Projective geometry for blueprints

Blueprints are a common generalization of commutative (semi)rings and monoids. The associated geometric objects, blue schemes, are therefore a common generalization of usual scheme theory and ${{\mathbb{F}}_{1}}$–geometry (as considered by Kato [5], Deitmar [3] and Connes-Consani [2]). The possibility of forming semiring schemes allows us to talk about idempotent schemes and tropical schemes (cf. [11]). All this is worked out in [9].

It is known, though not covered in literature yet, that the $\operatorname{Proj}$-construction from usual algebraic geometry has an analogue in ${{\mathbb{F}}_{1}}$-geometry (after Kato, Deitmar and Connes-Consani). In this note we describe a generalization of this to blueprints. In private communication, Koen Thas announced a treatment of $\operatorname{Proj}$ for monoidal schemes (see [13]).

We follow the notations and conventions of [10]. Namely, all blueprints that appear in this note are proper and with a zero. We remark that the following constructions can be carried out for the more general notion of a blueprint as considered in [9]; the reason that we restrict to proper blueprints with a zero is that this allows us to adopt a notation that is common in ${{\mathbb{F}}_{1}}$-geometry.

Namely, we denote by ${\mathbb{A}}^{n}_{B}$ the (blue) affine $n$-space $\operatorname{Spec}\bigl{(}B[T_{1},\dotsc,T_{n}]\bigr{)}$ over a blueprint $B$. In case of a ring, this does not equal the usual affine $n$-space since $B[T_{1},\dotsc,T_{n}]$ is not closed under addition. Therefore, we denote the usual affine $n$-space over a ring $B$ by ${\vphantom{{\mathbb{A}}}}^{+\!}{{\mathbb{A}}}^{n}_{B}=\operatorname{Spec}\bigl{(}B[T_{1},\dotsc,T_{n}]^{+}\bigr{)}$. Similarly, we use a superscript “$+$” for the usual projective space ${\vphantom{{\mathbb{P}}}}^{+}{{\mathbb{P}}}^{n}_{B}$ and the usual Grassmannian $\operatorname{Gr}(k,n)^{+}_{B}$ over a ring $B$.

Let $B$ be a blueprint and $M$ a subset of $B$. We say that $M$ is additively closed in $B$ if for all additive relations $b\equiv\sum a_{i}$ with $a_{i}\in M$ also $b$ is an element of $M$. Note that, in particular, $0$ is an element of $M$. A graded blueprint is a blueprint $B$ together with additively closed subsets $B_{i}$ for $i\in{\mathbb{N}}$ such that $1\in B_{0}$, such that for all $i,j\in{\mathbb{N}}$ and $a\in B_{i}$, $b\in B_{j}$, the product $ab$ is an element of $B_{i+j}$ and such that for every $b\in B$, there are a unique finite subset $I$ of ${\mathbb{N}}$ and unique non-zero elements $a_{i}\in B_{i}$ for every $i\in I$ such that $b\equiv\sum a_{i}$. An element of $\bigcup_{i\geq 0}B_{i}$ is called homogeneous. If $a\in B_{i}$ is non-zero, then we say, more specifically, that $a$ is homogeneous of degree $i$.

We collect some immediate facts for a graded blueprint $B$ as above. The subset $B_{0}$ is multiplicatively closed, i.e. $B_{0}$ can be seen as a subblueprint of $B$. The subblueprint $B_{0}$ equals $B$ if and only if for all $i>0$, $B_{i}=\{0\}$. In this case we say that $B$ is trivially graded. By the uniqueness of the decomposition into homogeneous elements, we have $B_{i}\cap B_{j}=\{0\}$ for $i\neq j$. This means that the union $\bigcup_{i\geq 0}B_{i}$ has the structure of a wedge product $\bigvee_{i\geq 0}B_{i}$. Since $\bigvee_{i\geq 0}B_{i}$ is multiplicatively closed, it can be seen as a subblueprint of $B$. We define $B_{\textup{hom}}=\bigvee_{i\geq 0}B_{i}$ and call the subblueprint $B_{\textup{hom}}$ the homogeneous part of $B$.

