Limits of one-parameter orbits and concentrator subschemes

Let $X$ be a separated $k$-scheme of finite type with an action of ${\mathbf{G}}_m$ (Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers), and let $Z\subseteq X$ be a ${\mathbf{G}}_m$-stable closed subscheme. For a morphism $\phi :{\mathbf{G}}_m{\times }_k{X}_0\to X$, the limit ${\lim }_{t\to 0}\phi (t)$ exists if $\phi$ extends to a morphism ${\mathbf{A}}^1\times X_0\to X$, in which case the extension and its value at $0$ are unique by separatedness; for a point $x\in X(R)$ one writes $tx$ for the orbit map and asks that $t\mapsto tx$ extend over ${\mathbf{A}}_R^1$. When $X=\mathrm{Spec}A$ is affine, the action is a $\mathbb{Z}$-gradation $A={⨁}_{n\in \mathbb{Z}}{A}_n$ and ${\lim }_{t\to 0}tx$ exists iff $f_n(x)=0$ for all $f_n\in A_n$ with $n<0$; the concentrator subscheme $X(Z)$ is the closed subscheme defined by the ideal generated by $A_0\cap \mathfrak{a}+{\sum }_{n<0}{A}_n$, where $\mathfrak{a}=\ker (A\to A(Z))$ is the graded ideal of $Z$. The associated functor sends $R$ to the set of $x\in X(R)$ with ${\lim }_{t\to 0}tx\in Z(R)$.

The uniqueness in the first sentence follows from separatedness (Separated S-scheme): the equalizer of two extensions is closed in ${\mathbf{A}}^1\times X_0$ and contains ${\mathbf{G}}_m\times X_0$. This open is schematically dense, since on every affine chart of $X_0$ the map $R[t]\to R[t,t^{-1}]$ is injective. The equalizer is therefore the whole source, also when $X_0$ is nonreduced; the affine description is the gradation induced by the coaction of $k[t,t^{-1}]$ on $A$ when $X$ is affine (Affine schemes and their coordinate rings), and the closed subscheme structure is that of the ideal sheaf generated by the listed homogeneous pieces (Ideal sheaves). The functor-of-points description of $X(Z)$ is stated here and proved in Representability and smoothness of concentrator subschemes; no representability is asserted by the definition itself.