Intermediate cases

While the smooth group case has been affirmed by Will Sawin and the most general case has been refuted by Angelo, there is quite some space in between where we can still find more affirmative cases.

1 A Simple Example

You may skip this example and read the further sections if you are only interested in the results. Otherwise, it serves three purposes:

• A more gentle start than abstract schemes (after all, it is not quite a reserach-level question and I'd like to make some allowances for a broader range of readers accordingly)
• A motivating example showing some (but not all) interesting behaviour investigated in general
• A “historical” fun fact as this was the example I kept in mind while working this out for my own purposes

1.1 Algebraic Setting Consider the ring $R = \mathbb{F}_2[T]/(T^2-1)$ as a $\mathbb{Z}/2\mathbb{Z}$-graded ring with the grading $\mathrm{deg}(T) = 1 + 2\mathbb{Z}$, inducing the ring homomorphism $$\begin{split}\alpha\colon R &\to R[S]/(S^2-1)\\T &\mapsto TS.\end{split}$$.

1.2 Geometric Setting Consider the group scheme $$G = \mathrm{Spec}(\mathbb{Z}[S]/(S^2-1)) = \mathrm{Spec}(\mathbb{Z}[\mathbb{Z}/2]),$$ the affine scheme $X = \mathrm{Spec}(R)$ and the group action $$G \times X = \mathrm{Spec}(R[S]/(S^2-1)) \to \mathrm{Spec}(R) = X$$ given by $\alpha$.

1.3 Classical Subobjects When working in the unmodified setting of the question, a somewhat unclean picture emerges:

• Open subschemes of $X$:
  1. $\emptyset$ (invariant)
  2. $X$ (invariant)
• Closed subschemes of $X$:
  1. $X$ (invariant, non-reduced)
  2. $\emptyset$ (invariant, reduced)
  3. $\mathrm{Spec}(\mathbb{F}_2)$ (non-invariant, reduced)
• Ideals of $R$:
  1. $(0)$ (homogeneous, non-radical)
  2. $(1)$ (homogeneous, radical)
  3. $(T+1)$ (non-homogeneous, radical)

So far, no real surprises. There are two ideals (resp. closed subschemes) corresponding to the empty open subscheme, but neither of them is homogeneous (resp. invariant) and radical (resp. reduced). However, as indicated by my choice of numbering, there still appears to be a nice correspondence between invariant open subschemes, invariant closed subschemes and homogeneous ideals, except that for a perfect correspondence we'd like to ignore the existence of the nilpotent non-homogeneous element $X+1$ so that $(0)$ would be a radical. So, we'll do just that.

1.4 Better Subojects When the closed subscheme is only required to be as reduced as an invariant subscheme can be and the ideal is considered to have the radical property only with respect to the homogeneous elements, we instead obtain the following picture:

• Open subschemes of $X$:
  1. $\emptyset$ (invariant)
  2. $X$ (invariant)
• Closed subschemes of $X$:
  1. $X$ (invariant, maximally reduced)
  2. $\emptyset$ (invariant, maximally reduced)
  3. $\mathrm{Spec}(\mathbb{F}_2)$ (non-invariant, whatever)
• Ideals of $R$:
  1. $(0)$ (homogeneous, homogeneously radical)
  2. $(1)$ (homogeneous, homogeneously radical)
  3. $(T+1)$ (non-homogeneous, whatever)

2 Diagonalizable and Affine

Consider the diagonalizable group schemes $\mathrm{D}(M) = \mathrm{Spec}(\mathbb{Z}[M])$ for Abelian groups $M$ (in particular, $\mathbb{G}_{\mathrm{m}} = \mathrm{D}(\mathbb{Z})$). Then, a $\mathrm{D}(M)$-action on an affine scheme $X$ is the same as an $M$-graded ring. Restricting ourselves to these, i.e. diagonalizable groups and affine schemes, your first two conjectures are nearly correct and the last one may even be strengthened slightly.

2.1 Definition A homogeneous ideal is called homogeneously radical iff it contains all homogeneous elements of its radical. An invariant closed subscheme $C$ is called maximally reduced if any of its invariant closed subschemes on the same points is already identical to $C$.

2.2 Lemma When a diagonalizable group acts on an affine scheme, there is a bijective three-way correspondence working exactly like you'd expect between

• homogeneously radical homogeneous ideals,
• maximally reduced invariant closed subschemes and
• invariant open subschemes.

Proof. The equivalence of the first two points is immediate from the correspondence between homogeneous ideals and invariant closed subschemes. The equivalence of the latter two points is a special case of the more general treatment in the next section.

