Let $S$ be a scheme and let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_S$-module. Then we can construct a graded quasi-coherent $\mathcal{O}_S$-algebra $\mathscr{A}:= Sym(\mathcal{E})$ and define the quasi-coherent bundle $\mathbb{V}(\mathcal{E}) := Spec(Sym(\mathcal{E}))$. we set $Sym_0(\mathcal{E}):= \mathcal{O}_S$ the $0$-degree elements and denote $f: \mathbb{V}(\mathcal{E}) \to S$ the structure morphism.

It's easy to show that $\mathbb{V}(\mathcal{E})$ represents the functor $F:(Sch/S) \to (Set), (h:T \to S) \mapsto \Gamma(T, h^* \mathcal{E}^{\vee})$.

this is the consequence of identifications $$Hom_S(T,\mathbb{V}(\mathcal{E})) = Hom_{(\mathcal{O}_S -Alg)}(Sym(\mathcal{E}), h_* \mathcal{O}_T)=Hom_{\mathcal{O}_S}(\mathcal{E}, h_* \mathcal{O}_T)=Hom_{\mathcal{O}_T}(h^*\mathcal{E}, \mathcal{O}_T)=\Gamma(T, h^* \mathcal{E}^{\vee})$$

the zero element $0 \in \Gamma(T, h^* \mathcal{E}^{\vee})$ corresponds via identifications above to zero section $i: S \to \mathbb{V}(\mathcal{E})$, i.e. $f \circ i= id_S$ (section property). going through the identifications it's easy to see that the zero section is induced by augmentation homomorphism $Sym(\mathcal{E}) \to Sym(\mathcal{E})/Sym(\mathcal{E})_+ = \mathcal{O}_S$ simply by "killing" the elements of grades $\ge 1$. this implies that the zero section $i$ is a closed immersion.

$C:=\mathbb{V}(\mathcal{E})$ is also called the "cone". we can "remove" the image of the zero section and obtain $C^0:= C \backslash i(S)$.

The structure morphism $f: C \to S$ corresponds once more by our identifications to $id_C$ and a homomorphism

$$u: f^*\mathcal{E} \to \mathcal{O}_C$$

of $\mathcal{O}_C$-modules.

Q: why is $C^0$ is exacly the locus, where $u$ is surjective? i.e. why $C^0= C \backslash (Coker(u))$ or equivalently $i(S) =Supp(Coker(u))$?

recall: $u$ is surjective at $c \in C$ if and only if $u_c: (f^*\mathcal{E})_c \to \mathcal{O}_{C,c}$ is surjective homomorphism of $\mathcal{O}_{C,c}$-modules.

P.S. 1) all previously used notations & definitions are from Görtz' and Wedhorn's Algebraic Geometry.

P.S. 2) I have already tried to ask similar question at MSE without succeeding in solving the problem.



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