Torified varieties and the limits of monoid schemes

Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. Public domain (CC0).

Introduction

A monoid scheme becomes an ordinary scheme after base change to the integers. This lesson asks which schemes arise in this way, and what to do with the ones that do not.

The answer to the first question is restrictive. The base change of a monoid scheme of finite type is, as a set, a disjoint union of spectra of group rings, one for each point of the monoid scheme. In the cases of interest these are tori, and they fit together as the orbits of a toric variety do. Section 4 turns this into two theorems.

A Grassmannian $\operatorname{Gr}(k,n)$ with $2 \le k \le n-2$ is not the base change of any monoid scheme, over the integers or over any field (Theorem 4.4).
If a group law is the base change of a morphism of monoid schemes, then the unit component of the group is diagonalizable. In particular the group law of $\mathrm{GL}_n$ with $n \ge 2$ is not the base change of a morphism of monoid schemes, whatever monoid scheme one tries (Theorem 4.6 and Corollary 4.7).

The second question leads to torified varieties, which were introduced in [López Peña–Lorscheid 2011b]. A torification of a scheme is a decomposition of the scheme into split tori, with no condition on how the tori fit together. Grassmannians, flag varieties and split reductive groups have torifications, through Schubert cells and the Bruhat decomposition. A torified variety has a counting polynomial, and the number of tori of each dimension can be read off from it.

The price is that few constructions are functorial. A torified morphism must map tori to tori by group homomorphisms. We prove that a group law is a torified morphism only for an extension of a finite group by a torus (Theorem 5.5). We also study affine torifications, which are the torifications that can be seen on an affine cover. They are the ones that produce varieties over $\mathbb{F}_1$ in the sense of Soulé and of Connes and Consani. Section 6 proves this comparison in the affine case, in a sharp form: the construction gives back the variety exactly when all its fibres over the primes are reduced (Theorems 6.5 and 6.9).

Section 7 states the problem that started the subject: to make sense of a split reductive group as a group over $\mathbb{F}_1$ whose group of $\mathbb{F}_1$-points is the Weyl group. We show why the direct formulation has no solution, and we state exactly what [Lorscheid 2012b] achieves.

What the lesson assumes. The lesson Monoid schemes, and with it Commutative monoids and their spectra. Schemes, at the level of the first chapters of [Stacks]; each fact we use is cited by tag. For the examples in Section 3 we use the Bruhat decomposition of a split reductive group, which we state as Fact 3.8 and do not prove. We prove it by hand for $\mathrm{GL}_2$ and $\mathrm{SL}_2$.

Basic references are [López Peña–Lorscheid 2011b], [López Peña–Lorscheid 2011a], [Lorscheid 2012b] and [Connes–Consani 2011a]. Result numbers of these works refer to their arXiv versions.

Conventions. Rings are commutative with 1. A monoid is a commutative monoid written multiplicatively, with a unit 1 and an absorbing element 0. Morphisms of monoids preserve 1 and 0. The group of units of $M$ is $M^\times$. For a commutative monoid or group $A$ without zero we write $A_0 = A \sqcup \{0\}$. An ideal of $M$ is a subset $I$ with $0 \in I$ and $MI \subseteq I$; it is prime when $I \ne M$ and $ab \in I$ implies $a \in I$ or $b \in I$. The set $M \smallsetminus M^\times$ is the unique maximal ideal of $M$. The space $\operatorname{Spec} M$ is the set of prime ideals with the topology generated by the sets $D(f) = \{\mathfrak{p} : f \notin \mathfrak{p}\}$.

For a ring $k$, the ring $k[M]$ is the monoid algebra of $M$ over $k$ modulo the ideal generated by the zero of $M$. It is a free $k$-module with basis $M \smallsetminus \{0\}$. A ring homomorphism $k[M] \to R$ of $k$-algebras is the same as a morphism of monoids from $M$ to the multiplicative monoid of $R$.

A monoid scheme is a topological space with a sheaf of monoids that is locally isomorphic to $(\operatorname{Spec} M, \mathcal{O})$. A morphism of monoid schemes is a continuous map with a map of sheaves of monoids that is local on stalks: non-units go to non-units. A monoid scheme is of finite type if it has a finite cover by open subsets $\operatorname{Spec} M_i$ with every $M_i$ finitely generated. The base change $X_k$ of a monoid scheme $X$ to a ring $k$ is the scheme glued from the schemes $\operatorname{Spec} k[M]$, where $\operatorname{Spec} M$ runs over the affine open subsets of $X$. For a ring homomorphism $k \to k'$ one has $X_{k'} = X_k \times_{\operatorname{Spec} k} \operatorname{Spec} k'$, because $k[M] \otimes_k k' = k'[M]$.

The product $X \times X'$ of two monoid schemes is glued from the spectra of the monoids $M \wedge M'$. Here $M \wedge M'$ is the product $M \times M'$ in which all pairs with a zero coordinate are identified with $0$. The underlying space of $X \times X'$ is the product of the underlying spaces with the product topology, and the stalk at $(x, x')$ is $\mathcal{O}_{X,x} \wedge \mathcal{O}_{X',x'}$. Since $k[M \wedge M'] = k[M] \otimes_k k[M']$, one has $(X \times X')_k = X_k \times_k X'_k$.

The split torus of dimension $d$ is $\mathbb{G}_m^d = \operatorname{Spec} \mathbb{Z}[t_1^{\pm 1}, \dots, t_d^{\pm 1}]$. Without coordinates, it is $\operatorname{Spec} \mathbb{Z}[A]$ for a free abelian group $A$ of rank $d$, and $A$ is its group of characters. It is a group scheme, with comultiplication $a \mapsto a \otimes a$ for $a \in A$. Its points with values in a ring $R$ are the group homomorphisms $A \to R^\times$. More generally, for an abelian group $H$ and a ring $k$, the scheme $D_k(H) = \operatorname{Spec} k[H]$ with the comultiplication $h \mapsto h \otimes h$ is a commutative group scheme over $k$, called diagonalizable.

Following [López Peña–Lorscheid 2011b], a variety in this lesson is a reduced scheme of finite type over $\mathbb{Z}$. For a scheme $X$ of finite type over $\mathbb{Z}$ and a prime power $q$ we write $N_X(q)$ for the number of points of $X$ with values in $\mathbb{F}_q$.

1. The base change of a monoid scheme is a union of tori

Let $M$ be a monoid and $k$ a ring. We write $m$ also for the basis element of $k[M]$ given by $m \in M \smallsetminus \{0\}$, and the zero of $M$ goes to $0 \in k[M]$. If $\mathfrak{P}$ is a prime ideal of the ring $k[M]$, then $\mathfrak{P} \cap M = \{m \in M : m \in \mathfrak{P}\}$ is a prime ideal of the monoid $M$. This defines a map

\pi \colon \operatorname{Spec} k[M] \longrightarrow \operatorname{Spec} M, \qquad \mathfrak{P} \longmapsto \mathfrak{P} \cap M .

It is continuous, because $\pi^{-1}(D(f))$ is the open set of $\operatorname{Spec} k[M]$ where $f$ does not vanish. The maps $\pi$ are compatible with localization. So they glue to a continuous map $\pi \colon X_k \to X$ for every monoid scheme $X$, with $\pi^{-1}(U) = U_k$ for every open $U \subseteq X$. If $f \colon X' \to X$ is a morphism of monoid schemes, then $\pi \circ f_k = f \circ \pi$ as maps of sets.

The fibres of $\pi$ are the building blocks of this lesson. For a point $x$ of a monoid scheme $X$ we write

H_x = \mathcal{O}_{X,x}^\times

for the group of units of the stalk at $x$.

Lemma 1.1. Let $X$ be a monoid scheme and $k$ a ring.

Let $x \in X$ have an affine open neighbourhood $\operatorname{Spec} M$ with $M$ finitely generated. Then the set $U_x$ of all generizations of $x$ is open in $X$, and $U_x \cong \operatorname{Spec} \mathcal{O}_{X,x}$. The fibre $\pi^{-1}(x)$ is the underlying set of the closed subscheme $Y_x = \operatorname{Spec} k[H_x] \subseteq \operatorname{Spec} k[\mathcal{O}_{X,x}] = (U_x)_k$ that is defined by the ideal spanned by the non-units of $\mathcal{O}_{X,x}$. So $Y_x$ is a locally closed subscheme of $X_k$.
For every field $L$ that is a $k$-algebra, there is a bijection, natural in $L$, $X_k(L) \;=\; \coprod_{x \in X} \operatorname{Hom}(H_x, L^\times),$ where $\operatorname{Hom}$ means group homomorphisms. The point of $X_k(L)$ given by $x$ and $\chi \colon H_x \to L^\times$ has image in $\pi^{-1}(x)$.

Proof. (1) Let $g_1, \dots, g_m$ generate $M$, let $\mathfrak{p}$ be the prime ideal of $M$ that is the point $x$, and let $f$ be the product of the $g_i$ that are not in $\mathfrak{p}$. A product of generators lies outside the prime ideal $\mathfrak{p}$ exactly when all its factors do. So $M \smallsetminus \mathfrak{p}$ is the submonoid generated by the $g_i \notin \mathfrak{p}$, and inverting $f$ is the same as inverting $M \smallsetminus \mathfrak{p}$. Hence $M_f = M_{\mathfrak{p}} = \mathcal{O}_{X,x}$. A prime ideal $\mathfrak{q}$ is determined by the generators it contains, so $f \notin \mathfrak{q}$ holds exactly when $\mathfrak{q} \subseteq \mathfrak{p}$. The primes $\mathfrak{q} \subseteq \mathfrak{p}$ are the generizations of $\mathfrak{p}$, because the closure of $\{\mathfrak{q}\}$ is the set of primes containing $\mathfrak{q}$. So $U_x = D(f) \cong \operatorname{Spec} M_f$ is open and affine.

Put $\mathcal{O} = \mathcal{O}_{X,x}$ and let $\mathfrak{m} = \mathcal{O} \smallsetminus H_x$ be its maximal ideal. The quotient of $k[\mathcal{O}]$ by the ideal spanned by $\mathfrak{m}$ is the group algebra $k[H_x]$. A prime ideal $\mathfrak{P}$ of $k[\mathcal{O}]$ lies over the closed point $\mathfrak{m}$ of $\operatorname{Spec} \mathcal{O}$ exactly when $\mathfrak{P} \supseteq \mathfrak{m}$: if $\mathfrak{P} \supseteq \mathfrak{m}$, then $\mathfrak{P} \cap \mathcal{O}$ is a prime ideal that contains the maximal ideal, so it is equal to it. This proves the claim on $\pi^{-1}(x)$.

(2) A morphism $\operatorname{Spec} L \to X_k$ has one point $y$ as its image. Put $x = \pi(y)$ and choose an affine open $U = \operatorname{Spec} M$ containing $x$. The morphism factors through the open subscheme $U_k = \operatorname{Spec} k[M]$, so it is a homomorphism of $k$-algebras $k[M] \to L$, that is, a morphism of monoids $\chi \colon M \to L$ into the multiplicative monoid of $L$. Since $L$ is a field, $\chi^{-1}(0)$ is a prime ideal $\mathfrak{p}$ of $M$, and $\mathfrak{p}$ is the point $x$. The morphism $\chi$ sends $M \smallsetminus \mathfrak{p}$ into $L^\times$. So it extends uniquely to $M_{\mathfrak{p}} \to L$, and this extension sends the maximal ideal of $M_{\mathfrak{p}}$ to $0$ and restricts to a group homomorphism $H_x = M_{\mathfrak{p}}^\times \to L^\times$. Conversely, a group homomorphism $H_x \to L^\times$, extended by zero to $M_{\mathfrak{p}}$ and composed with $M \to M_{\mathfrak{p}}$, is a morphism of monoids $M \to L$ whose zero set is $\mathfrak{p}$. The two constructions are inverse to each other, and they do not depend on the choice of $U$, because a smaller affine neighbourhood of $x$ has the same stalk at $x$. $\blacksquare$

If $H_x$ is a free abelian group of rank $r$, then $Y_x$ is the torus $\mathbb{G}_{m,k}^r$ over $k$. We call $r$ the rank of the point $x$ and write $r_x$.

Corollary 1.2 (point count). Let $X$ be a monoid scheme of finite type and $q$ a prime power. Then $N_{X_{\mathbb{Z}}}(q) \;=\; \sum_{x \in X} \#\operatorname{Hom}(H_x, \mathbb{F}_q^\times).$ If every $H_x$ is free of rank $r_x$, then $N_{X_{\mathbb{Z}}}(q) = \sum_{x \in X} (q-1)^{r_x}$.

Proof. A point of $X_{\mathbb{Z}}$ with values in $\mathbb{F}_q$ is the same as a point of $X_{\mathbb{F}_q}$ with values in $\mathbb{F}_q$. Apply Lemma 1.1 (2) with $k = L = \mathbb{F}_q$. The space $X$ is finite, because the spectrum of a finitely generated monoid is finite. If $H_x \cong \mathbb{Z}^{r}$, then $\operatorname{Hom}(H_x, \mathbb{F}_q^\times) = (\mathbb{F}_q^\times)^{r}$ has $(q-1)^{r}$ elements. $\blacksquare$

Example 1.3. (a) Let $M$ be the free monoid with zero on $t_1, \dots, t_n$, so that $\operatorname{Spec} M$ is the affine space $\mathbb{A}^n$ over $\mathbb{F}_1$. The prime ideals are the ideals $\mathfrak{p}_I$ generated by $\{t_i : i \in I\}$, for the subsets $I \subseteq \{1, \dots, n\}$. The group $H_x$ at $\mathfrak{p}_I$ is free on the $t_i$ with $i \notin I$. The fibre $Y_x$ is the torus of points of $\mathbb{A}^n$ whose coordinates vanish exactly on $I$. The count is $\sum_{l} \binom{n}{l} (q-1)^l = q^n$.

(b) Let $M = \{0, 1, \varepsilon\}$ with $\varepsilon^2 = 1$. This is the monoid $\mathbb{F}_{1^2}$. Its spectrum is one point with $H_x = \mathbb{Z}/2$. The base change is $\operatorname{Spec} \mathbb{Z}[\varepsilon]/(\varepsilon^2 - 1)$, and $N(q) = \#\operatorname{Hom}(\mathbb{Z}/2, \mathbb{F}_q^\times)$ is $2$ for odd $q$ and $1$ for even $q$. So the count of a monoid scheme is not always a polynomial in $q$. Torsion in the groups $H_x$ is the reason.

(c) Let $M = \{0, 1, \varepsilon\}$ with $\varepsilon^2 = 0$. The spectrum is again one point, now with $H_x = \{1\}$. The base change $\operatorname{Spec} \mathbb{Z}[\varepsilon]/(\varepsilon^2)$ is not reduced, and the fibre $Y_x = \operatorname{Spec} \mathbb{Z}$ is a proper closed subscheme with the same points.

So the base change of a monoid scheme of finite type is, as a set, a disjoint union of the schemes $\operatorname{Spec} k[H_x]$. When the groups $H_x$ are free, these are tori. The next section takes this property as a definition.

2. Decompositions and torifications

A reference for this section is [López Peña–Lorscheid 2011b, Sections 1.1 and 1.2].

An immersion of schemes is a morphism that is an isomorphism onto a locally closed subscheme. Immersions are monomorphisms [Stacks, Tag 01L7].

Definition 2.1. A decomposition of a scheme $X$ is a family $\{Y_i\}_{i \in I}$ of nonempty locally closed subschemes of $X$ whose underlying sets are pairwise disjoint and cover $X$.

Lemma 2.2. Let $\{Y_i\}_{i \in I}$ be a family of nonempty locally closed subschemes of a scheme $X$. The following are equivalent.

The family is a decomposition of $X$.
For every field $L$, the map $\coprod_{i} Y_i(L) \to X(L)$ is bijective.
For every algebraically closed field $L$, the map $\coprod_{i} Y_i(L) \to X(L)$ is bijective.

Proof. A morphism $\operatorname{Spec} L \to X$ is a point $x \in X$ together with an embedding of the residue field $\kappa(x)$ into $L$ [Stacks, Tag 01J6]. Let $Y \subseteq X$ be a locally closed subscheme. If $x \in Y$, then the residue field of $Y$ at $x$ is $\kappa(x)$, so the morphism factors through $Y$, and it does so in one way only, because $Y \to X$ is a monomorphism. If $x \notin Y$, it does not factor through $Y$. So an $L$-point of $X$ with image $x$ comes from $Y_i(L)$ exactly when $x \in Y_i$, and then from one element of $Y_i(L)$.

If (1) holds, every $x$ lies in exactly one $Y_i$, which gives (2). Clearly (2) implies (3). Assume (3), let $x \in X$, and let $L$ be an algebraic closure of $\kappa(x)$. The point $\operatorname{Spec} L \to \operatorname{Spec} \kappa(x) \to X$ comes from exactly one $Y_i$, so $x$ lies in exactly one $Y_i$. $\blacksquare$

For a nonempty scheme $S$ in place of $\operatorname{Spec} L$, the map $\coprod_i Y_i(S) \to X(S)$ is still injective, by the same argument applied to a point of $S$. It is not surjective in general, and for $S = \emptyset$ it is not injective when $I$ has more than one element.

Lemma 2.3 (finiteness). A decomposition of a noetherian scheme has finitely many pieces.

Proof. We show that a noetherian scheme $X$ has no partition into infinitely many nonempty locally closed subsets. Suppose it has one. Among the closed subsets of $X$ that have such an infinite partition, choose a minimal one, $Z$. Let $\eta$ be the generic point of an irreducible component $Z'$ of $Z$, and let $W$ be the union of the other irreducible components of $Z$. Let $Y_0$ be the piece that contains $\eta$, and write $Y_0 = O \cap C$ with $O$ open and $C$ closed in $Z$. Since $\eta \in C$, we get $Z' \subseteq C$. So $Y_0$ contains $V = O \cap (Z \smallsetminus W)$, which is open in $Z$ and contains $\eta$. Put $Z_1 = Z \smallsetminus V$. This is a proper closed subset of $Z$. Every piece other than $Y_0$ is disjoint from $V$, so it lies in $Z_1$. These pieces, together with $Y_0 \cap Z_1$ if it is nonempty, form an infinite partition of $Z_1$ into nonempty locally closed subsets. This contradicts the choice of $Z$. $\blacksquare$

Proposition 2.4 (the pieces at the generic points). Let $\{Y_i\}_{i \in I}$ be a decomposition of a noetherian scheme $X$ into irreducible pieces. For an irreducible component $Z'$ of $X$ with generic point $\eta$, let $i(Z')$ be the index with $\eta \in Y_{i(Z')}$.

The map $Z' \mapsto i(Z')$ is injective.
The piece $Y_{i(Z')}$ is contained in $Z'$. It is open and dense in $Z'$, and it contains a nonempty open subset of $X$.
If $X$ is irreducible and reduced, then $Y_{i(X)}$ is an open subscheme of $X$.

Proof. Write $Y = Y_{i(Z')}$ and let $\xi$ be the generic point of $Y$. Then $\eta$ is a specialization of $\xi$. Since $\eta$ is the generic point of an irreducible component of $X$, it has no generization other than itself, so $\xi = \eta$. This proves (1), because $\xi$ is determined by the piece. It also gives $Y \subseteq \overline{\{\eta\}} = Z'$. A locally closed subset of the irreducible space $Z'$ that contains its generic point is open and dense in $Z'$. The proof of Lemma 2.3 shows that $Y$ contains an open subset of $X$ that contains $\eta$. This proves (2). For (3), the set $Y$ is open in $X = Z'$ by (2). Let $O$ be the open subscheme of $X$ with these points. Then $Y$ is a closed subscheme of $O$ with the same points as $O$. Since $O$ is reduced, the ideal of $Y$ in $O$ is zero, so $Y = O$. $\blacksquare$

Example 2.5 (open in its component, but not open). Let $X = \operatorname{Spec} \mathbb{Z}[x,y]/(xy)$, the union of the lines $L_1 = V(y)$ and $L_2 = V(x)$. Let $P = V(x,y)$ and $Q = V(x-1, y)$. Put $Y_0 = Q, \qquad Y_1 = L_1 \smallsetminus Q = \operatorname{Spec} \mathbb{Z}[x, (x-1)^{-1}], \qquad Y_2 = L_2 \smallsetminus P = \operatorname{Spec} \mathbb{Z}[y, y^{-1}] .$ These three subschemes are locally closed, and they form a decomposition of $X$. The piece $Y_1$ contains the generic point of $L_1$ and the points of $P$. It is open in $L_1$. It is not open in $X$: an open subset of $X$ that contains a point of $P$ meets the irreducible line $L_2$ in a nonempty open subset of $L_2$, and this is not contained in $P$.