Let $S$ be a multiplicative subset of $B$. If $b/s$ is an element of the localization $S^{-1}B$ where $f$ is homogeneous of degree $i$ and $s$ is homogeneous of degree $j$, then we say that $b/s$ is a homogeneous element of degree $i-j$. We define $S^{-1}B_{0}$ as the subset of homogeneous elements of degree $0$. It is multiplicatively closed, and inherits thus a subblueprint structure from $S^{-1}B$. If $S$ is the complement of a prime ideal ${\mathfrak{p}}$, then we write $B_{({\mathfrak{p}})}$ for the subblueprint $(B_{\mathfrak{p}})_{0}$ of homogeneous elements of degree $0$ in $B_{\mathfrak{p}}$.

An ideal $I$ of a graded blueprint $B$ is called homogeneous if it is generated by homogeneous elements, i.e. if for every $c\in I$, there are homogeneous elements $p_{i},q_{j}\in I$ and elements $a_{i},b_{j}\in B$ and an additive relation $\sum a_{i}p_{i}+c\equiv\sum b_{j}q_{j}$ in $B$.

Let $B$ be a graded blueprint. Then we define $\operatorname{Proj}B$ as the set of all homogeneous prime ideals ${\mathfrak{p}}$ of $B$ that do not contain $B_{\textup{hom}}^{+}=\bigvee_{i>0}B_{i}$. The set $X=\operatorname{Proj}B$ comes together with the topology that is defined by the basis

where $h$ ranges through $B_{\textup{hom}}$ and with a structure sheaf ${\mathcal{O}}_{X}$ that is the sheafification of the association $U_{h}\mapsto B[h^{-1}]_{0}$ where $B[h^{-1}]$ is the localization of $B$ at $S=\{h^{i}\}_{i\geq q0}$.

Note that if $B$ is a ring, the above definitions yield the usual construction of $\operatorname{Proj}B$ for graded rings. In complete analogy to the case of graded rings, one proves the following theorem.

If $B$ is a graded blueprint, then the associated semiring $B^{+}$ inherits a grading. Namely, let $B_{\textup{hom}}=\bigvee_{i\geq 0}B_{i}$ the homogeneous part of $B$. Then we can define $B_{i}^{+}$ as the additive closure of $B_{i}$ in $B^{+}$, i.e. as the set of all $b\in B$ such that there is an additive relation of the form $b\equiv\sum a_{k}$ in $B$ with $a_{k}\in B_{i}$. Then $\bigvee B_{i}^{+}$ defines a grading of $B^{+}$. Similarly, the grading of $B$ induces a grading on a tensor product $B\otimes_{C}D$ with respect to blueprint morphisms $C\to B$ and $C\to D$ under the assumption that the image of $C\to B$ is contained in $B_{0}$. Consequently, a grading of $B$ implies a grading of $B_{\textup{inv}}=B\otimes_{{\mathbb{F}}_{1}}{{\mathbb{F}}_{1^{2}}}$ and of the ring $B^{+}_{\mathbb{Z}}=B_{\textup{inv}}^{+}$. Along the same lines, if both $B$ and $D$ are graded and the images of $C\to B$ and $C\to D$ lie in $B_{0}$ and $C_{0}$ respectively, then $B\otimes_{C}D$ inherits a grading obtained from the gradings of $B$ and $D$.

The functor $\operatorname{Proj}$ allows the definition of the projective space ${\mathbb{P}}^{n}_{B}$ over a blueprint $B$. Namely, the free blueprint $C=B[T_{0},\dotsc,T_{n}]$ over $B$ comes together with a natural grading (cf. [9, Section 1.12] for the definition of free blueprints). Namely, $C_{i}$ consists of all monomials $bT_{0}^{e_{0}}\dotsb T_{n}^{e_{n}}$ such that $e_{0}+\dotsb+e_{n}=i$ where $b\in B$. Note that $C_{0}=B$ and $C_{\textup{hom}}=C$. The projective space ${\mathbb{P}}^{n}_{B}$ is defined as $\operatorname{Proj}B[T_{0},\dotsc,T_{n}]$. It comes together with a structure morphism ${\mathbb{P}}^{n}_{B}\to\operatorname{Spec}B$.

