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Let $S$ be a noetherian scheme, $\mathcal{E}$ a quasi-coherent sheaf on $S$ and let $d \in \mathbb{N}$. There is a Plücker embedding $\omega : \mathrm{Grass}_d(\mathcal{E}) \hookrightarrow \mathbb{P}(\wedge^d \mathcal{E})$. A very elegant functorial construction can be found in EGA I, 9.8. My question is:

How can we describe the corresponding quasi-coherent ideal $I$ on $\mathbb{P}(\wedge^d \mathcal{E})$ globally? More precisely, if $\mathcal{E}$ is coherent, then by results of EGA II there is an epimorphism $\oplus_i M_i(n_i) M_i \otimes_{\mathcal{O}_S} \mathcal{O}(n_i) \twoheadrightarrow I$ for some coherent $\mathcal{O}_S$-modules $M_i$ and integers $n_i$. I would like know if one can write this down without using a presentation of $\mathcal{E}$.

The answer in the special case $\mathcal{E} = \mathcal{O}_S^{\oplus I}$ for some set $I$ is well-known (at least when $S$ is a field and $I$ is finite, but the general case works the same. Does anybody know a reference where this is done?): The Plücker relations generate $I$. More precisely, let $\mathcal{O}(1)$ \mathcal{O}_{\mathbb{P}}(1)$ be the universal invertible sheaf on $\mathbb{P}(\wedge^d \mathcal{E})$ together with its universal epimorphism $s : \wedge^d \mathcal{E} \otimes_{\mathcal{O}_S} \mathcal{O}_{\mathbb{P}} \twoheadrightarrow \mathcal{O}(1)$ mathcal{O}_{\mathbb{P}}(1)$ . Then define

$P : \wedge^{d-1}(\mathcal{E}) \otimes otimes_{\mathcal{O}_S} \wedge^{d+1}(\mathcal{E}) \otimes_{\mathcal{O}_S} \mathcal{O}_{\mathbb{P}} \to \mathcal{O}(2),$ mathcal{O}_{\mathbb{P}}(2),$ $${\small f_1 \wedge \dotsc \wedge f_{d-1} \otimes e_0 \wedge \dotsc \wedge e_d \mapsto \sum_{l=0}^{d} (-1)^l s(f_1 \wedge \dotsc \wedge f_{d-1} \wedge e_k) \otimes s(e_0 \wedge \dotsc \wedge \widehat{e_k} \wedge \dotsc \wedge e_d).}$$

Then $I$ is the image of $\check{P} : \wedge^{d-1}(\mathcal{E}) \otimes otimes_{\mathcal{O}_S} \wedge^{d+1}(\mathcal{E}) \otimes otimes_{\mathcal{O}_S} \mathcal{O}(-2) mathcal{O}_{\mathbb{P}}(-2) \to \mathcal{O}_X$.mathcal{O}_{\mathbb{P}}$.

For the general $\mathcal{E}$, these Plücker relations are also satisfied, but I couldn't prove the converse and meanwhile I'm convinced that we need more relations. If it helps, you may assume that $2$ is invertible on $S$.
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Let $S$ be a noetherian scheme, $\mathcal{E}$ a quasi-coherent sheaf on $S$ and let $d \in \mathds{N}$. mathbb{N}$. There is a Plücker embedding $\omega : \mathrm{Grass}_d(\mathcal{E}) \hookrightarrow \mathbb{P}(\wedge^d \mathcal{E})$. A very elegant functorial construction can be found in EGA I, 9.8. My question is:

How can we describe the corresponding quasi-coherent ideal $I$ on $\mathbb{P}(\wedge^d \mathcal{E})$ globally? More precisely, if $\mathcal{E}$ is of finite typecoherent, then by results of EGA II there is epimorphism $\oplus_i M_i(n_i) \twoheadrightarrow I$ for some coherent $\mathcal{O}_S$-modules $M_i$ and integers $n_i$. I would like know if one can write this down without using a presentation of $\mathcal{E}$.