Reference: [López Peña–Lorscheid 2011b, Lemma 1.3 (2)] states that the pieces that are open in $X$ correspond to the irreducible components when these pieces are irreducible; in this example only $Y_2$ is open in $X$, so Proposition 2.4 uses the pieces at the generic points instead.

Definition 2.6. Let $X$ be a scheme. A torification of $X$ is a family $T = \{\tau_i \colon T_i \to X\}_{i \in I}$ of immersions, where each $T_i = \mathbb{G}_m^{d_i}$ is a split torus over $\mathbb{Z}$ of some dimension $d_i \ge 0$, such that the images of the $\tau_i$ form a decomposition of $X$. The pair $(X,T)$ is a torified scheme. It is a torified variety if $X$ is reduced and of finite type over $\mathbb{Z}$. The torification is affine if $X$ has a cover by affine open subschemes each of which is a union of images of tori; then $(X,T)$ is affinely torified.

We write $A_i$ for the character group of $T_i$, so that $T_i = \operatorname{Spec} \mathbb{Z}[A_i]$ and $A_i \cong \mathbb{Z}^{d_i}$. We often write $T_i$ for the image of $\tau_i$ as well. By Lemma 2.2, a family of immersions of tori is a torification exactly when every point of $X$ with values in a field factors through exactly one $\tau_i$. This is the definition of [López Peña–Lorscheid 2011b, Definition 1.5]. By Lemma 2.3, a torified variety has finitely many tori. If an open subscheme $U$ of $X$ is a union of images of tori, then these tori form a torification of $U$.

The group structure of the tori is part of the data. It plays no role in this section. It enters with the morphisms in Section 5.

Proposition 2.7. Let $(X,T)$ be a torified variety.

For every field $K$, the tori $T_i \otimes K$ form a decomposition of $X \otimes K$.
The scheme $X$ is flat over $\mathbb{Z}$: its structure sheaf has no $\mathbb{Z}$-torsion.

Proof. (1) The base change of an immersion is an immersion, and a point of $X \otimes K$ with values in a field $L \supseteq K$ is a point of $X$ with values in $L$. Apply Lemma 2.2.

(2) Let $f$ be a function on an affine open $U \subseteq X$ with $Nf = 0$ for some integer $N \ge 1$. For each $i$, the pullback of $f$ to $\tau_i^{-1}(U)$ is a torsion function on an open subscheme of $T_i$. The ring $\mathbb{Z}[A_i]$ is a domain of characteristic zero, so this pullback is zero. Hence $f$ vanishes at every point of $U$, so $f$ is nilpotent, and $f = 0$ because $X$ is reduced. $\blacksquare$

Proposition 2.8 (counting polynomial). Let $(X,T)$ be a torified variety and let $\delta_l$ be the number of tori of dimension $l$ in $T$. Then for every prime power $q$, $N_X(q) \;=\; \sum_{l \ge 0} \delta_l\, (q-1)^l .$ In particular the numbers $\delta_l$ depend only on $X$, not on the torification.

Proof. By Lemma 2.2, $X(\mathbb{F}_q)$ is the disjoint union of the sets $T_i(\mathbb{F}_q) = (\mathbb{F}_q^\times)^{d_i}$. If $T'$ is another torification with numbers $\delta'_l$, then the polynomials $\sum \delta_l t^l$ and $\sum \delta'_l t^l$ agree at $t = q - 1$ for all prime powers $q$, so they are equal. $\blacksquare$

Reference: [López Peña–Lorscheid 2011b, Proposition 1.11].

So a torifiable variety has a counting function that is a polynomial in $q - 1$ with non-negative integer coefficients. Its value at $q = 1$ is the number $\delta_0$ of tori of dimension zero. Its order of vanishing at $q = 1$ is the smallest dimension of a torus. Both numbers return in Section 7.

Example 2.9 (varieties without torification). (a) The projective line over $\mathbb{Z}$ minus the sections $0$, $1$ and $\infty$ has $N(q) = q - 2 = (q-1) - 1$. The coefficient $-1$ is negative, so there is no torification.

(b) Let $X = \operatorname{Spec} \mathbb{Z}[x,y]/(x^2 + y^2 - 1)$. For odd $q$, the number $N_X(q)$ is $q - 1$ when $-1$ is a square in $\mathbb{F}_q$ and $q + 1$ when it is not, because $X \otimes \mathbb{F}_q$ is then a conic with two points, or no point, at infinity. So $N_X$ is not a polynomial, and $X$ has no torification. Over $\mathbb{Z}[i, 1/2]$ this scheme becomes the torus $\mathbb{G}_m$ with coordinate $x + iy$. A torification asks for split tori over $\mathbb{Z}$.

Lemma 2.10 (refinements and products).

Let $\{X_j\}$ be a decomposition of a scheme $X$, and let each $X_j$ have a torification. Then the tori of all the $X_j$ form a torification of $X$.
Let $(X,T)$ and $(X',T')$ be torified schemes. Then the immersions $\tau_i \times \tau'_j \colon T_i \times T'_j \to X \times X'$ form a torification $T \times T'$ of $X \times X'$. If $T$ and $T'$ are affine, so is $T \times T'$. If $X$ and $X'$ are torified varieties, so is $X \times X'$.

Proof. (1) A locally closed subscheme of a locally closed subscheme is locally closed, and $X(L) = \coprod_j X_j(L)$ for every field $L$. (2) A product of immersions is an immersion, $T_i \times T'_j$ is a split torus, and $(X \times X')(L) = X(L) \times X'(L)$. If $U$ and $U'$ are affine open unions of tori, so is $U \times U'$. For the last claim we must show that $X \times X'$ is reduced. Both factors are flat over $\mathbb{Z}$ by Proposition 2.7, so $X \times X'$ is flat over $\mathbb{Z}$, and its rings of functions embed into those of $(X \times X') \otimes \mathbb{Q}$. This scheme is the product of the reduced $\mathbb{Q}$-schemes $X \otimes \mathbb{Q}$ and $X' \otimes \mathbb{Q}$, and it is reduced because $\mathbb{Q}$ is perfect [Stacks, Tag 00I4]. $\blacksquare$

Reference: [López Peña–Lorscheid 2011b, Lemmas 1.8 and 1.9].

3. Examples

A reference for this section is [López Peña–Lorscheid 2011b, Section 1.3].

3.1 Tori, affine spaces and monoid schemes

A split torus $\mathbb{G}_m^d$ is torified by its identity map. The affine line is the union of the point $V(t) \cong \mathbb{G}_m^0$ and its complement $\mathbb{G}_m^1$. By Lemma 2.10, the affine space $\mathbb{A}^n$ is torified by the $2^n$ coordinate tori of Example 1.3 (a), and $\delta_l = \binom{n}{l}$. We call this the coordinate torification of $\mathbb{A}^n$. It depends on the coordinates.

The general statement behind these examples is the following.

Proposition 3.1. Let $X$ be a monoid scheme of finite type. Assume that for every affine open $\operatorname{Spec} M$ of $X$, the set $M \smallsetminus \{0\}$ is closed under multiplication and is a submonoid of a torsion-free abelian group. Then $X_{\mathbb{Z}}$ is a variety, the fibres $Y_x = \operatorname{Spec} \mathbb{Z}[H_x]$ of $\pi \colon X_{\mathbb{Z}} \to X$ are split tori, and they form an affine torification of $X_{\mathbb{Z}}$. The counting polynomial is $\sum_{x \in X} (q-1)^{r_x}$.

Proof. Let $\operatorname{Spec} M$ be an affine open of $X$ with $M$ finitely generated, and let $G$ be a torsion-free abelian group that contains $M \smallsetminus \{0\}$. Then $\mathbb{Z}[M]$ is a subring of the group ring $\mathbb{Z}[G]$, which is a domain. So $X_{\mathbb{Z}}$ is reduced, and it is of finite type. For a prime $\mathfrak{p}$ of $M$, the nonzero elements of $M_{\mathfrak{p}}$ also lie in $G$, so $H_x \subseteq G$ is torsion-free. It is finitely generated: in a finitely generated monoid, a product of generators is a unit only if every factor is a unit, so the units are generated by the generators that are units and their inverses. Hence $H_x \cong \mathbb{Z}^{r_x}$ and $Y_x \cong \mathbb{G}_m^{r_x}$. By Lemma 1.1 (1), $Y_x$ is a locally closed subscheme of $X_{\mathbb{Z}}$ with underlying set $\pi^{-1}(x)$. The fibres of $\pi$ are disjoint and cover $X_{\mathbb{Z}}$. So the $Y_x$ form a torification. For every affine open $U$ of $X$, the affine open $U_{\mathbb{Z}} = \pi^{-1}(U)$ is the union of the tori $Y_x$ with $x \in U$, so the torification is affine. The count is Corollary 1.2. $\blacksquare$

The hypothesis holds for the monoid scheme of a fan, which is glued from the monoids $(\sigma^\vee \cap \mathbb{Z}^n)_0$ of its cones $\sigma$ (see Monoid schemes). So every toric variety over $\mathbb{Z}$ is affinely torified by its torus orbits; this is [López Peña–Lorscheid 2011b, Propositions 1.12 and 1.14]. Proposition 3.1 also covers monoids that are not saturated. For example, $M = \{0\} \cup \{t^n : n = 0, 2, 3, 4, \dots\}$ gives the cuspidal curve $\operatorname{Spec} \mathbb{Z}[t^2, t^3]$, which is the union of the point $t = 0$ and the torus $\operatorname{Spec} \mathbb{Z}[t, t^{-1}]$.

3.2 Grassmannians and Schubert cells

Let $1 \le k \le n-1$. For a ring $R$, let $\operatorname{Gr}(k,n)(R)$ be the set of submodules $W \subseteq R^n$ such that $R^n/W$ is a finitely generated projective module of rank $n - k$. For a field $K$, this is the set of subspaces of $K^n$ of dimension $k$. The functor is represented by a scheme $\operatorname{Gr}(k,n)$ of finite type over $\mathbb{Z}$, the Grassmannian [Stacks, Tag 089T, applied to the quotients $R^n/W$]. We use the following description.

For a subset $J \subseteq \{1, \dots, n\}$ with $k$ elements, let $U_J$ be the subfunctor of those $W$ for which the coordinate projection $R^n \to R^J$ restricts to an isomorphism $W \to R^J$. Such a $W$ has a unique basis that maps to the standard basis of $R^J$. Write this basis as the rows of a $k \times n$ matrix. The columns with index in $J$ form the identity matrix, and the other $k(n-k)$ entries $a_{t,c}$, with $1 \le t \le k$ and $c \notin J$, are arbitrary. Conversely, the row space $W$ of such a matrix is a complement of $R^{J^c}$ in $R^n$. So $U_J \cong \mathbb{A}^{k(n-k)}$, with coordinates $a_{t,c}$.

The $U_J$ are open subschemes. Indeed, $W$ lies in $U_J(R)$ exactly when the composite $R^{J^c} \to R^n \to R^n/W$ is an isomorphism. This is a map between finitely generated projective modules of the same rank $n - k$, so it is an isomorphism as soon as it is surjective. The set of points of $\operatorname{Spec} R$ where it is surjective is open, and it is compatible with base change, because surjectivity at a point can be tested over the residue field. The $U_J$ cover $\operatorname{Gr}(k,n)$. Indeed, for a field $K$, a matrix whose rows are a basis of a subspace $W \subseteq K^n$ of dimension $k$ has $k$ columns that form an invertible matrix, and $W$ lies in $U_J(K)$ for the set $J$ of these columns. A family of open subschemes that contains all points with values in fields is a cover.

Lemma 3.2. Put $d = k(n-k)$. The scheme $\operatorname{Gr}(k,n)$ is separated. For every field $K$, the scheme $\operatorname{Gr}(k,n)_K$ is integral of dimension $d$, and every point of it has an open neighbourhood isomorphic to $\mathbb{A}^d_K$.

Proof. Let $J$ and $J'$ be two subsets with $k$ elements. On $U_J$, let $D_{J'}$ be the determinant of the columns with index in $J'$ of the matrix above. Then $U_J \cap U_{J'}$ is the open subscheme of $U_J$ where $D_{J'}$ is invertible, so it is affine, and its ring is generated by the coordinates of $U_J$ and $D_{J'}^{-1}$. On $U_J \cap U_{J'}$, the matrix of $W$ for $J'$ is obtained from the matrix for $J$ by multiplying on the left with the inverse of its block of columns $J'$. So the determinant $D_J$ of the columns $J$ of the matrix for $J'$, which is a polynomial in the coordinates of $U_{J'}$, restricts to $D_{J'}^{-1}$. Hence $\mathcal{O}(U_J) \otimes \mathcal{O}(U_{J'}) \to \mathcal{O}(U_J \cap U_{J'})$ is surjective, and $\operatorname{Gr}(k,n)$ is separated [Stacks, Tag 01KP].

Over a field $K$, the open subschemes $U_{J,K} \cong \mathbb{A}^d_K$ are integral of dimension $d$, and any two of them meet, because $D_{J'}$ is not the zero polynomial on $U_J$: there is a subspace that projects isomorphically to both $K^J$ and $K^{J'}$. (Choose a bijection $\beta \colon J \smallsetminus J' \to J' \smallsetminus J$, and take the span of the $e_j$ with $j \in J \cap J'$ and the $e_j + e_{\beta(j)}$ with $j \in J \smallsetminus J'$.) A scheme that is covered by integral open subschemes that meet pairwise is integral. $\blacksquare$

Now fix $J = \{j_1 < \dots < j_k\}$. The Schubert cell $C_J$ is the closed subscheme of $U_J$ defined by the equations $a_{t,c} = 0 \qquad \text{for all } t \text{ and all } c \notin J \text{ with } c > j_t .$ In words: row $t$ of the matrix ends with its entry $1$ in column $j_t$. The free coordinates of $C_J$ are the $a_{t,c}$ with $c \notin J$ and $c < j_t$. Among the $j_t - 1$ indices below $j_t$, exactly $t - 1$ belong to $J$. So $C_J \cong \mathbb{A}^{d(J)}, \qquad d(J) = \sum_{t=1}^{k} (j_t - t) .$

Theorem 3.3 (Schubert decomposition). The Schubert cells $C_J$, for the subsets $J \subseteq \{1, \dots, n\}$ with $k$ elements, form a decomposition of $\operatorname{Gr}(k,n)$.

Proof. Each $C_J$ is closed in the open subscheme $U_J$, so it is locally closed. By Lemma 2.2 it is enough to show that for every field $K$, every subspace $W \subseteq K^n$ of dimension $k$ lies in $C_J(K)$ for exactly one $J$.

For a nonzero vector $v$, let $\ell(v)$ be the largest index $c$ with $v_c \ne 0$. Put $J(W) = \{\ell(v) : v \in W,\ v \ne 0\}$. Let $F_c$ be the span of the first $c$ standard basis vectors. An index $c$ lies in $J(W)$ exactly when $W \cap F_c \ne W \cap F_{c-1}$, and then the dimension goes up by one, because $W \cap F_{c-1}$ is the kernel of the $c$-th coordinate on $W \cap F_c$. Summing over $c$ shows that $J(W)$ has $\dim W = k$ elements. Write $J(W) = \{j_1 < \dots < j_k\}$.

Choose $w_t \in W$ with $\ell(w_t) = j_t$ and with entry $1$ in column $j_t$. The entry of $w_t$ in column $j_s$ is zero for $s > t$. For $t = 2, \dots, k$ in turn, replace $w_t$ by $w_t - \sum_{s < t} (w_t)_{j_s} w_s$. Since the vectors $w_s$ with $s < t$ are already zero in the columns $j_{s'}$ with $s' \ne s$, the new $w_t$ is zero in all columns $j_s$ with $s \ne t$, and its entries in the columns $c \ge j_t$ have not changed. The matrix with rows $w_1, \dots, w_k$ now has the identity matrix in the columns $J(W)$. So its rows are a basis of $W$, the subspace $W$ lies in $U_{J(W)}(K)$, and the equations of $C_{J(W)}$ hold.

If $W$ lies in $C_J(K)$, then the rows $w_t$ of its matrix satisfy $\ell(w_t) = j_t$, so $J \subseteq J(W)$. Both sets have $k$ elements, so $J = J(W)$. $\blacksquare$

Corollary 3.4. Put $d = k(n-k)$.

$N_{\operatorname{Gr}(k,n)}(q) = \sum_J q^{d(J)}$. This is the Gaussian binomial coefficient $\binom{n}{k}_q$ of Counting over finite fields and the limit $q \to 1$.
The number of $J$ with $d(J) = d - c$ is the number of partitions of $c$ into at most $k$ parts, each at most $n - k$. It is $1$ for $c = 0$ and for $c = 1$. For $c = 2$ it is $2$ if $2 \le k \le n - 2$.
The coordinate tori of the cells $C_J \cong \mathbb{A}^{d(J)}$ form a torification of $\operatorname{Gr}(k,n)$, with $\delta_l = \sum_J \binom{d(J)}{l}$. We call it the Schubert torification.

Proof. (1) follows from Theorem 3.3 and Lemma 2.2; both sides count the subspaces of dimension $k$ of $\mathbb{F}_q^n$. (2) Put $\mu_t = (n - k) - (j_t - t)$. Then $n - k \ge \mu_1 \ge \dots \ge \mu_k \ge 0$ and $\sum_t \mu_t = d - d(J)$, and $J \mapsto \mu$ is a bijection onto the partitions with at most $k$ parts, each at most $n - k$. The partitions of $2$ are $(2)$, which needs $n - k \ge 2$, and $(1,1)$, which needs $k \ge 2$. (3) follows from Lemma 2.10 (1) and the fact that $\operatorname{Gr}(k,n)$ is reduced and of finite type. $\blacksquare$

Example 3.5 ($\operatorname{Gr}(2,4)$). The six cells have $d(\{1,2\}) = 0$, $d(\{1,3\}) = 1$, $d(\{1,4\}) = d(\{2,3\}) = 2$, $d(\{2,4\}) = 3$, $d(\{3,4\}) = 4$. So $N(q) = 1 + q + 2q^2 + q^3 + q^4 = 6 + 12(q-1) + 11(q-1)^2 + 5(q-1)^3 + (q-1)^4 ,$ and the Schubert torification has $6$, $12$, $11$, $5$ and $1$ tori of dimensions $0$ to $4$. The value at $q = 1$ is $6 = \binom{4}{2}$, the number of subsets with two elements of a set with four elements. We show in Example 5.7 that the Schubert torification of $\operatorname{Gr}(2,4)$ is not affine, and in Theorem 4.4 that $\operatorname{Gr}(2,4)$ is not the base change of a monoid scheme.

Reference: [López Peña–Lorscheid 2011b, Example 1.16] for the numbers of tori.

3.3 Flag varieties and split reductive groups

Facts 3.6 to 3.8 below are proved in the course on reductive group schemes; each proof names the result there and adds the short translation needed here. They are used in Proposition 3.9, in the reductive case of Corollary 5.6, and in the examples of Section 7.

Fact 3.6 (flag varieties). Let $d_1 + \dots + d_m = n$ with all $d_j \ge 1$. The variety of flags $0 \subset V_1 \subset \dots \subset V_m$ of type $(d_1, \dots, d_m)$ in an $n$-dimensional space has a decomposition into Schubert cells, which are affine spaces. The cells are indexed by the cosets of $S_{d_1} \times \dots \times S_{d_m}$ in the symmetric group $S_n$. With the coordinate torification of each cell, the flag variety is a torified variety. For complete flags the cells are indexed by the permutations $w \in S_n$, the cell of $w$ has dimension $\ell(w)$, the length of $w$, which is its number of inversions, and the counting polynomial is $\sum_{w \in S_n} q^{\ell(w)}$.

