Ring of functions on the Grassmannian Background: Let $k$ be a commutative ring. In general for a $k$-functor $X : \textbf{Alg}_k \to \textbf{Set}$ we define the ring of functions on $X$ to be $\textbf{Nat}(X , \mathbb{A}^1)$ with the ring operations given term-wise (where $\mathbb{A}^1$ denotes the forgetful functor).
For any $n \in \mathbb{N}$ the Grassmannian $\text{Gr}_n$ is defined to be the $k$-functor $\textbf{Alg}_k \to \textbf{Set}$ which assigns to every $k$-algebra $R$ the set of module direct summands of $R^n$, and to every $k$-algebra homomorphism $\varphi: R \to S$, the function which sends a direct summand $N \subseteq R^n$ to the image of $N \otimes_R S$ in $S^n$ under the obvious inclusion map (or equivalently the $S$-submodule of $S^n$ generated by $\varphi^{\oplus n}(N)$). It is well known that the Grassmannian is represented by a scheme.
Question: Is there a nice description of the ring of functions on the Grassmannian? I'm sure this is standard, but I couldn't find a reference. Thanks!!
 A: Claim: For any flat and proper morphism of schemes $X\to S$ with $S$ connected and having geometrically connected, geometrically reduced fibers, we have that $\Gamma(X,\mathcal{O}_X)\cong\Gamma(S,\mathcal{O}_S)$.
Proof: By properness and flatness, we have that $X\to S$ is both closed and open, so because $S$ is connected $X\to S$ is surjective. By Stein factorization, we can write $X\to S$ as $X\to S'\to S$, where $S'\to S$ is finite and has geometrically connected and reduced fibers. A finite surjective morphism with geometrically connected and geometrically reduced fibers is an isomorphism, so $S'\cong S$ and since the structure sheaf of $S'$ is the pushforward of the structure sheaf on $X$, we have that the global sections are equal. $\blacksquare$
To apply this in our scenario, we note that the Grassmanian over a ring $R$ is just the base change of the Grassmanian over $\Bbb Z$ along the spectrum of the canonical map $\Bbb Z\to R$. Since the Grassmanian over the integers is smooth and proper and the conditions about the fibers are preserved by base change, we have that the Grassmanian over any base satisfies our conditions, so the lemma applies and we have that the regular functions are just $R$.