In case of $B={{\mathbb{F}}_{1}}$, the projective space ${\mathbb{P}}^{n}_{{\mathbb{F}}_{1}}$ is the monoidal scheme that is known from ${{\mathbb{F}}_{1}}$-geometry (see [4], [1, Section 3.1.4]) and [10, Ex. 1.6]). The topological space of ${\mathbb{P}}^{n}_{{\mathbb{F}}_{1}}$ is finite. Its points correspond to the homogeneous prime ideals $(S_{i})_{i\in I}$ of ${{\mathbb{F}}_{1}}[S_{0},\dotsc,S_{n}]$ where $I$ ranges through all proper subsets of $\{0,\dotsc,n\}$.

In case of a ring $B$, the projective space ${\mathbb{P}}^{n}_{B}$ does not coincide with the usual projective space since the free blueprint $B[S_{0},\dotsc,S_{n}]$ is not a ring, but merely the blueprint of all monomials of the form $bS_{0}^{e_{0}}\dotsb S_{n}^{e_{n}}$ with $b\in B$. However, the associated scheme ${\vphantom{{\mathbb{P}}}}^{+}{{\mathbb{P}}}^{n}_{B}=({\mathbb{P}}^{n}_{B})^{+}$ coincides with the usual projective space over $B$, which equals $\operatorname{Proj}B[S_{0},\dotsc,S_{n}]^{+}$.

Let ${\mathcal{X}}$ be a scheme of finite type. By an ${{\mathbb{F}}_{1}}$-model of ${\mathcal{X}}$ we mean a blue scheme $X$ of finite type such that $X_{\mathbb{Z}}^{+}$ is isomorphic to ${\mathcal{X}}$. Since a finitely generated ${\mathbb{Z}}$-algebra is, by definition, generated by a finitely generated multiplicative subset as a ${\mathbb{Z}}$-module, every scheme of finite type has an ${{\mathbb{F}}_{1}}$-model. It is, on the contrary, true that a scheme of finite type possesses a large number of ${{\mathbb{F}}_{1}}$-models.

Given a scheme ${\mathcal{X}}$ with an ${{\mathbb{F}}_{1}}$-model $X$, we can associate to every closed subscheme ${\mathcal{Y}}$ of ${\mathcal{X}}$ the following closed subscheme $Y$ of $X$, which is an ${{\mathbb{F}}_{1}}$-model of ${\mathcal{Y}}$. In case that $X=\operatorname{Spec}B$ is the spectrum of a blueprint $B=A\!\sslash\!{\mathcal{R}}$, and thus ${\mathcal{X}}\simeq\operatorname{Spec}B_{\mathbb{Z}}^{+}$ is an affine scheme, we can define $Y$ as $\operatorname{Spec}C$ for $C=A\!\sslash\!{\mathcal{R}}(Y)$ where ${\mathcal{R}}(Y)$ is the pre-addition that contains $\sum a_{i}\equiv\sum b_{j}$ whenever $\sum a_{i}=\sum b_{j}$ holds in the coordinate ring $\Gamma{\mathcal{Y}}$ of ${\mathcal{Y}}$. This is a process that we used already in [10, Section 3].

Since localizations commute with additive closures, i.e. $(S^{-1}B)^{+}_{\mathbb{Z}}=S^{-1}(B^{+}_{\mathbb{Z}})$ where $S$ is a multiplicative subset of $B$, the above process is compatible with the restriction to affine opens $U\subset X$. This means that given $U=\operatorname{Spec}(S^{-1}B)$, which is an ${{\mathbb{F}}_{1}}$-model for ${\mathcal{X}}^{\prime}=U_{\mathbb{Z}}^{+}$, then the ${{\mathbb{F}}_{1}}$–model $Y^{\prime}$ that is associated to the closed subscheme ${\mathcal{Y}}^{\prime}={\mathcal{X}}^{\prime}\times_{\mathcal{X}}{\mathcal{Y}}$ of ${\mathcal{X}}^{\prime}$ by the above process is the spectrum of the blueprint $S^{-1}C$. Consequently, we can associate with every closed subscheme ${\mathcal{Y}}$ of a scheme ${\mathcal{X}}$ with an ${{\mathbb{F}}_{1}}$-model $X$ a closed subscheme $Y$ of $X$, which is an ${{\mathbb{F}}_{1}}$–model of ${\mathcal{Y}}$; namely, we apply the above process to all affine open subschemes of ${\mathcal{X}}$ and glue them together, which is possible since additive closures commute with localizations.