Proof. The flag scheme of type $d$ over $\mathbb{Z}$ is the quotient of $\mathrm{GL}_n$ by the block upper triangular parabolic subgroup with block sizes $d_1, \dots, d_m$, and Automorphisms, forms and parabolic subgroups, Theorem 7.1 and Section 7 stratifies it, compatibly with base change, by locally closed cells isomorphic to $\mathbb{A}^{\ell(w)}$, one for each minimal-length representative $w$ of a coset of $S_{d_1} \times \dots \times S_{d_m}$ in $S_n$. The minimal-length representative of a coset is the permutation that is increasing on each block: right multiplication by $S_{d_1} \times \dots \times S_{d_m}$ permutes the values within the blocks, and sorting them removes the inversions inside a block without changing those between blocks. The length is the number of inversions (Root data, Weyl chambers and the Bruhat decomposition, Proposition 3.1, and Exercise 6). Each cell with its coordinate torification is torified, so the flag scheme is a torified variety, and counting the points of the cells over $\mathbb{F}_q$ gives the counting polynomial. $\blacksquare$ The same decomposition is in [López Peña–Lorscheid 2011b, Section 1.3.5].

For the type $(k, n-k)$ the flag variety is $\operatorname{Gr}(k,n)$, and Theorem 3.3 proves the statement. The coset of $w$ corresponds to the subset $J = w(\{1, \dots, k\})$, and $d(J)$ is the smallest number of inversions of a permutation in the coset: it is attained by the permutation that is increasing on $\{1, \dots, k\}$ and on $\{k+1, \dots, n\}$, whose inversions are the pairs of an element $j_t$ of $J$ and a smaller element outside $J$.

Fact 3.7 (notation for split reductive groups). Let $G$ be a split reductive group scheme over $\mathbb{Z}$, for example $\mathrm{GL}_n$, $\mathrm{SL}_n$ or $\mathrm{Sp}_{2n}$. By definition it is affine and smooth over $\mathbb{Z}$. It has a split maximal torus $T \cong \mathbb{G}_m^r$ and a Borel subgroup $B \supseteq T$ with unipotent radical $U$. The normalizer $N$ of $T$ in $G$ has the finite Weyl group $W = N/T$ as quotient, and every $w \in W$ has a representative $n_w \in N(\mathbb{Z})$. The group $W$ acts on the characters of $T$ by $(w\chi)(t) = \chi(n_w^{-1}\, t\, n_w)$, and this action permutes the roots. Let $\Phi^+$ be the set of positive roots, with $s$ elements. Each root $\alpha$ has a root subgroup $X_\alpha \cong \mathbb{G}_a$, and the product of the $X_\alpha$ with $\alpha \in \Phi^+$, taken in increasing order for a fixed total order of the character lattice, is an isomorphism of schemes $\mathbb{A}^s \to U$. For $w \in W$, let $\Phi_w$ be the set of positive roots that $w$ sends to negative roots, and let $\ell(w)$ be its number of elements, the length of $w$. The product of the $X_\alpha$ with $\alpha \in \Phi_w$, in the same order, is a closed subscheme $U_w \cong \mathbb{A}^{\ell(w)}$ of $U$.

Proof. After a choice of pinning, the torus, the Borel subgroup and the root subgroups are those of the pinning (Pinnings and the classification of split reductive groups, Theorem 10.1, and Roots and reductive groups of rank one, Theorem 4.1 for the root subgroups). The quotient $N/T$ is a finite étale group scheme (Tori, maximal tori and their conjugacy, Theorem 5.1); over a field its points form the Weyl group of the root datum, which acts on the characters and permutes the roots (Root data, Weyl chambers and the Bruhat decomposition, Theorems 2.1 and 4.1). Pinnings and the classification of split reductive groups, Proposition 1.1 constructs the representatives $n_w \in N(\mathbb{Z})$ integrally from the pinning. Section 5 of the root-data lesson shows that the product of the positive root subgroups, in any order, is an isomorphism $\mathbb{A}^s \to U$, and that for a set of positive roots containing every root that is a sum of its members, the corresponding product is a closed subgroup isomorphic to an affine space. The set $\Phi_w$ has this property, since $w$ maps a root that is a sum of roots of $\Phi_w$ to a sum of negative roots; so $U_w$ is a closed subgroup isomorphic to $\mathbb{A}^{\ell(w)}$. $\blacksquare$ The notation is that of [Connes–Consani 2011a, Sections 4.3 and 4.4] and [López Peña–Lorscheid 2011b, Section 1.3.6].

For $\mathrm{GL}_n$, with the diagonal torus and the upper triangular matrices as $B$, the group $W$ is $S_n$, and $\ell(w)$ is the number of inversions of the permutation $w$ (Exercise 6).

Fact 3.8 (Bruhat decomposition). In the notation of Fact 3.7, for every $w \in W$ the morphism $U \times T \times U_w \longrightarrow G, \qquad (u, t, u') \longmapsto u\, t\, n_w\, u'$ is an immersion. Its image $C_w$ has the same points as $B n_w B$, and the subschemes $C_w$, $w \in W$, form a decomposition of $G$.

Proof. Root data, Weyl chambers and the Bruhat decomposition, Theorem 8.1 gives locally closed immersions $U_{\Psi_v} \times B \to G$, $(u, b) \mapsto u\, n_v\, b$, with $\Psi_v = \{\alpha \in \Phi^+ : v^{-1}\alpha < 0\}$, whose images form a finite disjoint stratification of $G$, compatible with base change. Take $v = w^{-1}$, so that $\Psi_v = \Phi_w$ and $U_{\Psi_v} = U_w$. Both $n_v^{-1}$ and $n_w$ represent $w$, so $n_v^{-1} = t_1 n_w$ with $t_1 \in T(\mathbb{Z})$. Since $(u\, n_v\, b)^{-1} = b^{-1} t_1\, n_w\, u^{-1}$, the inversion of $G$, an automorphism of the scheme $G$, composed with the immersion $(u, b) \mapsto u\, n_v\, b$, equals the morphism in the statement composed with the isomorphism $U_w \times B \to U \times T \times U_w$, $(u, b) \mapsto (u', t', u^{-1})$, where $b^{-1} t_1 = u' t'$ with $u' \in U$ and $t' \in T$. Hence that morphism is an immersion, and the images $C_w$ form a finite disjoint stratification of $G$. Over a field, $C_w$ consists of the inverses of the points of $B\, n_v\, B$, which are the points of $B\, n_v^{-1}\, B = B\, n_w\, B$. $\blacksquare$ For points with values in a field, this is [Connes–Consani 2011a, Theorem 4.5]. [López Peña–Lorscheid 2011b, Theorem 1.18] writes the cells in the opposite order; in that order the left factor is $U_{w^{-1}}$, because the cell of $w$ is the inverse of the cell of $w^{-1}$.

Proposition 3.9. Let $G$ be a split reductive group scheme over $\mathbb{Z}$, with the notation of Fact 3.7. Then $G$ has an affine torification in which the tori of smallest dimension are the $|W|$ cosets $T n_w$, of dimension $r$. Its counting polynomial is $N_G(q) = (q-1)^r\, q^s \sum_{w \in W} q^{\ell(w)} , \qquad \text{so} \qquad \lim_{q \to 1} \frac{N_G(q)}{(q-1)^r} = |W| .$

Proof. By Fact 3.8, the cell $C_w$ is isomorphic to $\mathbb{A}^s \times \mathbb{G}_m^r \times \mathbb{A}^{\ell(w)}$. Take the coordinate torifications of the two affine spaces. By Lemma 2.10, the products of tori form a torification of each $C_w$, and together they form a torification of $G$. The scheme $G$ is affine, so the torification is affine. It is reduced and of finite type, because it is smooth over $\mathbb{Z}$. The only torus of dimension $r$ in $C_w$ is the image of $\{0\} \times T \times \{0\}$, which is $T n_w$. The count follows from Proposition 2.8, or directly from the cells. $\blacksquare$

Reference: [López Peña–Lorscheid 2011b, Proposition 1.19]; the limit is [Lorscheid 2012b, equation (1)].

The union of the cosets $T n_w$ is the normalizer $N$. So the torification of $G$ restricts to a torification of $N$ by $|W|$ tori of dimension $r$.

Example 3.10 ($\mathrm{GL}_2$ and $\mathrm{SL}_2$ by hand). Let $G = \mathrm{GL}_2 = \operatorname{Spec} \mathbb{Z}[a,b,c,d, (ad-bc)^{-1}]$, with $a, b$ the first row. The closed subscheme $c = 0$ is the group $B$ of upper triangular matrices. It is $\operatorname{Spec} \mathbb{Z}[a^{\pm 1}, d^{\pm 1}, b] \cong \mathbb{G}_m^2 \times \mathbb{A}^1$. On the open subscheme $c \ne 0$, put $\delta = ad - bc$. Then $b = (ad - \delta)/c$, so this open subscheme is $\operatorname{Spec} \mathbb{Z}[a, d, c^{\pm 1}, \delta^{\pm 1}] \cong \mathbb{A}^2 \times \mathbb{G}_m^2$. These are the two Bruhat cells. The count is $(q-1)^2 q + (q-1)^2 q^2 = (q^2 - 1)(q^2 - q) ,$ the order of $\mathrm{GL}_2(\mathbb{F}_q)$. With the coordinate torifications of $\mathbb{A}^1$ and $\mathbb{A}^2$ we get $2$ tori of dimension $2$, $3$ of dimension $3$ and $1$ of dimension $4$. The two tori of dimension $2$ are the diagonal matrices ($b = c = 0$) and the antidiagonal matrices ($a = d = 0$). Their union is the group $N$ of monomial matrices.

For $\mathrm{SL}_2$, the same computation with $\delta = 1$ gives the cells $\mathbb{G}_m \times \mathbb{A}^1$ and $\mathbb{A}^2 \times \mathbb{G}_m$, the count $q(q-1)(q+1)$, and $2$ tori of dimension $1$, $3$ of dimension $2$ and $1$ of dimension $3$.

Example 3.11 (a torus that is not a coordinate torus). Let $Y \subseteq \mathbb{A}^3$ be the complement of the quadric cone $z^2 = xy$, so $Y = \operatorname{Spec} \mathbb{Z}[x,y,z,(z^2 - xy)^{-1}]$. The following four subschemes form a torification of $Y$.

$P_1 = \{x \ne 0,\ z \ne 0\}$. With $f = z^2 - xy$ we have $y = (z^2 - f)/x$, so $P_1 = \operatorname{Spec} \mathbb{Z}[x^{\pm 1}, z^{\pm 1}, f^{\pm 1}] \cong \mathbb{G}_m^3$.
$P_2 = \{x = 0,\ y \ne 0\} = \operatorname{Spec} \mathbb{Z}[y^{\pm 1}, z^{\pm 1}] \cong \mathbb{G}_m^2$. Here $z$ is invertible because $z^2 = z^2 - xy$ is.
$P_3 = \{x = 0,\ y = 0\} = \operatorname{Spec} \mathbb{Z}[z^{\pm 1}] \cong \mathbb{G}_m$.
$P_4 = \{z = 0,\ x \ne 0\} = \operatorname{Spec} \mathbb{Z}[x^{\pm 1}, y^{\pm 1}] \cong \mathbb{G}_m^2$. Here $y$ is invertible because $-xy$ is.

Each is locally closed. A point of $Y$ with values in a field has $x \ne 0$ and $z \ne 0$, or $x = 0$, or $x \ne 0$ and $z = 0$, and these cases exclude each other. So the four pieces form a decomposition, with $\delta_3 = 1$, $\delta_2 = 2$, $\delta_1 = 1$, and $N_Y(q) = (q-1)^3 + 2(q-1)^2 + (q-1) = q^3 - q^2 .$ The complement of the open torus $P_1$ in $\mathbb{A}^3$ is the union of two planes and the cone, not a union of three planes. One of the coordinate functions of $P_1$ is the quadratic form $z^2 - xy$.

Reference: [López Peña–Lorscheid 2011b, Remark 1.21] states that this variety $Y$ cannot be torified; the four pieces above are a torification.

4. What monoid schemes cannot describe

Grassmannians and split reductive groups are torified, like the base changes of monoid schemes. Their counting polynomials have the same shape. This section shows that they are nevertheless out of reach of monoid schemes. The obstruction is not in the counting polynomial alone. It is in the way the tori fit together, and in the group law.

4.1 The local structure of a monoid scheme with a smooth base change

Let $K$ be a field and $d \ge 0$. For a monoid scheme $X$ we consider the following conditions on $Y = X_K$.

(H1) $Y$ is integral, separated and of finite type over $K$.
(H2) Every $K$-rational point of $Y$ has an open neighbourhood that is isomorphic to an open subscheme of $\mathbb{A}^d_K$.

By Lemma 3.2, the Grassmannian $\operatorname{Gr}(k,n)_K$ satisfies both, with $d = k(n-k)$. Condition (H2) is used through one consequence: at a $K$-rational point with maximal ideal $\mathfrak{n}$, the $K$-vector space $\mathfrak{n}/\mathfrak{n}^2$ has dimension $d$, as it has at a $K$-rational point of $\mathbb{A}^d_K$.

In a monoid $P$ without zero whose only unit is $1$, an atom is an element $p \ne 1$ that is not a product of two elements different from $1$.

Theorem 4.1. Let $K$ be a field and $X$ a monoid scheme such that $Y = X_K$ satisfies (H1) and (H2).

$X$ is of finite type. There is a unique point $x_0 \in X$ that lies in every nonempty open subset, and $\mathcal{O}_{X,x_0} = G_0$ for a free abelian group $G$ of rank $d$.
For every $x \in X$, the generization map $\mathcal{O}_{X,x} \to \mathcal{O}_{X,x_0} = G_0$ is injective. So $S_x = \mathcal{O}_{X,x} \smallsetminus \{0\}$ is a submonoid of $G$. There is an isomorphism of monoids $S_x \cong H_x \times \mathbb{N}^{d - r_x}$, where $r_x$ is the rank of the free abelian group $H_x$.
The point $x_0$ is the only point of rank $d$. A point $x$ of rank $d - 2$ has exactly two generizations $y_1, y_2$ of rank $d - 1$, and $S_x = S_{y_1} \cap S_{y_2}$ inside $G$.
Two different points of rank $d - 2$ have different sets $\{y_1, y_2\}$.
Let $a_j$ be the number of points of $X$ of rank $j$. Then $a_d = 1$ and $a_{d-2} \le \binom{a_{d-1}}{2}$. If $K$ is finite, then $\#Y(F) = \sum_j a_j (|F| - 1)^j$ for every finite extension $F$ of $K$.

Proof. Step 1: finite type. Cover $X$ by affine opens $U = \operatorname{Spec} M$. The open subschemes $U_K$ cover $Y$, and finitely many suffice, because $Y$ is quasi-compact. The map $\pi$ is surjective, because every fibre has a $K$-rational point by Lemma 1.1 (2) (take the trivial homomorphism). Since $\pi^{-1}(U) = U_K$, the corresponding finitely many $U$ cover $X$. For each of them, $K[M] = \mathcal{O}(U_K)$ is a finitely generated $K$-algebra. Finitely many generators involve finitely many elements of $M$. These generate a submonoid $M'$ with $K[M'] = K[M]$, and comparing bases gives $M' = M$. So $M$ is finitely generated, and $X$ is a finite space.

Step 2: the generic point. Let $U = \operatorname{Spec} M$ be a nonempty affine open of $X$. Then $U_K$ is a nonempty open subscheme of the integral scheme $Y$, so $K[M]$ is a domain. The nonzero elements of $M$ are basis elements of $K[M]$. Hence $M$ has no zero divisors, and $ab = ac$ with $a \ne 0$ implies $b = c$, because $a(b - c) = 0$ in $K[M]$. So $\{0\}$ is a prime ideal of $M$, it lies in every nonempty open subset of $U$, and $M \smallsetminus \{0\}$ embeds into its group of fractions $G_U$. If $U'$ is a second nonempty affine open, then $U \cap U' \ne \emptyset$, because $U_K$ and $U'_K$ meet in the irreducible scheme $Y$. The point $\{0\}$ of $U$ and the point $\{0\}$ of $U'$ are then generizations of each other. The space $X$ is $T_0$, because two different primes of a monoid are separated by a set $D(f)$. So the two points are equal. This gives the point $x_0$, with $\mathcal{O}_{X,x_0} = M_{\{0\}} = (G_U)_0$. Put $G = G_U$.

For $x \in U$, given by the prime $\mathfrak{p}$ of $M$, the map $\mathcal{O}_{X,x} = M_{\mathfrak{p}} \to M_{\{0\}} = G_0$ is injective, because $M$ is cancellative without zero divisors. So $S_x$ is a submonoid of $G$. It is finitely generated and it generates $G$ as a group. If $g_1, \dots, g_m$ are nonzero generators of $M$, then $D(g_1 \cdots g_m) = \{x_0\}$. So $\{x_0\}$ is open in $X$, and $\operatorname{Spec} K[G] = \pi^{-1}(x_0)$ is an open subscheme of $Y$. Hence $K[G]$ is a domain, and $G$ is torsion-free: if $g \ne 1$ had finite order $m$, then $(g - 1)(1 + g + \dots + g^{m-1}) = 0$ with both factors nonzero. So $G$ and its subgroups $H_x$ are free abelian groups of finite rank.

Step 3: the local structure. Fix $x$ and write $S = S_x$, $H = H_x$, $r = r_x$. By Lemma 1.1 (1), $\operatorname{Spec} K[S]$ is an open subscheme of $Y$. The non-units of $S$ form an ideal of $S$. Let $J \subseteq K[S]$ be their span. Then $J$ is an ideal, $K[S] = K[H] \oplus J$, and $K[S]/J = K[H]$. Let $z$ be the $K$-rational point of $\operatorname{Spec} K[S]$ given by $K[S] \to K[H] \to K$, $h \mapsto 1$. Its maximal ideal is $\mathfrak{n} = I \oplus J$, where $I$ is the augmentation ideal of $K[H]$. Then $\mathfrak{n}^2 = I^2 + IJ + J^2$ with $I^2 \subseteq K[H]$ and $IJ + J^2 \subseteq J$, so $\mathfrak{n}/\mathfrak{n}^2 \;=\; I/I^2 \,\oplus\, J/(IJ + J^2) .$ The first summand has dimension $r$, because $K[H]$ is a Laurent polynomial ring in $r$ variables $t_i$ and $I$ is generated by the $t_i - 1$. For the second, the group $H$ acts freely on $S \smallsetminus H$ by multiplication, so $J$ is a free $K[H]$-module on a set of representatives of the orbits. Hence $J/IJ$ has a basis indexed by $P \smallsetminus \{1\}$, where $P = S/H$. The image of $J^2$ is spanned by the products of two elements of $P \smallsetminus \{1\}$. So $J/(IJ + J^2)$ has a basis indexed by the atoms of $P$. By (H2), we get $r + \#\{\text{atoms of } P\} = d .$ For $x = x_0$ we have $S = H = G$ and $P = \{1\}$, so the rank of $G$ is $d$. This completes (1).

The monoid $P$ is finitely generated and cancellative, and its only unit is $1$. We claim that it is generated by its atoms. Let $p_1, \dots, p_m$ be a generating set of $P$ that is minimal for inclusion. Suppose $p_i = ab$ with $a, b \ne 1$, and write $a = \prod p_l^{\alpha_l}$ and $b = \prod p_l^{\beta_l}$. If $\alpha_i + \beta_i = 0$, then $p_i$ is a product of the other generators, against minimality. Otherwise we cancel $p_i$ and get $1 = p_i^{\alpha_i + \beta_i - 1} \prod_{l \ne i} p_l^{\alpha_l + \beta_l}$. Since $1$ is the only unit and no $p_l$ is $1$, all exponents on the right are zero. Then $\{a, b\} = \{p_i, 1\}$, a contradiction. So every $p_i$ is an atom. Conversely, an atom that is written as a product of generators must be one of them. So the atoms are exactly $p_1, \dots, p_m$, and $m = d - r$.

