal{O}_S$$-algebras, $$\bigoplus_{d\geq 0} \text{Sym}^d_{\mathcal{O}_S} \left(\mathcal{F}\right).$$ In particular, the sheaf of relative differentials over $$S$$ for this affine $$S$$-scheme is canonically isomorphic to the pullback of $$\mathcal{F}$$. Thus, for every section over $$S$$ of the affine $$S$$-scheme, the pullback of the sheaf of relative differentials is isomorphic to $$\mathcal{F}$$. Of course there are sections, e.g., the zero section corresponding to the zero homomorphism from $$\mathcal{F}$$ to the structure sheaf.
• @PrimeRibeyeDeal For a scheme $S$ and a quasi-coherent sheaf of $\mathcal{O}_S$-modules $\mathcal{F}$, for the relative Spec $S$-scheme associated to the quasi-coherent sheaf of $\mathcal{O}_S$-algebras as above, the sheaf of relative differentials is canonically isomorphic to the pullback from $S$ of $\mathcal{F}$. Jun 21 at 11:10