The Stacks project

Definition 27.22.2. Let $0 < k < n$. The scheme $\mathbf{G}(k, n)$ representing the functor $G(k, n)$ is called Grassmannian over $\mathbf{Z}$. Its base change $\mathbf{G}(k, n)_ S$ to a scheme $S$ is called Grassmannian over $S$. If $R$ is a ring the base change to $\mathop{\mathrm{Spec}}(R)$ is denoted $\mathbf{G}(k, n)_ R$ and called Grassmannian over $R$.