The group of fractions of $P$ is $G/H$, of rank $d - r$, and it is generated by the $d - r$ atoms. A surjection $\mathbb{Z}^{d-r} \to G/H$ onto a group of rank $d - r$ has a kernel of rank zero, so it is an isomorphism. Hence the atoms are a basis of $G/H$, and $P$ is the free commutative monoid on them. Choose $e_1, \dots, e_{d-r} \in S$ that lift the atoms. The map $H \times \mathbb{N}^{d-r} \to S$, $(h, n) \mapsto h \prod e_i^{n_i}$, is surjective because $P$ is generated by the atoms. It is injective: compare the images in $P$, then cancel. This proves (2).

Step 4: generizations. Put $j = d - r$ and identify $S$ with $H \times \mathbb{N}^j$. The complement $F$ of a prime ideal of $S_0$ is a submonoid that contains $H$, and $ab \in F$ implies $a, b \in F$. So $F$ is generated by $H$ and the $e_i$ it contains. Hence the primes are the ideals $\mathfrak{p}_I$, for $I \subseteq \{1, \dots, j\}$, where $\mathfrak{p}_I$ consists of $0$ and the $(h,n)$ with $n_i > 0$ for some $i \in I$. The localization at $\mathfrak{p}_I$ inverts the $e_i$ with $i \notin I$, so its nonzero elements form $H \times \mathbb{Z}^{I^c} \times \mathbb{N}^{I}$, with a unit group of rank $d - |I|$. By Lemma 1.1 (1), these $2^j$ points are the generizations of $x$.

If $j = 0$, then $x$ is its only generization, so $x = x_0$. If $j = 2$, the generizations are $x$, $x_0$, and two points $y_1, y_2$ of rank $d - 1$, with $S_{y_1} = H \times \mathbb{N} \times \mathbb{Z}$ and $S_{y_2} = H \times \mathbb{Z} \times \mathbb{N}$ inside $G = H \times \mathbb{Z} \times \mathbb{Z}$. Their intersection is $S_x$. This proves (3).

Step 5: separation. Let $x \ne x'$ be points of rank $d - 2$ with the same pair $\{y_1, y_2\}$. The inclusions $S_y \subseteq G$ are the generization maps of the sheaf $\mathcal{O}_X$, so they do not depend on $x$. By (3), $S_x = S_{x'}$ as subsets of $G$. Let $U_x$ and $U_{x'}$ be the affine opens of generizations, and put $V = (U_x)_K \cap (U_{x'})_K$. Since $Y$ is separated, $V$ is affine and $\mathcal{O}((U_x)_K) \otimes \mathcal{O}((U_{x'})_K) \to \mathcal{O}(V)$ is surjective [Stacks, Tag 01KP]. All three schemes are open in the integral scheme $Y$ and contain the torus $\operatorname{Spec} K[G]$, so restriction embeds their rings into $K[G]$. The images of $\mathcal{O}((U_x)_K)$ and $\mathcal{O}((U_{x'})_K)$ are both $K[S_x]$. So the image of $\mathcal{O}(V)$ is $K[S_x]$ too, and the restriction $\mathcal{O}((U_x)_K) \to \mathcal{O}(V)$ is an isomorphism. The open immersion $V \to (U_x)_K$ of affine schemes is then an isomorphism, so $V = (U_x)_K$. In the same way $V = (U_{x'})_K$. Applying $\pi$ gives $U_x = U_{x'}$. But $x$ is the only point of $U_x$ of which all points of $U_x$ are generizations. So $x = x'$, a contradiction. This proves (4).

Step 6. By (3), $a_d = 1$. By (3) and (4), $x \mapsto \{y_1, y_2\}$ is an injective map from the points of rank $d - 2$ to the sets of two points of rank $d - 1$. The count is Lemma 1.1 (2) for $L = F$. $\blacksquare$

Part (2) says that $X$ satisfies the hypothesis of Proposition 3.1, and that every stalk looks like the stalk of an affine space times a torus. Parts (3) and (4) are the properties "a cone of dimension two of a fan has two rays, and it is determined by them" in the language of monoid schemes.

4.2 Grassmannians

Lemma 4.2. Let $X$ be a monoid scheme, $V$ a scheme of finite type over $\mathbb{Z}$, and $K$ a field with $X_K \cong V_K$ over $K$. Then there is a finite field $F$ with $X_F \cong V_F$ over $F$.

Proof. Step 1 of the proof of Theorem 4.1 uses only that $X_K$ is of finite type, so $X$ is of finite type, and $X_{\mathbb{Z}}$ is of finite type over $\mathbb{Z}$. The field $K$ is the directed union of its finitely generated subrings $R$. By [Stacks, Tag 01ZM], the isomorphism $X_K \to V_K$ and its inverse are base changes of morphisms $X_R \to V_R$ and $V_R \to X_R$ for some $R$, and after enlarging $R$ the two composites are the identities. So $X_R \cong V_R$. Let $\mathfrak{m}$ be a maximal ideal of $R$. The ring $\mathbb{Z}$ is a Jacobson ring [Stacks, Tag 00G4], so the maximal ideal $\mathfrak{m}$ of the finitely generated $\mathbb{Z}$-algebra $R$ lies over a maximal ideal $(p)$ of $\mathbb{Z}$, and the residue field $F = R/\mathfrak{m}$ is a finite extension of $\mathbb{F}_p$ [Stacks, Tag 00GB]. Base change along $R \to F$ gives $X_F \cong V_F$. $\blacksquare$

Theorem 4.3. Let $2 \le k \le n - 2$ and $d = k(n-k)$. Let $K$ be a field and $X$ a monoid scheme such that $Y = X_K$ satisfies (H1) and (H2) for this $d$. If $K$ is finite, then the numbers of points of $Y$ over the finite extensions $F$ of $K$ are not the numbers $N_{\operatorname{Gr}(k,n)}(|F|)$.

Proof. Suppose they are. By Theorem 4.1 (5) and Corollary 3.4 (1), $\sum_j a_j (Q - 1)^j \;=\; \sum_J Q^{d(J)}$ for all powers $Q$ of $|K|$, so this is an identity of polynomials in $Q$. By Corollary 3.4 (2), the right side is $Q^d + Q^{d-1} + 2Q^{d-2}$ plus terms of degree at most $d - 3$. In terms of $t = Q - 1$, $Q^d = t^d + d\,t^{d-1} + \tbinom{d}{2} t^{d-2} + \dots, \qquad Q^{d-1} = t^{d-1} + (d-1)\,t^{d-2} + \dots, \qquad 2Q^{d-2} = 2t^{d-2} + \dots .$ Comparing coefficients gives $a_d = 1, \qquad a_{d-1} = d + 1, \qquad a_{d-2} = \tbinom{d}{2} + (d - 1) + 2 = \tbinom{d+1}{2} + 1 .$ This contradicts $a_{d-2} \le \binom{a_{d-1}}{2} = \binom{d+1}{2}$. $\blacksquare$

Theorem 4.4 (Grassmannians are not base changes of monoid schemes). Let $2 \le k \le n - 2$.

For every field $K$, there is no monoid scheme $X$ with $X_K \cong \operatorname{Gr}(k,n)_K$.
There is no monoid scheme $X$ with $X_{\mathbb{Z}} \cong \operatorname{Gr}(k,n)$.

Proof. (1) By Lemma 4.2 we may assume that $K$ is finite. Then $Y = X_K \cong \operatorname{Gr}(k,n)_K$ satisfies (H1) and (H2) by Lemma 3.2, and it has the point counts of the Grassmannian. This contradicts Theorem 4.3. (2) follows from (1) for a finite field $K$, because $(X_{\mathbb{Z}}) \otimes K = X_K$; this case does not use Lemma 4.2. $\blacksquare$

Example 4.5. (a) For $\operatorname{Gr}(2,4)$ the numbers would be $a_4 = 1$, $a_3 = 5$, $a_2 = 11$, and $11 > \binom{5}{2} = 10$. These are the numbers of tori of the Schubert torification in Example 3.5. So there are torifications with these numbers, but no monoid scheme whose base change satisfies (H1) and (H2) has them. The counting polynomial alone does not show this. Its coefficients in $q - 1$ are non-negative, and it is the counting polynomial of a monoid scheme: the disjoint union of $6$, $12$, $11$, $5$ and $1$ copies of $\operatorname{Spec} (\mathbb{Z}^j)_0$ for $j = 0, \dots, 4$, whose base change is a disjoint union of tori.

(b) For the projective space $\mathbb{P}^d$ with $d \ge 2$, with $N(q) = q^d + q^{d-1} + \dots + 1$, the same computation gives $a_{d-1} = d + 1$ and $a_{d-2} = \binom{d}{2} + (d-1) + 1 = \binom{d+1}{2}$. The inequality of Theorem 4.1 (5) holds with equality. This must be so, because $\mathbb{P}^d$ is the base change of the monoid scheme $\mathbb{P}^d$ over $\mathbb{F}_1$ of Monoid schemes. The cases $k = 1$ and $k = n - 1$ of the Grassmannian are projective spaces.

4.3 Group laws

Theorem 4.6 (group laws of monoid schemes are diagonalizable). Let $K$ be a field, $X$ a monoid scheme and $\mu \colon X \times X \to X$ a morphism of monoid schemes. Assume that $X_K$ is of finite type over $K$ and that $\mu_K \colon X_K \times_K X_K \to X_K$ is the group law of a group scheme over $K$. We do not assume that the unit or the inverse come from morphisms of monoid schemes. Let $e \in X_K(K)$ be the unit and $x_e = \pi(e)$.

$X$ is a finite discrete space.
$\mathcal{O}_{X,x_e} = H_0$ for a finitely generated abelian group $H$. The open and closed subscheme $\pi^{-1}(x_e) = \operatorname{Spec} K[H]$ of $X_K$ is a subgroup scheme, and its group law is that of the diagonalizable group $D_K(H)$: the comultiplication is $h \mapsto h \otimes h$. The connected component of the unit of $X_K$ is the diagonalizable group $D_K(H/H')$, where $H' \subseteq H$ is the subgroup of the elements of finite order prime to the characteristic of $K$ (of all elements of finite order if the characteristic is zero).
If $X_K$ is connected, then $X = \operatorname{Spec} H_0$ and $(X_K, \mu_K) = D_K(H)$. In particular the group scheme is commutative. If $X_K$ is integral, it is a split torus.

Proof. As in Step 1 of the proof of Theorem 4.1, $X$ is of finite type, so it is a finite space, and every fibre of $\pi$ has a $K$-rational point.

Translations. Let $L \supseteq K$ be a field, $g \in X_K(L)$, and let $x_g \in X$ be the image under $\pi$ of the point of $g$. Define $\lambda_g \colon X \to X$ by $\lambda_g(x) = \mu(x_g, x)$. This map is continuous, because $\mu$ is continuous and $X \times X$ has the product topology. For $y \in X_K(L)$ over the point $x$, the pair $(g, y)$ is a point of $(X \times X)_K$ over $(x_g, x)$, and $\pi \circ \mu_K = \mu \circ \pi$. So the product $g y$ lies over $\lambda_g(x)$. Every $x$ has some $y \in X_K(L)$ over it. From $g^{-1}(g y) = y$ we get $\lambda_{g^{-1}} \circ \lambda_g = \mathrm{id}$, and in the same way $\lambda_g \circ \lambda_{g^{-1}} = \mathrm{id}$. So $\lambda_g$ is a homeomorphism of $X$. Since $g e = g$, it satisfies $\lambda_g(x_e) = x_g$.

(1). Let $x, x' \in X$ with $x'$ in the closure of $x$. Choose $g, g' \in X_K(K)$ over $x$ and $x'$, and put $\varphi = \lambda_{g'} \circ \lambda_g^{-1}$. This is a homeomorphism with $\varphi(x) = x'$. It maps the closure $Z$ of $x$ onto the closure $Z'$ of $x'$, and $Z' \subseteq Z$. Since $Z$ is finite, $Z' = Z$, so $x$ and $x'$ have the same closure, and $x = x'$ because $X$ is $T_0$. So every point of the finite space $X$ is closed, and $X$ is discrete.

(2). Put $M = \mathcal{O}_{X,x_e}$. Since $X$ is discrete, $\{x_e\} = \operatorname{Spec} M$ is open and closed, $M$ has only one prime ideal, which is its maximal ideal $\mathfrak{m}$, and $Y_e = \pi^{-1}(x_e) = \operatorname{Spec} K[M]$ is open and closed in $X_K$. Since $e \cdot e = e$, we have $\mu(x_e, x_e) = x_e$. So $\mu$ restricts to a morphism $\operatorname{Spec}(M \wedge M) \to \operatorname{Spec} M$, given by a morphism of monoids $\delta \colon M \to M \wedge M$, and $\mu_K$ maps $Y_e \times Y_e$ to $Y_e$ by $\operatorname{Spec} K[\delta]$. The unit $e$ is a homomorphism of $K$-algebras $\varepsilon \colon K[M] \to K$, that is, a morphism of monoids $\varepsilon \colon M \to K$. The rules $e y = y = y e$ on $Y_e$ say that $(\varepsilon \otimes \mathrm{id}) \circ K[\delta] = \mathrm{id} = (\mathrm{id} \otimes \varepsilon) \circ K[\delta] \quad \text{on } K[M] .$ Let $m \in M$, $m \ne 0$. If $\delta(m) = 0$, the left identity gives $m = 0$ in $K[M]$, which is false. So $\delta(m) = (a, b)$ with $a, b \ne 0$, and the two identities read $\varepsilon(a)\, b = m$ and $\varepsilon(b)\, a = m$ in $K[M]$. Since $a$, $b$ and $m$ are basis elements, $a = b = m$ and $\varepsilon(m) = 1$. So $\delta(m) = (m, m) \quad \text{and} \quad \varepsilon(m) = 1 \qquad \text{for all } m \ne 0 .$ The set $\varepsilon^{-1}(0)$ is a prime ideal of $M$, so it is $\mathfrak{m}$. Hence $\mathfrak{m} = \{0\}$, that is, $M = H_0$ with $H = M^\times$. The algebra $K[H] = \mathcal{O}(Y_e)$ is finitely generated, so $H$ is finitely generated, as in Step 1 of the proof of Theorem 4.1. The restriction of $\mu_K$ to $Y_e$ is the group law $h \mapsto h \otimes h$ of $D_K(H)$, and $e$ is its unit. Since inverses in a group are unique, $Y_e$ is a subgroup scheme of $X_K$.

The unit component. The group $H'$ is finite, because $H$ is finitely generated, and its order $n$ is invertible in $K$. Put $c = n^{-1} \sum_{h \in H'} h \in K[H]$. Then $hc = c$ for $h \in H'$, so $c^2 = c$, and $K[H]$ is the product of the rings $K[H]c$ and $K[H](1-c)$. The ideal generated by $1 - c$ is the ideal generated by the elements $h - 1$ with $h \in H'$, because $h - 1 = (h-1)(1-c)$ and $1 - c = n^{-1}\sum_{h \in H'} (1 - h)$. So $K[H]c = K[H/H']$, and $D_K(H/H')$ is an open and closed subgroup scheme of $Y_e = D_K(H)$ that contains the unit. It is connected. Indeed, $H/H' \cong \mathbb{Z}^r \times P$, where $P$ is trivial if $K$ has characteristic zero and a finite group of order a power of $p$ if $K$ has characteristic $p > 0$. For $g \in P$ of order $p^j$ we have $(g - 1)^{p^j} = g^{p^j} - 1 = 0$ in $K[H/H']$. So the elements $g - 1$, $g \in P$, generate a nilpotent ideal with quotient $K[\mathbb{Z}^r]$, a domain, and $\operatorname{Spec} K[H/H']$ has the same space as the irreducible scheme $\operatorname{Spec} K[\mathbb{Z}^r]$. A connected open and closed subset that contains the unit is the connected component of the unit.

(3). If $X_K$ is connected, then $Y_e = X_K$, and $X = \{x_e\}$ because $\pi$ is surjective. If $K[H]$ is a domain, then $H$ is torsion-free, as in Step 2 of the proof of Theorem 4.1, so $H \cong \mathbb{Z}^r$. $\blacksquare$

Corollary 4.7. Let $K$ be a field. Let $G$ be a connected group scheme of finite type over $K$ that is not commutative, for example $\mathrm{GL}_{n,K}$ with $n \ge 2$, or let $G$ be the additive group $\mathbb{G}_{a,K}$. Then there is no monoid scheme $X$ with a morphism of monoid schemes $\mu \colon X \times X \to X$ such that $(X_K, \mu_K)$ is isomorphic to $G$ with its group law. The same holds over $\mathbb{Z}$ for $\mathrm{GL}_n$ with $n \ge 2$ and for $\mathbb{G}_a$.

Proof. By Theorem 4.6 (3), such a group would be $D_K(H)$, which is commutative. The scheme $\mathrm{GL}_{n,K}$ is connected, because it is a nonempty open subscheme of $\mathbb{A}^{n^2}_K$. It is not commutative for $n \ge 2$: the matrices $1 + E_{12}$ and $1 + E_{21}$ do not commute. If $D_K(H) \cong \mathbb{G}_{a,K}$ as schemes, then $K[H] \cong K[t]$. The units of $K[t]$ are the nonzero constants, and the units of $K[H]$ contain $H$. So $H = \{1\}$ and $K[H] = K$, which is not $K[t]$. A pair $(X, \mu)$ over $\mathbb{Z}$ would give one over $\mathbb{Q}$ by base change. $\blacksquare$

Example 4.8. The scheme of $n \times n$ matrices is $\operatorname{Spec} \mathbb{Z}[x_{ij}]$, the base change of the affine space of dimension $n^2$ over $\mathbb{F}_1$. Matrix multiplication is given by $x_{ij} \mapsto \sum_k x_{ik} \otimes x_{kj}$. A morphism of affine monoid schemes induces a ring homomorphism that sends each basis element of $\mathbb{Z}[M]$ to a basis element or to zero. A sum of $n \ge 2$ different basis elements is neither. So matrix multiplication is not the base change of a morphism of monoid schemes for these coordinates. Corollary 4.7 says that no other monoid scheme with base change $\mathrm{GL}_n$ can repair this. Extensions of finite groups by tori do occur: the group of monomial matrices is an example (Exercise 4).

Reference: [López Peña–Lorscheid 2011b, Section 6.1.3] states that a Chevalley scheme is a group object in monoid schemes only when it is a split torus; Theorem 4.6 proves this for every monoid scheme and every group law.

5. Torified morphisms and affine torifications

5.1 Torified morphisms

Definition 5.1. Let $(X,T)$ and $(Y,S)$ be torified schemes, with $T = \{\tau_i \colon T_i \to X\}_{i \in I}$ and $S = \{\sigma_j \colon S_j \to Y\}_{j \in J}$. A torified morphism $(X,T) \to (Y,S)$ is a morphism of schemes $\varphi \colon X \to Y$ for which there are a map $\tilde\varphi \colon I \to J$ and homomorphisms of group schemes $\varphi_i \colon T_i \to S_{\tilde\varphi(i)}$ with $\varphi \circ \tau_i = \sigma_{\tilde\varphi(i)} \circ \varphi_i \qquad \text{for all } i \in I .$

This is [López Peña–Lorscheid 2011b, Definition 1.6]. The map $\tilde\varphi$ and the $\varphi_i$ are determined by $\varphi$, because the images of the $\sigma_j$ are disjoint and the $\sigma_j$ are monomorphisms. So "torified" is a property of $\varphi$: it maps each torus into a torus, by a homomorphism of tori. A homomorphism $\varphi_i$ is the same as a homomorphism of character groups in the other direction. Torified morphisms can be composed, and the product of two torified morphisms is torified for the product torifications. Torified schemes with torified morphisms form a category, and so do the torified varieties and the affinely torified varieties.

