**Preparation.**


Let $R$ be a noetherian ring (perhaps we have to assume $2 \in R^\*$), $1 \leq d \leq n$ and consider the Plücker embedding $\omega : G \hookrightarrow P$, where $G = \text{Grass}\_d(R^n)$ and $P = \mathbb{P}(R^{\binom{n}{d}}) = \text{Proj}(R[\{x\_{i\_1,...,i\_d}\}])$. On $X$-valued points points, this maps a quotient $s: \mathcal{O}\_X^n \twoheadrightarrow \mathcal{F}$, where $\mathcal{F}$ is locally free of rank $d$, to the $d$th exterior power $x : \mathcal{O}\_X^{\binom{n}{d}} \twoheadrightarrow \wedge^d \mathcal{F}$; see EGA I, § 9.


Let's make the convention that every expression of the form $p\_{i\_1,...,i\_d}$ is meant to be alternating in the indices. Now $\omega$ is a closed immersion, which is cut out by the quasi-coherent ideal $I$ corresponding to the graded ideal of $R[\{x\_{i\_1,...,i\_d}\}]$, generated by the Plücker relations


$\sum\_{\lambda=1}^{d} (-1)^{\lambda} x\_{i\_1,...,i\_{d-1},k\_\lambda} x\_{k\_0,...,\hat{k\_\lambda},...,k\_d}$


for all $1 \leq i\_1,...,i\_{d-1},k\_0,...,k\_d \leq n$. Thus $\omega$ induces an isomorphism $G \cong V(I) \subseteq P$. This gives another interpretation (and also proof) of the Plücker embedding, see [here](https://mathoverflow.net/questions/51739/functorial-way-of-showing-that-the-segre-or-plucker-morphism-is-a-closed-embedd) for a related topic.


Now considering the identity of $G$ as a $G$-valued point of $G$, we get a (universal) locally free sheaf $\mathcal{F}$ of rank $d$ on $G$ (together with an epimorphism $s : \mathcal{O}^n \twoheadrightarrow \mathcal{F}$). Consider this on $V(I)$. Then $\omega\_\* \mathcal{F}$ is a coherent sheaf on $P$. If we apply Serre's Theorem twice, we see that $\omega\_\* \mathcal{F}$ fits into an exact sequence of the form


$\mathcal{O}\_P(q\_1)^{r\_1} \to \mathcal{O}\_P(q\_2)^{r\_2} \to \omega\_\\* \mathcal{F} \to 0.$


This may be viewn as a "twisted presentation". Here the map on the left is given by a $r\_2 \times r\_1$-matrix of global sections of $\mathcal{O}\_P(q\_2 - q\_1)$, which are just homogenuous polynomials of degree $q\_2 - q\_1$ in the variables $x\_{i\_1,...,i\_d}$.


**Question.** How does such a matrix look like explicitly?


Here is what I've done so far: Using the identifications, calculate explicitely $\mathcal{F}$ by means of cocycles on the covering of basic-opens $D\_+(x\_{i\_1,...,i\_d})$. This works. Then I wanted to calculate the graded module $\Gamma\_\*(\mathcal{F})$ and try to see twisted generators and relations. But it's all a big mess. By the way, I need this calculation for a sort of Plücker embedding in another context.



Calculation of twisted presentation of universal locally free sheaf on Grassmannian under Plücker embedding
To get a simpler answer it is better to fix a free $R$-module $V$ of rank $n$ so that $G = Gr(d,V)$ and $P = P(\Lambda^dV)$. First we have a tautological epimorphism $V\otimes O\_G \to F$ on $G$ which by adjunction gives an epimorphism $V\otimes O\_P \to i\_\*F$. Denote the kernel by $K$. Twisting the short exact sequence
$$
0 \to K \to V\otimes O\_P \to i\_\*F \to 0
$$
by $O(1)$ and taking cohomology one gets
$$
H^0(K) = Ker(V\otimes \Lambda^d V) \to \Sigma^{2,1^{d-1}}V),
$$
where $\Sigma$ is the Schur functor. It follows that $H^0(K) = \Lambda^{d+1}V$, so we have
$$
\Lambda^{d+1}V\otimes O\_P(-1) \to V\otimes O\_P \to i\_\*F \to 0.
$$
I guess this is what you need.