The answer in the special case $\mathcal{E} = \mathcal{O}_S^{\oplus I}$ for some set $I$ is well-known (at least when $S$ is a field and $I$ is finite, does but the general case works the same. Does anybody know a reference where this is done for arbitrary $S$?): done?): The Plücker relations generate $I$. More precisely, let $\mathcal{O}(1)$ be the universal invertible sheaf on $\mathbb{P}(\wedge^d \mathcal{E})$ and let together with its universal epimorphism $s : \wedge^d \mathcal{E} \to twoheadrightarrow \mathcal{O}(1)$ be its universal epimorphiusm. Then define

$P : \wedge^{d-1}(\mathcal{E}) \otimes \wedge^{d+1}(\mathcal{E}) \to \mathcal{O}(2)$ by mathcal{O}(2),$ $P(f_1 ${\small f_1 \wedge \dotsc \wedge f_{d-1} \otimes e_0 \wedge \dotsc \wedge e_d ) = \mapsto \sum_{l=0}^{d} (-1)^l s(f_1 \wedge \dotsc \wedge f_{d-1} \wedge e_k) \otimes s(e_0 \wedge \dotsc \wedge \widehat{e_k} \wedge \dotsc \wedge e_d)$. e_d).}$$

Then $I$ is the image of $P \check{P} : \wedge^{d-1}(\mathcal{E}) \otimes \wedge^{d+1}(\mathcal{E}) \otimes \mathcal{O}(-2) \to \mathcal{O}_X$.

For the general $\mathcal{E}$, these Plücker relations are also satisfied, but I couldn't prove the converse and meanwhile I'm convinced that we need more relations. If it helps, you may assume that $2$ is invertible on $S$.
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What is the ideal corresponding to the Plücker embedding?

Let $S$ be a noetherian scheme, $\mathcal{E}$ a quasi-coherent sheaf on $S$ and let $d \in \mathds{N}$. There is a Plücker embedding $\omega : \mathrm{Grass}_d(\mathcal{E}) \hookrightarrow \mathbb{P}(\wedge^d \mathcal{E})$. A very elegant functorial construction can be found in EGA I, 9.8. My question is:

How can we describe the corresponding quasi-coherent ideal $I$ on $\mathbb{P}(\wedge^d \mathcal{E})$ globally? More precisely, if $\mathcal{E}$ is of finite type, then by results of EGA II there is epimorphism $\oplus_i M_i(n_i) \twoheadrightarrow I$ and I would like know if one can write this down without using a presentation of $\mathcal{E}$.

The answer in the special case $\mathcal{E} = \mathcal{O}_S^{\oplus I}$ for some set $I$ is well-known (at least when $S$ is a field and $I$ is finite, does anybody know a reference where this is done for arbitrary $S$?): The Plücker relations generate $I$. More precisely, let $\mathcal{O}(1)$ be the universal invertible sheaf on $\mathbb{P}(\wedge^d \mathcal{E})$ and let $s : \wedge^d \mathcal{E} \to \mathcal{O}(1)$ be its universal epimorphiusm. Then define $P : \wedge^{d-1}(\mathcal{E}) \otimes \wedge^{d+1}(\mathcal{E}) \to \mathcal{O}(2)$ by $P(f_1 \wedge \dotsc \wedge f_{d-1} \otimes e_0 \wedge \dotsc \wedge e_d) = \sum_{l=0}^{d} (-1)^l s(f_1 \wedge \dotsc \wedge f_{d-1} \wedge e_k) \otimes s(e_0 \wedge \dotsc \wedge \widehat{e_k} \wedge \dotsc \wedge e_d)$. Then $I$ is the image of $P : \wedge^{d-1}(\mathcal{E}) \otimes \wedge^{d+1}(\mathcal{E}) \otimes \mathcal{O}(-2) \to \mathcal{O}_X$.

For the general $\mathcal{E}$, these Plücker relations are also satisfied, but I couldn't prove the converse and meanwhile I'm convinced that we need more relations.