Proposition 5.2. Let $f \colon X' \to X$ be a morphism between monoid schemes that satisfy the hypothesis of Proposition 3.1. Then $f_{\mathbb{Z}} \colon X'_{\mathbb{Z}} \to X_{\mathbb{Z}}$ is a torified morphism for the torifications by the fibres of $\pi$.

Proof. Let $x' \in X'$ and $x = f(x')$. The map on stalks $f^\# \colon \mathcal{O}_{X,x} \to \mathcal{O}_{X',x'}$ sends units to units, and non-units to non-units because $f$ is local. So it restricts to a homomorphism $H_x \to H_{x'}$, which is a homomorphism of tori $Y_{x'} \to Y_x$. The morphism $f$ maps the generizations of $x'$ to generizations of $x$, and on the affine opens of Lemma 1.1 (1) the morphism $f_{\mathbb{Z}}$ is given by $\mathbb{Z}[f^\#] \colon \mathbb{Z}[\mathcal{O}_{X,x}] \to \mathbb{Z}[\mathcal{O}_{X',x'}]$. The composite with the quotient map to $\mathbb{Z}[H_{x'}]$ sends the non-units of $\mathcal{O}_{X,x}$ to zero. So it factors through $\mathbb{Z}[H_x]$, by the homomorphism above. This is the required commutative square. $\blacksquare$

So base change is a functor from these monoid schemes to affinely torified varieties. In particular toric morphisms between toric varieties are torified.

Example 5.3. (a) The multiplication $\mathbb{G}_m^n \times \mathbb{G}_m^n \to \mathbb{G}_m^n$ is a homomorphism of tori, so it is torified for the torifications by one torus. It is the base change of the morphism of monoids $t_i \mapsto (t_i, t_i)$.

(b) Let $T$ be the coordinate torification of $\mathbb{A}^1$. The translation $x \mapsto x + 1$ is not a torified morphism $(\mathbb{A}^1, T) \to (\mathbb{A}^1, T)$: it maps the torus $x \ne 0$ onto the set $x \ne 1$, which is not contained in a torus of $T$. Let $T'$ be the torification by the point $x = 1$ and its complement, with coordinate $x - 1$. Then the translation is an isomorphism of torified varieties $(\mathbb{A}^1, T) \to (\mathbb{A}^1, T')$, and the identity of $\mathbb{A}^1$ is not a torified morphism between them. So a variety has many torifications, and the functor $(X,T) \mapsto X$ to schemes is faithful but not full.

(c) The point $x = 1$ of $\mathbb{G}_m$, as a torus of dimension zero, maps to $\mathbb{G}_m$ by a torified morphism. The point $x = -1$ does not, because a homomorphism from the trivial group has the unit as its image. Torified morphisms respect the unit sections of the tori.

Lemma 5.4. Let $\varphi \colon (X,T) \to (Y,S)$ be a torified morphism between affinely torified schemes, and let $Y$ be separated. Then $X$ has a cover by affine open unions of tori $U$, each of which is mapped by $\varphi$ into an affine open union of tori $V$ of $Y$.

Proof. Let $\tilde U \subseteq X$ and $\tilde V \subseteq Y$ be affine open unions of tori. The open set $\varphi^{-1}(\tilde V)$ is a union of tori, because $\varphi$ maps each torus of $X$ into a torus of $Y$, and this torus lies in $\tilde V$ or is disjoint from it. So $U = \tilde U \cap \varphi^{-1}(\tilde V)$ is an open union of tori. It is affine: $U = \tilde U \times_Y \tilde V$ is a closed subscheme of the affine scheme $\tilde U \times \tilde V$, because $Y$ is separated. These sets $U$ cover $X$, and $\varphi(U) \subseteq \tilde V$. $\blacksquare$

Reference: [López Peña–Lorscheid 2011b, Lemma 1.7] has no separation hypothesis; its proof uses that all sets $\tilde U \cap \varphi^{-1}(\tilde V)$ are affine, which fails for the inclusion of the plane into the plane with a doubled origin, so we assume that $Y$ is separated.

5.2 Group laws are not torified

Example 5.3 (a) is essentially the only group law that is a torified morphism.

Theorem 5.5. Let $G$ be a group scheme over $\mathbb{Z}$ with group law $m$, and let $T = \{\tau_i \colon T_i \to G\}_{i \in I}$ be a torification with $I$ finite. Assume that $m \colon (G \times G, T \times T) \to (G, T)$ is a torified morphism. Let $g_i = \tau_i(1) \in G(\mathbb{Z})$ be the image of the unit section of $T_i$.

The set $E = \{g_i : i \in I\}$ is a finite subgroup of $G(\mathbb{Z})$ with $|I|$ elements.
Let $0 \in I$ be the index with $g_0 = 1$. Then $\tau_0 \colon T_0 \to G$ is a homomorphism of group schemes.
All tori $T_i$ have the same dimension $r$, and $\tau_i(T_i(R)) = g_i\, \tau_0(T_0(R))$ for every ring $R$.
For every field $L$, the group $G(L)$ is the semidirect product of its normal subgroup $\tau_0(T_0(L)) \cong (L^\times)^r$ and the finite group $E$. The counting polynomial is $N_G(q) = |I|\,(q-1)^r$.

Proof. Let $\tilde m \colon I \times I \to I$ and $m_{ij} \colon T_i \times T_j \to T_{\tilde m(i,j)}$ be the data of the torified morphism $m$. The $m_{ij}$ are homomorphisms, so $m_{ij}(1,1) = 1$, and therefore $g_i g_j = g_{\tilde m(i,j)}$. The $g_i$ are different for different $i$, because the tori are disjoint. So $E$ is a finite subset of the group $G(\mathbb{Z})$ that is closed under multiplication. Such a subset is a subgroup: the powers of $g \in E$ repeat, so some power of $g$ with positive exponent is $1$, and $g^{-1}$ is a power of $g$ too. This proves (1).

For (2), $m_{00} \colon T_0 \times T_0 \to T_0$ satisfies $\tau_0(m_{00}(s,t)) = \tau_0(s)\tau_0(t)$. With $t = 1$ this gives $\tau_0(m_{00}(s,1)) = \tau_0(s)$, so $m_{00}(s,1) = s$, because $\tau_0$ is a monomorphism. In the same way $m_{00}(1,t) = t$. Since $m_{00}$ is a homomorphism, $m_{00}(s,t) = m_{00}(s,1)\, m_{00}(1,t) = st$. So $\tau_0(st) = \tau_0(s)\tau_0(t)$.

For (3), fix $i$ and let $i'$ be the index with $g_{i'} = g_i^{-1}$. Then $\tilde m(i, 0) = i$ and $\tilde m(i', i) = 0$. Define homomorphisms of tori $\rho \colon T_0 \to T_i,\quad \rho(t) = m_{i0}(1, t), \qquad\qquad \sigma \colon T_i \to T_0,\quad \sigma(s) = m_{i'i}(1, s) .$ They satisfy $\tau_i(\rho(t)) = g_i\, \tau_0(t)$ and $\tau_0(\sigma(s)) = g_i^{-1} \tau_i(s)$. So $\tau_0(\sigma\rho(t)) = \tau_0(t)$ and $\tau_i(\rho\sigma(s)) = \tau_i(s)$. Since $\tau_0$ and $\tau_i$ are monomorphisms, $\rho$ and $\sigma$ are inverse isomorphisms. So $T_i$ has the dimension of $T_0$, and $\tau_i(T_i(R)) = g_i\, \tau_0(T_0(R))$.

For (4), let $L$ be a field and write $T_0(L)$ for $\tau_0(T_0(L))$. By Lemma 2.2 and (3), $G(L)$ is the disjoint union of the cosets $g_i T_0(L)$. The homomorphism $m_{0i} \colon T_0 \times T_i \to T_i$ shows that $T_0(L)\, g_i\, T_0(L) \subseteq g_i T_0(L)$. Hence $g_i^{-1} T_0(L)\, g_i \subseteq T_0(L)$ for all $i$, and $T_0(L)$ is normal. The subgroup $E$ meets $T_0(L)$ only in $g_0 = 1$. So $G(L) = T_0(L) \rtimes E$, and $G(\mathbb{F}_q)$ has $|I|(q-1)^r$ elements. $\blacksquare$

Corollary 5.6. Let $G$ be $\mathrm{GL}_n$ with $n \ge 2$, or $\mathbb{G}_a$, or a split reductive group scheme over $\mathbb{Z}$ that is not a torus. Then there is no torification $T$ of $G$ for which the group law is a torified morphism $(G \times G, T \times T) \to (G, T)$.

Proof. By Theorem 5.5 the counting polynomial would be $|I|(q-1)^r$, which does not vanish at $q = 0$. But $N_{\mathrm{GL}_n}(q) = \prod_{i=0}^{n-1}(q^n - q^i)$ and $N_{\mathbb{G}_a}(q) = q$ are divisible by $q$, and so is the polynomial $(q-1)^r q^s \sum_w q^{\ell(w)}$ of Proposition 3.9 when $s \ge 1$, that is, when there are roots. $\blacksquare$

Reference: [Lorscheid 2012b, Remark 7.7] says that semidirect products of diagonalizable groups with finite constant groups seem to be the only algebraic groups with a group law of this kind; Theorems 4.6 and 5.5 prove statements of this form.

5.3 An affine torification is a strong condition

Example 5.7 (the Schubert torification of $\operatorname{Gr}(2,4)$ is not affine). Let $K$ be a field. We show that no affine open union of tori of the Schubert torification of $\operatorname{Gr}(2,4)_K$ contains the point $P_0 = C_{\{1,2\}}$, the span of $e_1$ and $e_2$. If the Schubert torification of $\operatorname{Gr}(2,4)$ were affine, it would give such an open set after base change to $K$.

Suppose $U$ is such an open set, and let $Z$ be its complement, a closed union of tori. Since $\operatorname{Gr}(2,4)_K$ is separated and $U$ is affine and dense, every irreducible component of $Z$ has codimension one [Stacks, Tag 0BCV]. The generic point of a component lies in a torus contained in $Z$, and the component is the closure of this torus. So every component of $Z$ is the closure of a torus of dimension $3$.

There are five tori of dimension $3$. Four lie in the cell $C_{\{3,4\}}$, whose points are the row spaces of $\begin{pmatrix} a & b & 1 & 0 \\ c & d & 0 & 1 \end{pmatrix} :$ they are given by the vanishing of exactly one of $a, b, c, d$. The fifth is the open torus $\alpha\beta\gamma \ne 0$ of the cell $C_{\{2,4\}}$, whose points are the row spaces of $\begin{pmatrix} \alpha & 1 & 0 & 0 \\ \beta & 0 & \gamma & 1 \end{pmatrix} .$ The point $P_0$ is the origin of the chart $U_{\{1,2\}}$, whose points are the row spaces of the matrices $(\,1_2 \mid A\,)$ with a $2 \times 2$ matrix $A$. If the first two columns of a $2 \times 4$ matrix $(B \mid B')$ form an invertible matrix $B$, then its row space is the point $A = B^{-1}B'$ of this chart.

For the first four tori, let $B_0$ be an invertible matrix with entries $0$ and $1$ and exactly one entry $0$, in the prescribed position. For $s \ne 0$, the matrix $(sB_0 \mid 1_2)$ is a point of the torus, and its chart coordinate is $A = s^{-1}B_0^{-1}$. For the fifth torus, take $\alpha = \gamma = 1$ and $\beta = s$; a direct computation gives $A = \begin{pmatrix} s^{-1} & s^{-1} \\ -s^{-1} & -s^{-1} \end{pmatrix} .$ In each case this is a morphism from $\mathbb{G}_m$, with coordinate $s$, into the torus, and it extends to a morphism from $\operatorname{Spec} K[s^{-1}]$ to the chart that sends the point $s^{-1} = 0$ to $A = 0$, which is $P_0$. So $P_0$ lies in the closure of each of the five tori.

Hence every irreducible component of $Z$ contains $P_0$. Since $P_0 \notin Z$, the set $Z$ is empty, and $\operatorname{Gr}(2,4)_K = U$ is affine. This is false. A function on $\operatorname{Gr}(2,4)_K$ is a polynomial $F(B)$ on the chart $U_{\{3,4\}}$ and a polynomial $F'(A)$ on the chart $U_{\{1,2\}}$, with $F(B) = F'(B^{-1})$ for invertible $B$. Then $F(sB)$ is a polynomial in $s$ and in $s^{-1}$, so it does not depend on $s$, and $F$ is constant. An affine scheme whose only functions are the constants is a point, but $\operatorname{Gr}(2,4)_K$ has dimension $4$.

Reference: [López Peña–Lorscheid 2011b, Example 1.16] shows that the cover by the six charts $U_J$ is not compatible with this torification; the argument above excludes every affine cover.

What is functorial and what is not. Base change is a functor from the monoid schemes of Proposition 3.1 to affinely torified varieties (Propositions 3.1 and 5.2). Products of torified varieties and of torified morphisms are torified (Lemma 2.10). The following constructions are not functorial.

The torification. A variety has many torifications (Examples 3.11 and 5.3 (b)). Only the numbers $\delta_l$ are invariants of the variety.
Morphisms. Translations, and the group laws of all groups other than extensions of finite groups by tori, are not torified (Example 5.3 (b), Theorem 5.5).
Subschemes. An open or closed subscheme of a torified variety need not be torifiable (Example 2.9).
Affine covers. A torification need not be affine (Example 5.7). Affine torifications are the ones for which the constructions of the next section work.

6. Torified varieties as varieties over $\mathbb{F}_1$

The lesson Varieties over the field with one element after Soulé and Connes–Consani studies two definitions of a variety over $\mathbb{F}_1$. Both describe a variety over $\mathbb{F}_1$ by a functor of "points with values in roots of unity", a complex object, and a universal property that produces a scheme over $\mathbb{Z}$. This section shows how an affinely torified variety gives such an object. We restate the definitions that we use. A reference for this section is [López Peña–Lorscheid 2011b, Sections 2 and 3].

6.1 Gadgets

Definition 6.1. A gadget is a triple $\mathcal{X} = (\underline{X}, X_{\mathbb{C}}, \mathrm{ev})$ of

a functor $\underline{X}$ from finite abelian groups to sets,
a reduced scheme $X_{\mathbb{C}}$ of finite type over $\mathbb{C}$,
a natural transformation $\mathrm{ev} \colon \underline{X}(D) \to \operatorname{Hom}(\operatorname{Spec} \mathbb{C}[D], X_{\mathbb{C}})$.

It is finite if all sets $\underline{X}(D)$ are finite, and graded if $\underline{X}$ is a disjoint union of functors $\underline{X}^{(l)}$, $l \ge 0$. A morphism of gadgets $\mathcal{X} \to \mathcal{Y}$ is a pair $(\underline{\varphi}, \varphi_{\mathbb{C}})$ of a natural transformation $\underline{X} \to \underline{Y}$ and a morphism $X_{\mathbb{C}} \to Y_{\mathbb{C}}$ with $\mathrm{ev}_{\mathcal{Y}} \circ \underline{\varphi} = \varphi_{\mathbb{C}} \circ \mathrm{ev}_{\mathcal{X}}$. It is an immersion if all maps $\underline{\varphi}(D)$ are injective and $\varphi_{\mathbb{C}}$ is an immersion.

A variety $V$ defines the gadget $\mathcal{G}(V) = (D \mapsto V(\mathbb{Z}[D]),\ V \otimes \mathbb{C},\ \text{base change to } \mathbb{C})$, and a morphism $f \colon V \to V'$ defines a morphism $\mathcal{G}(f)$.

Definition 6.2. An affine variety over $\mathbb{F}_1$, in the sense of Connes and Consani, is a finite graded gadget $\mathcal{X}$ for which there are an affine variety $X_{\mathbb{Z}}$ and an immersion $i \colon \mathcal{X} \to \mathcal{G}(X_{\mathbb{Z}})$ with the following universal property: for every affine variety $V$ and every morphism of gadgets $\varphi \colon \mathcal{X} \to \mathcal{G}(V)$, there is a unique morphism of schemes $\varphi_{\mathbb{Z}} \colon X_{\mathbb{Z}} \to V$ with $\varphi = \mathcal{G}(\varphi_{\mathbb{Z}}) \circ i$. The variety $X_{\mathbb{Z}}$ is the extension of scalars of $\mathcal{X}$. It is unique up to a unique isomorphism.

Reference: [Connes–Consani 2011a, Definitions 2.5 to 2.8]; we use the form of [López Peña–Lorscheid 2011b, Definitions 2.1 and 2.2], in which "variety over $\mathbb{Z}$" means a reduced scheme of finite type.

Definition 6.3. Let $(X,T)$ be a torified variety. The gadget $\mathcal{L}(X,T) = (\underline{X}, X_{\mathbb{C}}, \mathrm{ev})$ is defined by $\underline{X}^{(l)}(D) = \coprod_{i \,:\, d_i = l} \operatorname{Hom}(A_i, D), \qquad X_{\mathbb{C}} = X \otimes \mathbb{C},$ and $\mathrm{ev}$ sends $a \colon A_i \to D$ to the composite of $\operatorname{Spec} \mathbb{C}[a] \colon \operatorname{Spec} \mathbb{C}[D] \to T_i \otimes \mathbb{C}$ with $\tau_i$.

This gadget is finite and graded, and $\underline{X}(D)$ has $\sum_l \delta_l |D|^l$ elements. By Proposition 2.8 this is $N_X(q)$ when $|D| = q - 1$: the functor counts the points. There is a morphism of gadgets $i \colon \mathcal{L}(X,T) \longrightarrow \mathcal{G}(X)$ that is the identity on $X_{\mathbb{C}}$ and sends $a \colon A_i \to D$ to the point $\tau_i \circ \operatorname{Spec} \mathbb{Z}[a]$ of $X(\mathbb{Z}[D])$. It is an immersion: the index $i$ can be read off from the torus that contains the complex points of $\tau_i \circ \operatorname{Spec} \mathbb{Z}[a]$, and then $a$ is determined because $\tau_i$ is a monomorphism.

Lemma 6.4. Let $A$ be a free abelian group of finite rank and $F \in \mathbb{C}[A]$. Assume that for every $n \ge 1$ the image of $F$ in $\mathbb{C}[A/nA]$ lies in $\mathbb{Z}[A/nA]$. Then $F \in \mathbb{Z}[A]$.

Proof. Write $F = \sum_{\alpha \in \Sigma} c_\alpha \alpha$ with a finite set $\Sigma \subseteq A$. Identify $A$ with $\mathbb{Z}^r$ and choose $n$ larger than the absolute values of all coordinates of all differences $\alpha - \alpha'$ with $\alpha, \alpha' \in \Sigma$. Then the classes of the $\alpha \in \Sigma$ in $A/nA$ are pairwise different. The image of $F$ in $\mathbb{C}[A/nA]$ is $\sum c_\alpha \bar\alpha$ with different basis elements $\bar\alpha$, and it lies in $\mathbb{Z}[A/nA]$. So all $c_\alpha$ are integers. $\blacksquare$

6.2 The affine comparison theorem

Theorem 6.5. Let $(X,T)$ be a torified variety with $X = \operatorname{Spec} R$ affine. Let $\rho_i \colon R \to \mathbb{Z}[A_i]$ be the restriction to the torus $T_i$, and put $R^\sharp = \{ f \in R \otimes \mathbb{C} \;:\; (\rho_i \otimes \mathbb{C})(f) \in \mathbb{Z}[A_i] \text{ for all } i \in I \} .$

$R \subseteq R^\sharp \subseteq R \otimes \mathbb{Q}$. For every affine variety $V = \operatorname{Spec} B$ and every morphism of gadgets $\varphi \colon \mathcal{L}(X,T) \to \mathcal{G}(V)$, the homomorphism $\varphi_{\mathbb{C}}^\# \colon B \otimes \mathbb{C} \to R \otimes \mathbb{C}$ maps $B$ into $R^\sharp$.
The following are equivalent.
- (a) The immersion $i \colon \mathcal{L}(X,T) \to \mathcal{G}(X)$ has the universal property of Definition 6.2. So $\mathcal{L}(X,T)$ is an affine variety over $\mathbb{F}_1$ with extension of scalars $X$.
- (b) $R^\sharp = R$.
- (c) For every prime $p$, the scheme $X \otimes \mathbb{F}_p$ is reduced.