2.3 Corollary From this, we get the following alterations of your three conjectures:

2.4 Fixed Conjecture 1 The assignment $$\{\textrm{ideals of \(R\)}\} \to \{\textrm{open subschemes of \(X\)}\}$$ restricts to a bijective correspondence $$\{\textrm{homogeneously radical homogeneous ideals of \(R\)}\} \cong \{\textrm{invariant open subschemes of \(X\)}\}.$$

2.5 Fixed Conjecture 2 The open-complement assignment $$\{\textrm{closed subschemes of \(X\)}\} \to \{\textrm{open subschemes of \(X\)}\}$$ restricts to a bijective correspondence $$\{\textrm{maximally reduced invariant closed subschemes of \(X\)}\} \cong \{\textrm{invariant open subschemes of \(X\)}\}.$$

2.6 Fixed Conjecture 3 Any invariant open subscheme is the union of invariant principal opens (and thus in particular of invariant affine open subschemes).

3 General Case

I have not yet thought about the adaptations which would enable Conjecture 1 to be considered for general schemes and Conjecture 3 has been refuted by Angelo. However, in light of the diagonalizable and affine case it seems straightforward to salvage Conjecture 2. Throughout this section, let $G$ be a group scheme acting on a scheme $X$ via $\alpha\colon G \times X \to X$.

3.1 Lemma When a quasi-compact flat group scheme acts on a scheme, the “open complement” assigment restricts to a bijection of

• maximally reduced invariant closed subschemes and
• invariant open subschemes.

Proof. The remainder of this section.

3.2 Definition Given a closed subscheme $C$ of $X$, its invariant hull is the scheme-theoretic image of $\alpha^*C$ under the projection $\pi\colon G \times X \to X$.

3.3 Lemma Let $C \subseteq X$ be a closed subscheme and $C'$ its invariant hull. Then $C \subseteq C'$.

Proof. Using the unit of $G$, the inclusion of $C$ into $X$ factors through both the projection and the action with a single morphism $f\colon C \to G \times X$. Consequently, seen as $\alpha\circ f$ the inclusion of $C$ into $X$ factors through $\alpha^*C$ and seen as $\pi\circ f$ it factors through the image $C'$ of $\alpha^*C$ under $\pi$.

3.4 Lemma Let $C \subseteq X$ be a closed subscheme, $C'$ its invariant hull and $D \subseteq X$ a $G$-invariant closed subscheme containing $C$. Then $C' \subseteq D$.

Proof. We have $\alpha^*C \subseteq \alpha^*D = \pi^*D$ by monotonicity and invariance, yielding the result.

3.5 Lemma Let $C \subseteq X$ be a closed subscheme and $C'$ its invariant hull. If $G$ is flat and qc, $C'$ is invariant.

Proof. Since scheme-theoretic images of qc morphisms commute with flat base change, $\pi^*C'$ is the image of $D = G \times \alpha^*C \subseteq G \times G \times C$ under the projection $\pi_{\blacktriangle\!\triangledown\!\blacktriangle}$ eliding the middle component and $\alpha^*C'$ is the image of $$E = {\underbrace{\langle\pi_{\vartriangle\!\blacktriangledown\!\vartriangle}, \alpha\pi_{\blacktriangle\!\triangledown\!\blacktriangle}\rangle}_g}^*\alpha^*C$$ under $\pi_{\blacktriangle\!\triangledown\!\blacktriangle}$. As $$f = \langle\pi_{\blacktriangle\!\triangledown\!\vartriangle}, \mu\langle\pi_{\vartriangle\!\blacktriangledown\!\vartriangle}, \iota\pi_{\blacktriangle\!\triangledown\!\vartriangle}\rangle, \pi_{\vartriangle\!\triangledown\!\blacktriangle}\rangle$$ restricts to an isomorphism $D \to E$ over $G \times X$, they agree. (This boils down to a calculation that $$\alpha\pi_{\blacktriangle\!\triangledown\!\vartriangle} = \alpha gf$$ and thus $$D = \pi_{\blacktriangle\!\triangledown\!\vartriangle}^*\alpha^*C = (\alpha\pi_{\blacktriangle\!\triangledown\!\vartriangle})^*C = (\alpha gf)^*C = f^*E,$$ which is tedious in this formalism but obvious when spelled out on points.)

3.6 Lemma Let $C \subseteq X$ be a closed subscheme, $C'$ its invariant hull and $U \subseteq X$ a $G$-invariant open subscheme not meeting $C$. If $G$ is quasi-compact, then $U$ doesn't meet $C'$, either.

Proof. We show the contrapositive. As $G$ is qc, so is $\pi$, meaning that by [02JQ] for any field-valued point $p$ in $C'$ there is a point $q$ of $\alpha^*C$ such that $p$ is a specialization of $\pi(q)$. We note that if $p$ lies in both $C'$ and $U$ then so does $\pi(q)$, meaning $q$ lies in both $\alpha^*C$ and $\pi^*U = \alpha^*U$. Therefore, $\alpha(q)$ is a common point of $C$ and $U$.

3.7 Corollary The claim of 3.1 Lemma holds.

Proof. Given an invariant open subscheme, consider its reduced closed complement $C$ and its invariant hull $C'$. Now, $C'$ is the maximally reduced (3.4 Lemma) invariant (3.5 Lemma) closed subscheme on the same support as $C$ (3.3 Lemma and 3.6 Lemma), providing the inverse assignment.