In case of a projective variety, i.e. a closed subscheme ${\mathcal{Y}}$ of a projective space ${\vphantom{{\mathbb{P}}}}^{+}{{\mathbb{P}}}^{n}_{\mathbb{Z}}$, we derive the following description of the associated ${{\mathbb{F}}_{1}}$-model $Y$ in ${\mathbb{P}}^{n}_{{\mathbb{F}}_{1}}$ by homogeneous coordinate rings. Let $C$ be the homogeneous coordinate ring of ${\mathcal{Y}}$, which is a quotient of ${\mathbb{Z}}[S_{0},\dotsc,S_{n}]^{+}$ by a homogeneous ideal $I$. Let ${\mathcal{R}}$ be the pre-addition on ${{\mathbb{F}}_{1}}[S_{0},\dotsc,S_{n}]$ that consists of all relations $\sum a_{i}\equiv\sum b_{j}$ such that $\sum a_{i}=\sum b_{j}$ in $C$. Then $B={{\mathbb{F}}_{1}}[S_{0},\dotsc,S_{n}]\!\sslash\!{\mathcal{R}}$ inherits a grading from ${{\mathbb{F}}_{1}}[S_{0},\dotsc,S_{n}]$ by defining $B_{i}$ as the image of ${{\mathbb{F}}_{1}}[S_{0},\dotsc,S_{n}]_{i}$ in $B$. Note that $B\subset C$ and that the sets $B_{i}$ equal the intersections $B_{i}=C_{i}\cap B$ for $i\geq 0$ where $C_{i}$ is the homogeneous part of degree $i$ of $C$. Then the ${{\mathbb{F}}_{1}}$-model $Y$ of ${\mathcal{Y}}$ equals $\operatorname{Proj}B$.

One of the simplest examples of projective varieties that is not a toric variety (and in particular, not a projective space) is the Grassmann variety $\operatorname{Gr}(2,4)$. The problem of finding models over ${{\mathbb{F}}_{1}}$ for Grassmann varieties was originally posed by Soulè in [12], and solved by the authors by obtaining a torification from the Schubert cell decomposition (cf. [8, 7]).

In this note, we present ${{\mathbb{F}}_{1}}$-models for Grassmannians as projective varieties defined through (homogeneous) blueprints. The proposed construction for the Grassmannians fits within a more general framework for obtaining blueprints and totally positive blueprints from cluster data (cf. the forthcoming preprint [6]).

Classically, the homogeneous coordinate ring for the Grassmannian $\operatorname{Gr}(k,n)$ is obtained by quotienting out the homogeneous coordinate ring of the projective space $\mathbb{P}^{\binom{n}{k}-1}$ by the homogeneous ideal generated by the Plücker relations. A similar construction can be carried out using the framework of (graded) blueprints. In what follows, we make that construction explicit for the Grassmannian $\operatorname{Gr}(2,4)$.

Define the blueprint $\mathcal{O}_{{{\mathbb{F}}_{1}}}(\operatorname{Gr}(2,4))={{\mathbb{F}}_{1}}[x_{12},x_{13},x_{14},x_{23},x_{24},x_{34}]\!\sslash\!{\mathcal{R}}$ where the congruence ${\mathcal{R}}$ is generated by the Plücker relation $x_{12}x_{34}+x_{14}x_{23}\equiv x_{13}x_{24}$ (the signs have been picked to ensure that the totally positive part of the Grassmannian is preserved, cf. [6]). Since ${\mathcal{R}}$ is generated by a homogeneous relation, $\mathcal{O}_{{{\mathbb{F}}_{1}}}(\operatorname{Gr}(2,4))$ inherits a grading from the canonical morphism

Let $\operatorname{Gr}(2,4)_{{\mathbb{F}}_{1}}:=\operatorname{Proj}(\mathcal{O}_{{{\mathbb{F}}_{1}}}(\operatorname{Gr}(2,4)))$. The base extension $\operatorname{Gr}(2,4)^{+}_{\mathbb{Z}}$ is the usual Grassmannian, and $\pi$ defines a closed embedding of $\operatorname{Gr}(2,4)_{{\mathbb{F}}_{1}}$ into $\mathbb{P}^{5}_{{\mathbb{F}}_{1}}$, which extends to the classical Plücker embedding $\operatorname{Gr}(2,4)_{\mathbb{Z}}^{+}\hookrightarrow{\vphantom{{\mathbb{P}}}}^{+}{{\mathbb{P}}}_{\mathbb{Z}}^{5}$.