Proof. By Proposition 2.7, $R$ has no torsion, so $R \subseteq R \otimes \mathbb{Q} \subseteq R \otimes \mathbb{C}$, and $R \subseteq R^\sharp$ is clear.

(1) Choose a basis $(c_j)$ of $\mathbb{C}$ over $\mathbb{Q}$ with $c_0 = 1$. An element $f \in R \otimes \mathbb{C}$ is a finite sum $\sum f_j c_j$ with $f_j \in R \otimes \mathbb{Q}$, and $(\rho_i \otimes \mathbb{C})(f) = \sum (\rho_i \otimes \mathbb{Q})(f_j)\, c_j$. If $f \in R^\sharp$, this lies in $\mathbb{Z}[A_i] \subseteq \mathbb{Q}[A_i]\, c_0$, so $(\rho_i \otimes \mathbb{Q})(f_j) = 0$ for all $j \ne 0$ and all $i$. An element of $R \otimes \mathbb{Q}$ that restricts to zero on every torus vanishes at every point of $X \otimes \mathbb{Q}$, by Proposition 2.7 (1). So it is nilpotent, and it is zero because $R \otimes \mathbb{Q}$ is reduced. Hence $f = f_0 \in R \otimes \mathbb{Q}$.

Now let $\varphi = (\underline{\varphi}, \varphi_{\mathbb{C}})$ be a morphism to $\mathcal{G}(V)$, let $b \in B$ and $f = \varphi_{\mathbb{C}}^\#(b \otimes 1)$. Fix $i$ and $n \ge 1$, put $D = A_i/nA_i$, and let $a \colon A_i \to D$ be the quotient map. The compatibility of $\varphi$ with the evaluations says that the point $\varphi_{\mathbb{C}} \circ \mathrm{ev}(a)$ of $V$ with values in $\mathbb{C}[D]$ is the base change of the point $\underline{\varphi}(D)(a)$ with values in $\mathbb{Z}[D]$. Evaluating both at $b$ gives $\mathbb{C}[a]\bigl((\rho_i \otimes \mathbb{C})(f)\bigr) \in \mathbb{Z}[D] .$ By Lemma 6.4, $(\rho_i \otimes \mathbb{C})(f) \in \mathbb{Z}[A_i]$. So $f \in R^\sharp$.

(2) (b) implies (a). Let $\varphi \colon \mathcal{L}(X,T) \to \mathcal{G}(V)$ be a morphism. By (1) and (b), $\varphi_{\mathbb{C}}^\#$ maps $B$ into $R$. Let $\psi \colon X \to V$ be the morphism with $\psi^\# = \varphi_{\mathbb{C}}^\#|_B$. Then $\psi \otimes \mathbb{C} = \varphi_{\mathbb{C}}$. For $a \in \operatorname{Hom}(A_i, D)$, the two points $\underline{\varphi}(D)(a)$ and $\psi \circ i(D)(a)$ of $V(\mathbb{Z}[D])$ have the same base change to $\mathbb{C}[D]$, namely $\varphi_{\mathbb{C}} \circ \mathrm{ev}(a)$. Since $\mathbb{Z}[D] \subseteq \mathbb{C}[D]$, they are equal. So $\varphi = \mathcal{G}(\psi) \circ i$. If $\psi'$ is another morphism with this property, then $\psi' \otimes \mathbb{C} = \varphi_{\mathbb{C}}$, and $\psi' = \psi$ because $R \to R \otimes \mathbb{C}$ is injective.

(a) implies (b). Let $f \in R^\sharp$. Let $\mathbb{A}^1 = \operatorname{Spec} \mathbb{Z}[u]$. Define $\varphi_{\mathbb{C}} \colon X_{\mathbb{C}} \to \mathbb{A}^1_{\mathbb{C}}$ by $u \mapsto f$, and let $\underline{\varphi}(D)$ send $a \in \operatorname{Hom}(A_i, D)$ to the element $\mathbb{Z}[a](\rho_i(f))$ of $\mathbb{Z}[D] = \mathbb{A}^1(\mathbb{Z}[D])$. This is natural in $D$ and compatible with the evaluations, so it is a morphism of gadgets $\mathcal{L}(X,T) \to \mathcal{G}(\mathbb{A}^1)$. By (a) it is $\mathcal{G}(\psi) \circ i$ for a morphism $\psi \colon X \to \mathbb{A}^1$. Then $f = \psi^\#(u) \in R$.

(c) implies (b). Let $f \in R^\sharp$. By (1) we can write $f = g/N$ with $g \in R$ and an integer $N \ge 1$. We use induction on $N$. If $N > 1$, let $p$ be a prime factor of $N$. Then $\rho_i(g) = N \rho_i(f) \in p\,\mathbb{Z}[A_i]$ for all $i$. So the image of $g$ in $R/pR$ restricts to zero on every torus $T_i \otimes \mathbb{F}_p$. By Proposition 2.7 (1) it vanishes at every point of $X \otimes \mathbb{F}_p$, so it is nilpotent, and it is zero by (c). Hence $g = p g_1$ with $g_1 \in R$, and $f = g_1/(N/p)$.

(b) implies (c). Suppose $R/pR$ is not reduced. Choose $g \in R$ with $g \notin pR$ and $g^m \in pR$ for some $m$. Then $g$ vanishes at every point of $X \otimes \mathbb{F}_p$, so the image of $\rho_i(g)$ in the domain $\mathbb{F}_p[A_i]$ is nilpotent, hence zero. So $\rho_i(g) \in p\,\mathbb{Z}[A_i]$ for all $i$, and $f = g/p$ lies in $R^\sharp$. It does not lie in $R$, because $g \notin pR$ and $R$ has no torsion. $\blacksquare$

Reference: [López Peña–Lorscheid 2011b, Theorem 2.10] states (a) for every affinely torified variety; this fails in Example 6.6, where a fibre is not reduced, so we prove the equivalence with (c).

The proof above does not pass through an equality of the form $R[\delta^{-1}] \cap (R \otimes \mathbb{C}) = R$, on which the proofs of [López Peña–Lorscheid 2011b, Theorem 2.10] and [Connes–Consani 2011a, Theorem 4.10] rest. In the first, $\delta$ is a function that vanishes on the tori that are not open; in the second, $R$ is the ring of a Chevalley group scheme and $\delta$ is a function whose set of non-zeros is the big cell. Such an equality needs a condition at the primes: for a ring $R$ without torsion, it holds when $\delta$ is a non-zero-divisor in $R \otimes \mathbb{Q}$ and in $R/pR$ for every prime $p$. The lesson Varieties over the field with one element after Soulé and Connes–Consani proves it under this condition and applies it to Chevalley groups. In Example 6.6 the equality fails for $\delta = t^2$.

Condition (c) holds for the affine pieces $\operatorname{Spec} \mathbb{Z}[M]$ of Proposition 3.1, because $\mathbb{F}_p[M]$ is a subring of a domain. It holds for every $X$ that is smooth over $\mathbb{Z}$, in particular for a split reductive group with the torification of Proposition 3.9. So a split reductive group scheme is the extension of scalars of an affine variety over $\mathbb{F}_1$. [Connes–Consani 2011a, Theorem 4.10] obtains this over $\mathbb{F}_{1^2}$, with a different functor. In both cases the statement is about the variety, not about the group law; see Section 7.

Example 6.6 (a fibre that is not reduced). Let $p$ be a prime and $R = \mathbb{Z}[pt, t^2, t^3] = \{ f \in \mathbb{Z}[t] : p \text{ divides } f'(0) \}, \qquad X = \operatorname{Spec} R .$ As a group, $R = \mathbb{Z} \oplus \mathbb{Z}\,pt \oplus t^2\mathbb{Z}[t]$. The ideal $(pt, t^2, t^3)$ is the kernel of $f \mapsto f(0)$, so $T_0 = V(pt, t^2, t^3) \cong \operatorname{Spec} \mathbb{Z}$. The open subscheme $T_1 = D(t^2)$ is $\operatorname{Spec} \mathbb{Z}[t, t^{-1}]$, because $t = t^3/t^2$. A prime ideal that contains $t^2$ contains $pt$ and $t^3$, since $(pt)^2 = p^2 t^2$ and $(t^3)^2 = (t^2)^3$. So $T_0$ and $T_1$ form a torification of the affine variety $X$.

The fibre over $p$ is not reduced: $pt \notin pR$, but $(pt)^2 = p \cdot p t^2 \in pR$. Accordingly $f = t = (pt)/p$ lies in $R^\sharp$ and not in $R$; in fact $R^\sharp = \mathbb{Z}[t]$. By Theorem 6.5, the immersion $\mathcal{L}(X,T) \to \mathcal{G}(X)$ does not have the universal property. Concretely, the morphism of gadgets to $\mathcal{G}(\mathbb{A}^1)$ given by the function $t$ does not come from a morphism $X \to \mathbb{A}^1$.

The gadget $\mathcal{L}(X,T)$ is still an affine variety over $\mathbb{F}_1$, with another extension of scalars. The inclusion $R \subseteq \mathbb{Z}[t]$ is a morphism $\nu \colon \mathbb{A}^1 \to X$ that maps the two coordinate tori of $\mathbb{A}^1$ isomorphically onto $T_0$ and $T_1$, and $\nu \otimes \mathbb{C}$ is an isomorphism because $R \otimes \mathbb{C} = \mathbb{C}[t]$. So the morphism of gadgets $\mathcal{L}(\nu)$ of Proposition 6.7 is an isomorphism from the gadget of $\mathbb{A}^1$, with its coordinate torification, to $\mathcal{L}(X,T)$. The fibres of $\mathbb{A}^1$ are reduced. Hence the extension of scalars of $\mathcal{L}(X,T)$ is $\mathbb{A}^1$, and $\mathbb{A}^1$ is not isomorphic to $X$, whose fibre over $p$ is not reduced.

6.3 Functoriality, and the dependence on the torification

Proposition 6.7.

Let $\varphi \colon (X,T) \to (X',T')$ be a torified morphism of torified varieties, with data $\tilde\varphi$ and $\varphi_i$. Let $\varphi_i^\ast \colon A'_{\tilde\varphi(i)} \to A_i$ be the map on character groups. Then $\underline{\varphi}(D) \colon \operatorname{Hom}(A_i, D) \to \operatorname{Hom}(A'_{\tilde\varphi(i)}, D), \quad a \mapsto a \circ \varphi_i^\ast, \qquad \varphi_{\mathbb{C}} = \varphi \otimes \mathbb{C}$ is a morphism of gadgets $\mathcal{L}(\varphi) \colon \mathcal{L}(X,T) \to \mathcal{L}(X',T')$. This makes $\mathcal{L}$ a functor, and $i_{X'} \circ \mathcal{L}(\varphi) = \mathcal{G}(\varphi) \circ i_X$.
Let $X$ be a monoid scheme as in Proposition 3.1, and give $X_{\mathbb{Z}}$ the torification by the fibres of $\pi$. Then, naturally in the finite abelian group $D$, $\underline{X_{\mathbb{Z}}}(D) \;=\; \operatorname{Hom}(\operatorname{Spec} D_0, X),$ the set of morphisms of monoid schemes from the spectrum of the monoid $D_0 = D \sqcup \{0\}$ to $X$.

Proof. (1) The evaluation of $a \circ \varphi_i^\ast$ is $\tau'_{\tilde\varphi(i)} \circ \varphi_i \circ \operatorname{Spec} \mathbb{C}[a]$, and $\tau'_{\tilde\varphi(i)} \circ \varphi_i = \varphi \circ \tau_i$. So the pair is compatible with the evaluations. The other claims follow from the definitions.

(2) The monoid $D_0$ has one prime ideal, $\{0\}$. A morphism $\operatorname{Spec} D_0 \to X$ is a point $x$ of $X$ and a local morphism of monoids $\mathcal{O}_{X,x} \to D_0$. Such a morphism sends the non-units to $0$ and restricts to a group homomorphism $H_x \to D$, and every group homomorphism $H_x \to D$ arises once. The character group of the torus $Y_x = \operatorname{Spec} \mathbb{Z}[H_x]$ is $H_x$. $\blacksquare$

Reference: [López Peña–Lorscheid 2011b, Propositions 2.8 and 5.1].

For a cyclic group $D = \mu_n$, the monoid $D_0$ is $\mathbb{F}_{1^n}$. So for the base change of a monoid scheme, the functor of the gadget is the functor of points of the monoid scheme with values in the extensions $\mathbb{F}_{1^n}$. The gadget of a torified variety extends this functor of points to varieties that do not come from monoid schemes.

Part (1) and Theorem 6.5 show that for affine torified varieties with reduced fibres, the extension of scalars of $\mathcal{L}(\varphi)$ is $\varphi$. The gadget is functorial in the torified variety. It is not determined by the variety.

Example 6.8 (two torifications of the plane with different gadgets). Let $T$ be the coordinate torification of $\mathbb{A}^2$, with tori $\{(0,0)\}, \qquad \{x \ne 0,\ y = 0\}, \qquad \{x = 0,\ y \ne 0\}, \qquad \{x \ne 0,\ y \ne 0\} .$ Let $T'$ be the torification with tori $\{(1,0)\}, \qquad \{x \ne 1,\ y = 0\}, \qquad \{x = 0,\ y \ne 0\}, \qquad \{x \ne 0,\ y \ne 0\},$ where the second torus has the coordinate $x - 1$. We claim that $\mathcal{L}(\mathbb{A}^2, T)$ and $\mathcal{L}(\mathbb{A}^2, T')$ are not isomorphic.

Let $(\underline{\varphi}, \varphi_{\mathbb{C}})$ be an isomorphism. For the trivial group, $\underline{X}(1)$ is the index set, so $\underline{\varphi}(1)$ is a bijection $\psi$ between the index sets. By naturality for $D \to 1$, the map $\underline{\varphi}(D)$ sends $\operatorname{Hom}(A_i, D)$ to $\operatorname{Hom}(A'_{\psi(i)}, D)$, and comparing cardinalities shows that $\psi$ preserves the dimension of the tori. Take $D = \mathbb{Z}/n$ and a character $\chi \colon D \to \mathbb{C}^\times$. The evaluations at the point $\chi$ of $\operatorname{Spec} \mathbb{C}[D]$ are points of $T_i(\mathbb{C})$ whose coordinates are $n$-th roots of unity, and every such point arises. So $\varphi_{\mathbb{C}}$ maps the points of finite order of $T_i(\mathbb{C})$ into $T'_{\psi(i)}(\mathbb{C})$. These points are dense in $T_i \otimes \mathbb{C}$: a Laurent polynomial that vanishes at all of them is zero, because for large $n$ its monomials restrict to different characters of the finite group of points with coordinates in $\mu_n$, and different characters are linearly independent. Hence $\varphi_{\mathbb{C}}$ maps the closure of $T_i \otimes \mathbb{C}$ into the closure of $T'_{\psi(i)} \otimes \mathbb{C}$, and the same holds for the inverse. So $\varphi_{\mathbb{C}}$ maps the closure of each torus of $T$ onto the closure of the corresponding torus of $T'$.

For both torifications, the closures of the two tori of dimension one are the two coordinate axes. So $\varphi_{\mathbb{C}}$ permutes the two axes, and it fixes their intersection point $(0,0)$. But $\varphi_{\mathbb{C}}$ also maps the torus $\{(0,0)\}$ of $T$ to the torus $\{(1,0)\}$ of $T'$. This is a contradiction.

Reference: [López Peña–Lorscheid 2011b, Remark 2.9] gives this pair of torifications.

6.4 Soulé's gadgets

The same method works for the definition of Soulé. We use it in the form of [López Peña–Lorscheid 2011b, Definitions 3.1 and 3.2]. Let $\mathcal{R}$ be the category of nonzero rings whose additive group is free of finite rank. For $R \in \mathcal{R}$ let $\mu(R)$ be the group of roots of unity of $R$.

An S-gadget is a triple $(\underline{X}, \mathcal{A}, e)$ of a functor $\underline{X} \colon \mathcal{R} \to \text{Sets}$, a $\mathbb{C}$-algebra $\mathcal{A}$, and a natural transformation $e \colon \underline{X}(R) \to \operatorname{Hom}(\mathcal{A}, R \otimes \mathbb{C})$. A morphism $(\underline{X}, \mathcal{A}, e) \to (\underline{X}', \mathcal{A}', e')$ is a natural transformation $\underline{\varphi} \colon \underline{X} \to \underline{X}'$ with a homomorphism $\varphi^\ast \colon \mathcal{A}' \to \mathcal{A}$ such that $e'(\underline{\varphi}(x)) = e(x) \circ \varphi^\ast$. It is an immersion if $\varphi^\ast$ and all $\underline{\varphi}(R)$ are injective. An affine scheme $V = \operatorname{Spec} B$ of finite type over $\mathbb{Z}$ defines the S-gadget $\mathcal{T}(V) = (R \mapsto \operatorname{Hom}(B, R),\ B \otimes \mathbb{C},\ \text{base change})$. An affine variety over $\mathbb{F}_1$ in the sense of Soulé is an S-gadget $\mathcal{X}$ with finite sets $\underline{X}(R)$, for which there are an affine scheme $X_{\mathbb{Z}}$ of finite type over $\mathbb{Z}$ and an immersion $i \colon \mathcal{X} \to \mathcal{T}(X_{\mathbb{Z}})$ that is universal among the morphisms from $\mathcal{X}$ to the S-gadgets $\mathcal{T}(V)$.

For a torified variety $(X,T)$ with $X = \operatorname{Spec} R_X$ affine, let $\mathcal{S}(X,T) = \Bigl( R \mapsto \coprod_{i \in I} \operatorname{Hom}(A_i, \mu(R)),\ \ R_X \otimes \mathbb{C},\ \ e \Bigr),$ where $e$ sends $a \colon A_i \to \mu(R)$ to the composite of $\rho_i \otimes \mathbb{C} \colon R_X \otimes \mathbb{C} \to \mathbb{C}[A_i]$ with the homomorphism $\mathbb{C}[A_i] \to R \otimes \mathbb{C}$ given by $a$. Let $i \colon \mathcal{S}(X,T) \to \mathcal{T}(X)$ be the identity on $R_X \otimes \mathbb{C}$, and let it send $a$ to the composite of $\rho_i$ with $\mathbb{Z}[A_i] \to R$.

Theorem 6.9. Let $(X,T)$ be a torified variety with $X = \operatorname{Spec} R_X$ affine. Then the sets of $\mathcal{S}(X,T)$ are finite and $i$ is an immersion. The immersion $i \colon \mathcal{S}(X,T) \to \mathcal{T}(X)$ has the universal property, so that $\mathcal{S}(X,T)$ is an affine variety over $\mathbb{F}_1$ in the sense of Soulé with extension of scalars $X$, if and only if $X \otimes \mathbb{F}_p$ is reduced for every prime $p$.

Proof. Finiteness. Let $R \in \mathcal{R}$ have rank $N$. The algebra $R \otimes \mathbb{C}$ is a finite product of local algebras with residue field $\mathbb{C}$. A root of unity $\zeta \in R$ has in each factor the form $\lambda + \nu$, with a root of unity $\lambda \in \mathbb{C}$ and a nilpotent $\nu$. From $(1 + \nu/\lambda)^m = 1$ we get $\nu \cdot u = 0$ for a unit $u$, so $\nu = 0$. Each $\lambda$ is a root of the characteristic polynomial of multiplication by $\zeta$ on $R \cong \mathbb{Z}^N$, which is monic of degree $N$ with integer coefficients. There are finitely many roots of unity of degree at most $N$ over $\mathbb{Q}$. So $\mu(R)$ is finite.

Immersion. The map on algebras is the identity. Let $a \colon A_i \to \mu(R)$. Since $R \ne 0$, there is a homomorphism $R \to \mathbb{C}$, and the composite $R_X \to \mathbb{Z}[A_i] \to R \to \mathbb{C}$ is a complex point of $X$ in $T_i$. This determines $i$. Then $a$ is determined, because $\tau_i$ is a monomorphism.

Universal property. Let $V = \operatorname{Spec} B$ and let $(\underline{\varphi}, \varphi^\ast)$ be a morphism $\mathcal{S}(X,T) \to \mathcal{T}(V)$. Let $b \in B$ and $f = \varphi^\ast(b \otimes 1) \in R_X \otimes \mathbb{C}$. For $n \ge 1$ and $D = A_i/nA_i$, the ring $\mathbb{Z}[D]$ lies in $\mathcal{R}$, and $D \subseteq \mu(\mathbb{Z}[D])$. Applying the compatibility with $e$ to the quotient map $a \colon A_i \to D$ gives, as in the proof of Theorem 6.5, that the image of $(\rho_i \otimes \mathbb{C})(f)$ in $\mathbb{C}[D]$ lies in $\mathbb{Z}[D]$. By Lemma 6.4, $f \in R^\sharp$. If $R^\sharp = R_X$, then $\varphi^\ast$ restricts to a homomorphism $B \to R_X$, which defines $\psi \colon X \to V$. The equality $(\underline{\varphi}, \varphi^\ast) = \mathcal{T}(\psi) \circ i$ holds on the algebras by construction, and on the functors because $R \to R \otimes \mathbb{C}$ is injective for $R \in \mathcal{R}$. The morphism $\psi$ is unique because $R_X \to R_X \otimes \mathbb{C}$ is injective. Conversely, an element $f \in R^\sharp \smallsetminus R_X$ defines a morphism to $\mathcal{T}(\mathbb{A}^1)$ that does not come from a morphism $X \to \mathbb{A}^1$: on algebras it is $u \mapsto f$, and it sends $a \colon A_i \to \mu(R)$ to the image of $\rho_i(f) \in \mathbb{Z}[A_i]$ in $R$. So the universal property holds exactly when $R^\sharp = R_X$. By Theorem 6.5 this is the condition on the fibres. $\blacksquare$

Reference: [López Peña–Lorscheid 2011b, Theorem 3.11] states, in the second step of its proof, that the universal property holds for every affine torified variety; Example 6.6 shows that the condition on the fibres is needed.

6.5 The non-affine case

Let $\mathcal{X} = (\underline{X}, X_{\mathbb{C}}, \mathrm{ev})$ be a finite graded gadget and $U_{\mathbb{C}} \subseteq X_{\mathbb{C}}$ an open subscheme. The restriction of $\mathcal{X}$ to $U_{\mathbb{C}}$ is the gadget with the complex variety $U_{\mathbb{C}}$ and the functor of those $x \in \underline{X}(D)$ for which $\mathrm{ev}(x)$ factors through $U_{\mathbb{C}}$. Following [López Peña–Lorscheid 2011b, Definition 2.4], a finite graded gadget is a variety over $\mathbb{F}_1$ if $X_{\mathbb{C}}$ has a cover by affine open subschemes $U_{\alpha,\mathbb{C}}$ such that every restriction is an affine variety over $\mathbb{F}_1$ whose functor counts the points of its extension of scalars, and such that every element of every $\underline{X}(D)$ lies in one of the restrictions.

Proposition 6.10. Let $(X,T)$ be an affinely torified variety such that $X \otimes \mathbb{F}_p$ is reduced for every prime $p$. Then $\mathcal{L}(X,T)$ is a variety over $\mathbb{F}_1$ in this sense. For every affine open union of tori $U \subseteq X$, the restriction of $\mathcal{L}(X,T)$ to $U \otimes \mathbb{C}$ is $\mathcal{L}(U, T|_U)$, an affine variety over $\mathbb{F}_1$ with extension of scalars $U$.

Proof. The evaluation of $a \in \operatorname{Hom}(A_i, D)$ has its image in $T_i \otimes \mathbb{C}$, and the torus $T_i$ lies in $U$ or is disjoint from it. So the restriction to $U \otimes \mathbb{C}$ has the functor $\coprod_{T_i \subseteq U} \operatorname{Hom}(A_i, D)$, which is that of $\mathcal{L}(U, T|_U)$. The fibres of $U$ are open in those of $X$, so they are reduced, and Theorem 6.5 applies to $U$. The functor of $\mathcal{L}(U, T|_U)$ counts the points of $U$ by Proposition 2.8. The sets $U$ cover $X$, and every torus lies in one of them. $\blacksquare$

[López Peña–Lorscheid 2011b, Theorem 2.10] also identifies the extension of scalars of the non-affine gadget $\mathcal{L}(X,T)$ with $X$, and [López Peña–Lorscheid 2011b, Theorem 3.11] treats the non-affine case for Soulé's varieties. The lesson does not prove these two statements; by Theorems 6.5 and 6.9 they need the condition on the fibres.

For a torification that is not affine, the gadget $\mathcal{L}(X,T)$ is still defined and still counts the points, but Proposition 6.10 does not apply. The Schubert torification of $\operatorname{Gr}(2,4)$ is such a case, by Example 5.7. [López Peña–Lorscheid 2011b, Section 6.2] proposes its gadget as a candidate for a model of $\operatorname{Gr}(2,4)$ over $\mathbb{F}_1$ in the theory of Connes and Consani, and records that the question of [Soulé 2004], how to realize $\operatorname{Gr}(2,4)$ over $\mathbb{F}_1$, stays open.

7. Algebraic groups over $\mathbb{F}_1$: the problem of Tits

7.1 The problem

By Proposition 3.9, the counting polynomial of a split reductive group $G$ of rank $r$ satisfies $N_G(q)/(q-1)^r \to |W|$ for $q \to 1$. For $\mathrm{GL}_n$ the limit is $n!$, the order of the symmetric group. The lesson Counting over finite fields and the limit $q \to 1$ describes the proposal of Tits that lies behind this: the Weyl group $W$ should be the group of points of $G$ over a field with one element. [Connes–Consani 2011a, Introduction] and [Lorscheid 2012b, Introduction] recall the proposal in this form.

[Lorscheid 2012b, Introduction, Problem B] turns the proposal into a precise problem. One looks for a category $\mathrm{Sch}_{\mathbb{F}_1}$ with finite products and a terminal object $\ast$, and a functor $\mathcal{X} \mapsto \mathcal{X}_{\mathbb{Z}}$ to schemes that preserves finite products and the terminal object, such that every split reductive group scheme $G$ with group law $m$ has a group object $(\mathcal{G}, \mu)$ in $\mathrm{Sch}_{\mathbb{F}_1}$ with two properties.

(i) $(\mathcal{G}_{\mathbb{Z}}, \mu_{\mathbb{Z}})$ is isomorphic to $(G, m)$.
(ii) The group $\mathcal{G}(\mathbb{F}_1) = \operatorname{Hom}(\ast, \mathcal{G})$ is isomorphic to $W = N(\mathbb{Z})/T(\mathbb{Z})$, in such a way that the composite $\sigma \colon W \cong \mathcal{G}(\mathbb{F}_1) \longrightarrow \mathcal{G}_{\mathbb{Z}}(\mathbb{Z}) \cong G(\mathbb{Z})$ sends every coset $n T(\mathbb{Z})$ to one of its elements.

Proposition 7.1. The problem has no solution for $G = \mathrm{SL}_2$.

Proof. A functor that preserves finite products and the terminal object maps group objects to group objects. For two points $a, b \colon \ast \to \mathcal{G}$, the product is $\mu \circ (a, b)$, and the functor sends it to $\mu_{\mathbb{Z}} \circ (a_{\mathbb{Z}}, b_{\mathbb{Z}})$. So $\mathcal{G}(\mathbb{F}_1) \to G(\mathbb{Z})$ is a group homomorphism, and $\sigma$ is a group homomorphism $W \to N(\mathbb{Z})$ that is a section of $N(\mathbb{Z}) \to W$. For $\mathrm{SL}_2$ and the diagonal torus, $T(\mathbb{Z}) = \{\pm 1\}$ and $N(\mathbb{Z}) = \{\pm 1, \pm w\}$ with $w = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \qquad w^2 = -1 .$ So $N(\mathbb{Z})$ is cyclic of order $4$, the group $W$ has order $2$, and both elements of the nontrivial coset have order $4$. There is no section that is a homomorphism. $\blacksquare$

Reference: [Lorscheid 2012b, Introduction].

For $\mathrm{GL}_n$ the permutation matrices are a section. So the obstruction of Proposition 7.1 is absent for $\mathrm{GL}_n$, but the theories of this lesson fail for $\mathrm{GL}_n$ too, for the reasons of the next subsection.

7.2 What fails in the theories of this lesson

For monoid schemes, Corollary 4.7 shows that the group law of $\mathrm{GL}_n$ is not a morphism. For torified varieties, Corollary 5.6 shows the same. For gadgets, the group law fails already on the normalizer of the torus. The product of two gadgets has the product of the functors, the product of the complex varieties and the product of the evaluations.

Proposition 7.2. Give $\mathrm{SL}_2$ the torification of Example 3.10, with six tori, and give the normalizer $N$ of the diagonal torus the torification by its two components. Let $\mathcal{X}$ be $\mathcal{L}(\mathrm{SL}_2)$ or $\mathcal{L}(N)$. Then there is no morphism of gadgets $\mathcal{X} \times \mathcal{X} \to \mathcal{X}$ whose complex part is the group law.

Proof. Let $I$ be the index set, with $6$ or $2$ elements. For the trivial group, $\underline{X}(1) = I$, and the evaluation sends $i$ to the complex point $g_i = \tau_i(1)$. A morphism of gadgets as in the statement gives a map $I \times I \to I$, $(i,j) \mapsto k$, with $g_i g_j = g_k$. So $E = \{g_i\}$ is a finite subset of $\mathrm{SL}_2(\mathbb{C})$ that is closed under multiplication, hence a subgroup, of order $6$ or $2$. One of the tori is the set of antidiagonal matrices. Every antidiagonal matrix in $\mathrm{SL}_2(\mathbb{C})$ has square $-1$, so it has order $4$. But $4$ divides neither $6$ nor $2$. $\blacksquare$

Reference: [López Peña–Lorscheid 2011b, Section 6.1.1], for the normalizer.

[Connes–Consani 2011a] avoids this obstruction for the normalizer by working over $\mathbb{F}_{1^2}$: there the functor is defined on pairs $(D, \epsilon)$ of a finite abelian group and an element $\epsilon$ of order $2$, which plays the role of $-1$. [Connes–Consani 2011a, Theorem 4.10] shows that a split Chevalley group scheme is a variety over $\mathbb{F}_{1^2}$, and the last paragraph of that work states that this holds for the variety and not for the group operation: only the terms of lowest degree form a group.

7.3 Two kinds of morphisms

[Lorscheid 2012b] solves a modified form of the problem. The objects are the schemes over $\mathbb{F}_1$ of [Connes–Consani 2010], in the form used in [Lorscheid 2012b, Section 2].

Definition 7.3. An $\mathbb{F}_1$-scheme is a triple $\mathcal{X} = (\tilde X, X, e_X)$ of a monoid scheme $\tilde X$, a scheme $X$ that is locally of finite type over $\mathbb{Z}$, and a morphism of schemes $e_X \colon \tilde X_{\mathbb{Z}} \to X$ such that $\tilde X_{\mathbb{Z}}(k) \to X(k)$ is bijective for every field $k$. Its base extension is $\mathcal{X}_{\mathbb{Z}} = X$.

A monoid scheme $\tilde X$ gives the $\mathbb{F}_1$-scheme $(\tilde X, \tilde X_{\mathbb{Z}}, \mathrm{id})$. A torified scheme $(X,T)$, with $X$ locally of finite type over $\mathbb{Z}$, gives the $\mathbb{F}_1$-scheme $(\tilde X_T, X, e)$, where $\tilde X_T$ is the disjoint union of the one-point monoid schemes $\operatorname{Spec}(A_i)_0$, so that $(\tilde X_T)_{\mathbb{Z}}$ is the disjoint union of the tori, and $e$ is given by the $\tau_i$. In this way Grassmannians and split reductive groups are $\mathbb{F}_1$-schemes [Lorscheid 2012b, Lemma 3.2 and Proposition 3.6].

Only the bijection on points with values in fields is used here. [Lorscheid 2012b, Section 3] therefore calls any morphism $e \colon \coprod_i \mathbb{G}_m^{d_i} \to X$ with this property a torification, and leaves open whether the restrictions of $e$ to the tori are then immersions, as Definition 2.6 requires. They are not always.

Example 7.4 (a bijection on points that is not a torification). Let $B = \{ g \in \mathbb{Z}[x,y] : g(1,1) = g(0,0) \}, \qquad X = \operatorname{Spec} B,$ and let $\nu \colon \mathbb{A}^2 \to X$ be the morphism given by $B \subseteq \mathbb{Z}[x,y]$. Let $\mathfrak{a} = (x,y) \cap (x-1, y-1)$. Then $B = \mathbb{Z} \oplus \mathfrak{a}$, and $\mathfrak{a}$ is an ideal of $B$ and of $\mathbb{Z}[x,y]$. For every $g \in \mathbb{Z}[x,y]$, the polynomial $g - (g(1,1) - g(0,0))\,x$ lies in $B$. So $\mathbb{Z}[x,y]$ is generated by $1$ and $x$ as a $B$-module, and $B$ is a finitely generated ring [Stacks, Tag 00IS]. It is a domain. So $X$ is a variety.

Let $k$ be a field and $\beta \colon B \to k$ a homomorphism. If $\beta(\mathfrak{a}) = 0$, then $\beta$ factors through $B/\mathfrak{a} = \mathbb{Z}$; there is one such $\beta$, and it is the image of both points $(1,1)$ and $(0,0)$ of $\mathbb{A}^2(k)$. If $\beta(h) \ne 0$ for some $h \in \mathfrak{a}$, then $\beta$ extends uniquely to $B[h^{-1}] = \mathbb{Z}[x,y][h^{-1}]$, so it comes from exactly one point of $\mathbb{A}^2(k)$, and this point is different from $(1,1)$ and $(0,0)$. Hence $\nu$ is bijective on $k$-points, except that it identifies $(1,1)$ with $(0,0)$.

Let $e$ be the restriction of $\nu$ to the three coordinate tori $\{x \ne 0, y \ne 0\}$, $\{x \ne 0, y = 0\}$ and $\{x = 0, y \ne 0\}$ of $\mathbb{A}^2$. Their union is the complement of $(0,0)$. So $e$ is bijective on $k$-points for every field $k$, and $X$ is torified in the sense of [Lorscheid 2012b, Section 3].

The image of the torus $\{x \ne 0, y \ne 0\}$ is not locally closed in $X$. Suppose it is. It contains the generic point, so it is then open. The morphism $\nu$ is finite, hence closed [Stacks, Tag 01WM], and surjective, so a subset of $X$ is open exactly when its preimage in $\mathbb{A}^2$ is open. The preimage of the image of the torus is the union of the open set $D(xy)$ and the closed set $V(x,y)$. Its complement $V(xy) \smallsetminus V(x,y)$ is not closed. So the restriction of $e$ to this torus is not an immersion, and $e$ is not a torification in the sense of Definition 2.6.

Reference: [Lorscheid 2012b, Remarks 3.1 and 5.3] leaves this question open; the example answers it.

Now let $\mathcal{X} = (\tilde X, X, e_X)$ be an $\mathbb{F}_1$-scheme. The rank of a point $x \in \tilde X$ is the rank of $H_x$, and the rank of $\tilde X$ is the smallest rank of its points. Let $\tilde X^{\mathrm{rk}} \;=\; \coprod_{x \text{ of smallest rank}} \operatorname{Spec} (H_x)_0 ,$ a monoid scheme with a discrete space, and let $X^{\mathrm{rk}}$ be the image of $\tilde X^{\mathrm{rk}}_{\mathbb{Z}}$ in $X$. For a torified variety, $\tilde X^{\mathrm{rk}}_{\mathbb{Z}}$ is the union of the tori of smallest dimension. For a split reductive group with the torification of Proposition 3.9, it is the normalizer $N$.

Definition 7.5. Let $\mathcal{X}$ and $\mathcal{Y}$ be $\mathbb{F}_1$-schemes. Consider pairs $(\tilde f, f)$ of a morphism of monoid schemes $\tilde f \colon \tilde X^{\mathrm{rk}} \to \tilde Y^{\mathrm{rk}}$ and a morphism of schemes $f \colon X \to Y$.

The pair is a strong morphism if $f \circ e_X = e_Y \circ \tilde f_{\mathbb{Z}}$ on $\tilde X^{\mathrm{rk}}_{\mathbb{Z}}$.
The pair is a weak morphism if $f$ maps $X^{\mathrm{rk}}$ to $Y^{\mathrm{rk}}$ and, for every point $x$ of $\tilde X^{\mathrm{rk}}$, the morphism $f$ maps the image of the component of $x$ into the image of the component of $\tilde f(x)$.

Reference: [Lorscheid 2012b, Definitions 4.1 and 5.4].

[Lorscheid 2012b, Definition 5.4] states condition 2 through a decomposition of $X^{\mathrm{rk}}$, as a scheme, into the disjoint union of the images of the components of $\tilde X^{\mathrm{rk}}_{\mathbb{Z}}$, and through the map between the sets of components that $f$ then induces. When such a decomposition exists and each image is connected, as for the tori of a torified variety, the two formulations agree. The decomposition exists for split reductive groups, where the images are the cosets $Tn_w$ of the torus in the normalizer $N$, and for the Grassmannians, where they are points. It does not exist in general: for the triangle of lines of Exercise 3, all three tori have rank one, so $X^{\mathrm{rk}}=X$; over $\mathbb{Q}$ the three lines meet pairwise, so $X_{\mathbb{Q}}$ is connected, while the disjoint union of the three tori has three components. Condition 2 as stated here makes sense for every $\mathbb{F}_1$-scheme. The results quoted in Section 7.4 are proved in [Lorscheid 2012b] for its formulation; they concern split reductive groups, their parabolic subgroups and the Grassmannians, where the two formulations agree.

Every strong morphism is weak. A strong morphism asks that $f$ agrees with a morphism of monoid schemes on the part of smallest rank. A weak morphism only asks that $f$ and $\tilde f$ induce the same map between the sets of components of smallest rank. On the rest of $X$, the morphism $f$ is free in both cases. This is what makes room for group laws.

The terminal object is $\ast = (\operatorname{Spec} \mathbb{F}_1, \operatorname{Spec} \mathbb{Z}, \mathrm{id})$, and the set of $\mathbb{F}_1$-points of $\mathcal{Y}$ is the set $\mathcal{Y}(\mathbb{F}_1)$ of strong morphisms $\ast \to \mathcal{Y}$.

Lemma 7.6. The set $\mathcal{Y}(\mathbb{F}_1)$ is the set of points of $\tilde Y^{\mathrm{rk}}$.

Proof. The monoid scheme $\operatorname{Spec} \mathbb{F}_1$ has one point, and it is equal to its part of smallest rank. A morphism $\operatorname{Spec} \mathbb{F}_1 \to \tilde Y^{\mathrm{rk}}$ is a point $y$ and a local morphism $(H_y)_0 \to \{0, 1\}$. There is exactly one: it sends $H_y$ to $1$. For a strong morphism $(\tilde f, f)$ from $\ast$, the morphism $f \colon \operatorname{Spec} \mathbb{Z} \to Y$ is $e_Y \circ \tilde f_{\mathbb{Z}}$, so it is determined by $\tilde f$, and every $\tilde f$ gives the strong morphism $(\tilde f, e_Y \circ \tilde f_{\mathbb{Z}})$. $\blacksquare$

Reference: [Lorscheid 2012b, Lemma 4.3].