Homogeneous prime ideals in $\mathcal{O}_{{\mathbb{F}}_{1}}(Gr(2,4))$ are described by their generators as the proper subsets $I\subsetneq\{x_{12},x_{13},x_{14},x_{23},x_{24},x_{25}\}$ such that $I$ is either contained in one of the sets $\{x_{12},x_{34}\}$, $\{x_{14},x_{23}\}$, $\{x_{13},x_{24}\}$, or otherwise $I$ has a nonempty intersection with all three of them. In other words, $I$ cannot contain elements in two of the above sets without also containing an element of the third one.

The structure of the set of (homogeneous) prime ideals of $\mathcal{O}_{{\mathbb{F}}_{1}}(\operatorname{Gr}(2,4))$ is depicted in Figure 1. It consists of $6+12+11+6+1=36$ prime ideals of ranks $0$, $1$, $2$, $3$ and $4$, respectively (cf. [10, Def. 2.3] for the definition of the rank of a prime ideal), thus resulting in a model essentially different to the one presented in [8] by means of torifications, which had $6+12+11+5+1=35$ points, in correspondence with the coefficients of the counting polynomial $N_{\operatorname{Gr}(2,4)}(q)=6+12(q-1)+11(q-1)^{2}+5(q-1)^{3}+1(q-1)^{4}$. It is worth noting that despite arising from different constructions, both ${{\mathbb{F}}_{1}}$-models for $\operatorname{Gr}(2,4)$ have $6=\binom{4}{2}$ closed points, corresponding to the combinatorial interpretation of $\operatorname{Gr}(2,4)_{{\mathbb{F}}_{1}}$ as the set of all subsets with two elements inside a set with four elements. These six points correspond to the ${{\mathbb{F}}_{1}}$-rational Tits points of $\operatorname{Gr}(2,4)_{{\mathbb{F}}_{1}}$, which reflect the naive notion of ${{\mathbb{F}}_{1}}$-rational points of an ${{\mathbb{F}}_{1}}$-scheme (cf. [10, Section 2.2]).

Like in the classical geometrical setting, the Grassmannian $\operatorname{Gr}(2,4)_{{{\mathbb{F}}_{1}}}$ does admit a covering by six ${{\mathbb{F}}_{1}}$-models of affine $4$-space, which correspond to the open subsets of $\operatorname{Gr}(2,4)_{{{\mathbb{F}}_{1}}}$ where one of $x_{12}$, $x_{34}$, $x_{14}$, $x_{23}$, $x_{13}$ or $x_{24}$ is non-zero. However, these ${{\mathbb{F}}_{1}}$-models of affine $4$-space are not the standard model ${\mathbb{A}}^{4}_{{\mathbb{F}}_{1}}=\operatorname{Spec}\bigl{(}{{\mathbb{F}}_{1}}[a,b,c,d]\bigr{)}$, but the “$2\times 2$-matrices” $M_{2,{{\mathbb{F}}_{1}}}=\operatorname{Spec}\bigl{(}{{\mathbb{F}}_{1}}[a,b,c,d]\!\sslash\!\langle ad\equiv bc+D\rangle\bigr{)}$ in case that one of $x_{12}$, $x_{34}$, $x_{14}$ or $x_{23}$ is non-zero, and the “twisted $2\times 2$-matrices” $M_{2,{{\mathbb{F}}_{1}}}^{\tau}=\operatorname{Spec}\bigl{(}{{\mathbb{F}}_{1}}[a,b,c,d]\!\sslash\!\langle ad+bc\equiv D\rangle\bigr{)}$ in case that one of $x_{13}$ or $x_{24}$ is non-zero.