So the $\mathbb{F}_1$-points of a torified variety are its tori of smallest dimension, and their number is the leading coefficient of the counting polynomial at $q = 1$: it is $\lim_{q \to 1} N_X(q)/(q-1)^\rho$, where $\rho$ is the smallest dimension of a torus. For the Schubert torification of $\operatorname{Gr}(k,n)$ the $\mathbb{F}_1$-points are the $\binom{n}{k}$ tori of dimension zero, one in each Schubert cell $C_J$. They correspond to the subsets $J$ with $k$ elements of $\{1, \dots, n\}$. For a split reductive group they are the $|W|$ cosets $T n_w$. This is [Lorscheid 2012b, Theorem 4.4].

7.4 What is achieved

The following results of [Lorscheid 2012b] are stated here without proof. An algebraic group over $\mathbb{F}_1$ is a group object $(\mathcal{G}, \mu)$ in the category of $\mathbb{F}_1$-schemes with weak morphisms such that $\mathcal{G}_{\mathbb{Z}}$ is an algebraic group, that is, a group scheme that is a variety. It is a model of a group scheme $G$ if $\mathcal{G}_{\mathbb{Z}} \cong G$ as group schemes, and a canonical model if $\mu$ is a strong morphism [Lorscheid 2012b, Definition 7.1]. The set $\mathcal{G}(\mathbb{F}_1)$ is a group, through the part of smallest rank [Lorscheid 2012b, Lemma 6.1].

Theorem 7.7 ([Lorscheid 2012b, Theorem 7.9]). Let $G$ be a split reductive group scheme over $\mathbb{Z}$ with Weyl group $W = N(\mathbb{Z})/T(\mathbb{Z})$.

$G$ has a model $\mathcal{G} = (\tilde G, G, e_G)$ over $\mathbb{F}_1$, and there is an isomorphism of groups $\mathcal{G}(\mathbb{F}_1) \cong W$ such that the map $\sigma \colon W \cong \mathcal{G}(\mathbb{F}_1) \to G(\mathbb{Z})$ sends every coset $n T(\mathbb{Z})$ to one of its elements.
If $N(\mathbb{Z}) \to W$ has a section $\sigma' \colon W \to N(\mathbb{Z})$ that is a group homomorphism, then $G$ has a canonical model, and the isomorphism can be chosen such that $\sigma = \sigma'$.

Theorem 7.8 ([Lorscheid 2012b, Lemma 8.1 and Theorem 8.2]). The group $\mathrm{GL}_n$ has a canonical model $\mathcal{G}$ over $\mathbb{F}_1$ with $\mathcal{G}(\mathbb{F}_1) \cong S_n$. A parabolic subgroup $P$ of type $(k, n-k)$ has a canonical model $\mathcal{P}$ with $\mathcal{P}(\mathbb{F}_1) \cong S_k \times S_{n-k}$. The action of $\mathcal{P}$ on $\mathcal{G}$ by multiplication has a quotient $\mathcal{Q}$ in the category with strong morphisms, with $\mathcal{Q}_{\mathbb{Z}} \cong \operatorname{Gr}(k,n)$, and $\mathcal{Q}(\mathbb{F}_1)$ is the set of subsets with $k$ elements of $\{1, \dots, n\}$, with the natural action of $S_n = \mathcal{G}(\mathbb{F}_1)$.

Theorem 7.7 does not contradict Proposition 7.1. In the problem of Section 7.1, the $\mathbb{F}_1$-points and the group law live in one category, and this forces $\sigma$ to be a homomorphism. In Theorem 7.7 the $\mathbb{F}_1$-points are defined by strong morphisms and the group law is a weak morphism. For $\mathrm{SL}_2$, the map $\sigma$ of part (1) is a map of sets that is not a homomorphism, by the computation in the proof of Proposition 7.1, and there is no canonical model given by part (2).

The cost is that the theory needs two classes of morphisms, and that a weak morphism forgets the monoid scheme outside the part of smallest rank. Consider instead the category in which a morphism $\mathcal{X} \to \mathcal{Y}$ is a pair $(\tilde f, f)$ of a morphism of monoid schemes $\tilde f \colon \tilde X \to \tilde Y$ on all of $\tilde X$ and a morphism of schemes $f \colon X \to Y$ with $f \circ e_X = e_Y \circ \tilde f_{\mathbb{Z}}$. Composites and products in this category are formed componentwise: the product of $\mathcal{X}$ and $\mathcal{Y}$ is $(\tilde X \times \tilde Y, X \times Y, e_X \times e_Y)$, and the terminal object is $\ast$. So the functor $\mathcal{X} \mapsto \tilde X$ to monoid schemes preserves finite products and the terminal object, and it maps a group object $\mathcal{G} = (\tilde G, G, e_G)$ with group law $(\tilde\mu, \mu)$ to a group object $(\tilde G, \tilde\mu)$ in monoid schemes. Base change preserves finite products and the terminal object as well, so $\tilde\mu_{\mathbb{Z}}$ is a group law on $\tilde G_{\mathbb{Z}}$ that comes from a morphism of monoid schemes. If $\tilde G_{\mathbb{Q}}$ is of finite type over $\mathbb{Q}$, Theorem 4.6 applies: the unit component of $\tilde G_{\mathbb{Q}}$ is diagonalizable. [Lorscheid 2012b, Remark 7.7] expects this: with such morphisms, only semidirect products of diagonalizable groups with finite constant groups seem to have models. The lesson Blueprints and blue schemes describes a category with one class of morphisms in which the group laws of $\mathrm{SL}_n$, $\mathrm{GL}_n$ and further split reductive groups are morphisms, and in which the Weyl group is obtained by a functor.

8. Exercises

Exercise 1 (projective space). For a nonempty subset $S \subseteq \{0, \dots, n\}$, let $T_S \subseteq \mathbb{P}^n$ be the subscheme of points whose nonzero homogeneous coordinates are exactly the $x_j$ with $j \in S$. Show that the $T_S$ form an affine torification of $\mathbb{P}^n$, find the numbers $\delta_l$, and check Proposition 2.8.

Solution. On the chart $U_j = \{x_j \ne 0\} \cong \mathbb{A}^n$, with coordinates $x_i/x_j$, the sets $T_S$ with $j \in S$ are the coordinate tori of Example 1.3 (a). So each $T_S$ is a locally closed subscheme isomorphic to $\mathbb{G}_m^{|S|-1}$, and each $U_j$ is an affine open union of tori. A point with values in a field has a well-defined set $S$ of nonzero coordinates, so the $T_S$ form a decomposition. Hence $\delta_l = \binom{n+1}{l+1}$, and $\sum_{l=0}^{n} \binom{n+1}{l+1} (q-1)^l = \frac{(1 + (q-1))^{n+1} - 1}{q-1} = \frac{q^{n+1} - 1}{q - 1} = N_{\mathbb{P}^n}(q) .$ There are $n+1$ tori of dimension zero. This torification is the one of Proposition 3.1 for the monoid scheme $\mathbb{P}^n$ over $\mathbb{F}_1$.

Exercise 2 (the quadric cone as a monoid scheme). Let $M$ be the monoid with zero generated by $a, b, c$ with the one relation $ab = c^2$, and $X = \operatorname{Spec} M$. (a) Find the points of $X$, their ranks, and the counting polynomial. (b) Show that $X_K$ satisfies (H1) but not (H2) for $d = 2$, and explain this with the proof of Theorem 4.1.

Solution. (a) Every nonzero element of $M$ has a unique form $a^i b^j c^\epsilon$ with $\epsilon \in \{0,1\}$. Sending it to $(2i + \epsilon, 2j + \epsilon)$ identifies $M \smallsetminus \{0\}$ with the monoid $S$ of pairs $(u,v)$ of non-negative integers with $u \equiv v \bmod 2$, inside the group $G \cong \mathbb{Z}^2$ of all pairs of integers with $u \equiv v \bmod 2$. The complement $F$ of a prime ideal is a submonoid such that a product lies in $F$ only if its factors do. If $c \in F$, then $ab = c^2 \in F$, so $a, b \in F$ and $F = S$. If $a, b \in F$, then $c^2 \in F$, so $c \in F$. So the possible $F$ are $\{1\}$, the powers of $a$, the powers of $b$, and $S$. The four primes are $(a,b,c)$, $(b,c)$, $(a,c)$ and $\{0\}$, and the groups $H_x$ are trivial, generated by $a$, generated by $b$, and $G$. The ranks are $0, 1, 1, 2$, and $N(q) = 1 + 2(q-1) + (q-1)^2 = q^2$.

(b) $K[M] = K[a,b,c]/(ab - c^2)$ is a domain of finite type, so (H1) holds. At the closed point, $H = \{1\}$ and $P = S$ has the three atoms $a$, $b$, $c$. By Step 3 of the proof of Theorem 4.1, the space $\mathfrak{n}/\mathfrak{n}^2$ at the vertex has dimension $0 + 3 = 3$. At a point of $\mathbb{A}^2_K$ it has dimension $2$. So (H2) fails: the vertex of the cone is a singular point. Accordingly the stalk at the closed point is not of the form $H \times \mathbb{N}^2$. The base change $X_{\mathbb{Z}}$ is still an affinely torified variety, by Proposition 3.1.

Exercise 3 (a triangle of lines). Let $X \subseteq \mathbb{A}^2$ be the union of the lines $L_1 = \{y = 0\}$, $L_2 = \{x = 0\}$ and $L_3 = \{x + y = 1\}$. (a) Compute $N_X(q)$. (b) Find a torification of $X$. (c) Show that no torification of $X$ contains a torus of dimension zero.

Solution. (a) The lines meet pairwise in the three points $v_{12} = (0,0)$, $v_{13} = (1,0)$, $v_{23} = (0,1)$, which are different over every field. So $N_X(q) = 3q - 3 = 3(q-1)$.

(b) Take $T_1 = L_1 \smallsetminus v_{12}$, $T_2 = L_2 \smallsetminus v_{23}$, $T_3 = L_3 \smallsetminus v_{13}$. Each is a line minus a section, so it is isomorphic to $\mathbb{G}_m$; for example $T_1 = \operatorname{Spec} \mathbb{Z}[x, x^{-1}]$. Each is open in a closed subscheme of $X$, so it is locally closed. The vertex $v_{12}$ lies in $T_2$ only, $v_{23}$ in $T_3$ only, $v_{13}$ in $T_1$ only, and every other point lies on one line. So the three tori form a decomposition.

(c) By Proposition 2.8, the numbers $\delta_l$ are determined by $N_X(q) = 3(q-1)$: $\delta_1 = 3$ and $\delta_0 = 0$. Note that each torus of the torification in (b) is open in its line but not open in $X$, as in Example 2.5, and that $N_X(1) = 0$ although $X$ is not empty.

Exercise 4 (monomial matrices). Let $N \subseteq \mathrm{GL}_n$ be the group scheme of monomial matrices, the matrices $\operatorname{diag}(t_1, \dots, t_n) P_w$ with a permutation matrix $P_w$, where $P_w e_j = e_{w(j)}$. Find a monoid scheme $X$ and a morphism $\mu \colon X \times X \to X$ of monoid schemes such that $(X_{\mathbb{Z}}, \mu_{\mathbb{Z}})$ is $N$ with its group law. Compare with Theorems 4.6 and 5.5.

Solution. Since $P_w \operatorname{diag}(s) P_w^{-1} = \operatorname{diag}(s_{w^{-1}(1)}, \dots, s_{w^{-1}(n)})$, the product is $\operatorname{diag}(t) P_w \cdot \operatorname{diag}(s) P_v = \operatorname{diag}\bigl(t_i\, s_{w^{-1}(i)}\bigr) P_{wv} .$ Let $A$ be the free abelian group on $u_1, \dots, u_n$, and let $X$ be the disjoint union of copies $X_w$ of $\operatorname{Spec} A_0$, one for each $w \in S_n$. Define $\mu$ on $X_w \times X_v$ as the morphism to $X_{wv}$ given by the morphism of monoids $A_0 \to A_0 \wedge A_0$, $u_i \mapsto (u_i, u_{w^{-1}(i)})$. It is local, because it maps units to units and $0$ to $0$. Its base change is the formula above. So $X_{\mathbb{Z}} = N$ and $\mu_{\mathbb{Z}}$ is the group law.

The space $X$ is discrete with $n!$ points, and the unit component is the diagonal torus $D(A)$, as Theorem 4.6 predicts. The components $\operatorname{diag}(t)P_w$ form a torification of $N$ for which the group law is torified. In Theorem 5.5, $E$ is the group of permutation matrices and $N(L) = (L^\times)^n \rtimes S_n$. For the normalizer in $\mathrm{SL}_2$ there is no such structure, by Proposition 7.2: there the unit points of the components cannot be chosen to form a group.

Exercise 5 (the gadget of the cuspidal curve). Let $X = \operatorname{Spec} \mathbb{Z}[t^2, t^3]$ with the torification by the point $t = 0$ and its complement. (a) Show that $\mathcal{L}(X,T)$ is an affine variety over $\mathbb{F}_1$ with extension of scalars $X$. (b) Show that $\mathcal{L}(X,T)$ and the gadget of $\mathbb{A}^1$ with its coordinate torification have the same functor but are not isomorphic. (c) Compare with Example 6.6.

Solution. (a) Put $R = \mathbb{Z}[t^2, t^3] = \mathbb{Z} \oplus t^2\mathbb{Z}[t]$. Then $pR = R \cap p\mathbb{Z}[t]$, so $R/pR$ is a subring of $\mathbb{F}_p[t]$, and it is reduced. Theorem 6.5 applies.

(b) Both functors are $D \mapsto \{\ast\} \sqcup D$. An isomorphism of gadgets contains an isomorphism of complex varieties between $\operatorname{Spec} \mathbb{C}[t^2, t^3]$ and $\mathbb{A}^1_{\mathbb{C}}$. There is none: the ring $\mathbb{C}[t^2,t^3]$ is not integrally closed, because $t$ lies in its fraction field and satisfies $T^2 - t^2 = 0$, while $\mathbb{C}[t]$ is integrally closed.

(c) In Example 6.6 the variety $X = \operatorname{Spec} \mathbb{Z}[pt, t^2, t^3]$ and $\mathbb{A}^1$ have the same complex points, the same tori and the same functor, and the gadget cannot tell them apart. They differ only in the fibre over $p$. Here the two varieties differ over $\mathbb{C}$, and the gadget sees it. A gadget records the complex variety and the points with coordinates in roots of unity. In Example 6.6 the universal property then produces the ring $R^\sharp$ of all functions that are integral on these points.

Exercise 6 (a check of the Bruhat count for $\mathrm{GL}_n$). Show that $\sum_{w \in S_n} q^{\ell(w)} = \prod_{i=1}^{n} [i]_q$, where $\ell(w)$ is the number of inversions of $w$ and $[i]_q = 1 + q + \dots + q^{i-1}$. Deduce that the polynomial of Proposition 3.9 for $\mathrm{GL}_n$, with $r = n$ and $s = \binom{n}{2}$, is the order of $\mathrm{GL}_n(\mathbb{F}_q)$.

Solution. Write a permutation of $\{1, \dots, n\}$ as a word. Removing the letter $n$ gives a permutation of $\{1, \dots, n-1\}$, and the number of inversions drops by the number of letters to the right of $n$, which is any number from $0$ to $n - 1$. So the sum for $n$ is $[n]_q$ times the sum for $n - 1$, and the formula follows by induction. Then $(q-1)^n q^{\binom{n}{2}} \prod_{i=1}^{n} [i]_q = q^{\binom{n}{2}} \prod_{i=1}^{n} (q^i - 1) = \prod_{i=0}^{n-1} q^{i}(q^{n-i} - 1) = \prod_{i=0}^{n-1} (q^n - q^i) ,$ which is the number of bases of $\mathbb{F}_q^n$. For $n = 2$ this is the count of Example 3.10.

It remains to see that the number $\ell(w)$ of Fact 3.7 is the number of inversions. Take the diagonal torus, the upper triangular matrices as $B$, and the permutation matrices $P_w$ with $P_w e_j = e_{w(j)}$ as $n_w$. The root subgroups are the groups of matrices $1 + \xi E_{ij}$ with $i \ne j$. The torus acts on such a matrix by the character $t_i/t_j$, and the positive roots are the $t_i/t_j$ with $i < j$. Since $P_w^{-1} \operatorname{diag}(t_1, \dots, t_n) P_w = \operatorname{diag}(t_{w(1)}, \dots, t_{w(n)})$, the permutation $w$ sends $t_i/t_j$ to $t_{w(i)}/t_{w(j)}$. This root is negative exactly when $w(i) > w(j)$. So $\Phi_w$ is the set of inversions of $w$.

What this lesson does not prove

Facts about schemes, each cited by tag: immersions are monomorphisms [Stacks, Tag 01L7]; morphisms from the spectrum of a field [Stacks, Tag 01J6]; the criterion for separated morphisms [Stacks, Tag 01KP]; reducedness of products over a perfect field [Stacks, Tag 00I4]; the Grassmannian is a scheme [Stacks, Tag 089T]; morphisms of finite presentation over a limit [Stacks, Tag 01ZM]; residue fields of finitely generated rings [Stacks, Tags 00G4 and 00GB]; the complement of an affine open subscheme has codimension one [Stacks, Tag 0BCV]; the theorem of Artin and Tate [Stacks, Tag 00IS]; finite morphisms are closed [Stacks, Tag 01WM].
The Gaussian binomial coefficient as the count of the Grassmannian (Corollary 3.4 (1), second sentence). See Counting over finite fields and the limit $q \to 1$. The proofs of this lesson use only the sum over Schubert cells.
The non-affine comparison theorems: that the extension of scalars of the gadget $\mathcal{L}(X,T)$ of a non-affine, affinely torified variety is $X$ [López Peña–Lorscheid 2011b, Theorem 2.10], the corresponding statement for Soulé's varieties, which uses Soulé's objects and their gluing [López Peña–Lorscheid 2011b, Theorem 3.11], and the comparison diagram of [López Peña–Lorscheid 2011b, Theorem 5.11] between these theories and monoid schemes. Theorems 6.5 and 6.9 show that these statements need the condition that the fibres over the primes are reduced.
Chevalley groups over $\mathbb{F}_{1^2}$: [Connes–Consani 2011a, Theorem 4.10]. See Varieties over the field with one element after Soulé and Connes–Consani.
The results of [Lorscheid 2012b] stated in Section 7.4: Lemma 6.1, Theorem 7.9, Lemma 8.1 and Theorem 8.2 of that work, and the parts of its Theorem 4.4 that rest on Facts 3.6 and 3.8.

References

[López Peña–Lorscheid 2011b] J. López Peña, O. Lorscheid, Torified varieties and their geometries over $\mathbb{F}_1$, arXiv:0903.2173.
[López Peña–Lorscheid 2011a] J. López Peña, O. Lorscheid, Mapping $\mathbb{F}_1$-land: an overview of geometries over the field with one element, arXiv:0909.0069.
[Lorscheid 2012b] O. Lorscheid, Algebraic groups over the field with one element, arXiv:0907.3824.
[Connes–Consani 2011a] A. Connes, C. Consani, On the notion of geometry over $\mathbb{F}_1$, arXiv:0809.2926. Free at https://alainconnes.org/wp-content/uploads/GeomoverF1.pdf
[Connes–Consani 2010] A. Connes, C. Consani, Schemes over $\mathbb{F}_1$ and zeta functions, arXiv:0903.2024. Free at https://alainconnes.org/wp-content/uploads/schemesF1zeta.pdf
[Soulé 2004] C. Soulé, Les variétés sur le corps à un élément, arXiv:math/0304444.
[Stacks] The Stacks project, cited by tag. Each tag links to the same place in AI Integrated Stacks Project, the programme's edition of the Stacks project, which agrees with it modulo corrections made by GPT-6 Astra (OpenAI, Ultra setting) on suggestions of GPT-5.6 Sol (OpenAI, Ultra setting). None of the tags cited in this lesson carries such a correction.
[SGA3] M. Demazure, A. Grothendieck (eds.), Schémas en groupes (SGA 3), Lecture Notes in Mathematics 151–153, Springer. Cited through [Connes–Consani 2011a] and [López Peña–Lorscheid 2011b]. Free at https://webusers.imj-prg.fr/~patrick.polo/SGA3/ (re-edition by P. Gille and P. Polo)