In this note, we generalize the Proj-construction from usual schemes to blue
schemes. This yields the definition of projective space and projective
varieties over a blueprint. In particular, it is possible to descend closed
subvarieties of a projective space to a canonical F_1-model. We discuss this
explicitly in case of the Grassmannian Gr(2,4).Comment: 5 pages, 1 figur

Lorscheid, Oliver

Peña, Javier López

English

arXiv

López Peña, Javier

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 350 (2012) 455–458Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comAlgebra/Algebraic GeometryProjective geometry for blueprintsGeometrie projective pour les canevas bleusJavier López Peña a,1, Oliver Lorscheid ba Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, United Kingdomb Department of Mathematics, University of Wuppertal, Gaußstr. 20, 42097 Wuppertal, Germanya r t i c l e i n f o a b s t r a c tArticle history:Received 8 March 2012Accepted after revision 27 April 2012Available online 17 May 2012Presented by Christophe SouléIn this Note, we generalize the Proj-construction from usual schemes to blue schemes.This yields the definition of projective space and projective varieties over a blueprint. Inparticular, it is possible to descend closed subvarieties of a projective space to a canonicalF1-model. We discuss this in case of the Grassmannian Gr(2,4).© 2012 Published by Elsevier Masson SAS on behalf of Académie des sciences.r é s u m éDans cette Note, nous généralisons la Proj-construction des schémas usuels aux schémasbleus. Cela entraine la définition d’espace projectif et de variétés projectives sur uncanevas bleu. En particulier, il est possible de descendre une sous-variété fermée d’unespace projectif en un F1-modèle canonique. Nous discutons cela dans le cas de laGrassmannienne Gr(2,4).© 2012 Published by Elsevier Masson SAS on behalf of Académie des sciences.1. IntroductionBlueprints 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 F1-geometry (as considered by Kato[5], Deitmar [3] and Connes–Consani [2]). The possibility of forming semiring schemes allows us to talk about idempotentschemes and tropical schemes (cf. [11]). All this is worked out in [9].It is known, though not covered in literature yet, that the Proj-construction from usual algebraic geometry has ananalogue in F1-geometry (after Kato, Deitmar and Connes–Consani). In this note we describe a generalization of this toblueprints. Privately, Koen Thas has announced a treatment of 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 azero. We remark that the following constructions can be carried out for the more general notion of a blueprint as consideredin [9]; the reason that we restrict to proper blueprints with a zero is that this allows us to adopt a notation that is commonin F1-geometry.Namely, we denote by AnB the (blue) affine n-space Spec(B[T1, . . . , Tn]) over a blueprint B . In case of a ring, this doesnot equal the usual affine n-space since B[T1, . . . , Tn] is not closed under addition. Therefore, we denote the usual affineE-mail addresses: jlp@math.ucl.ac.uk (J. López Peña), lorscheid@math.uni-wuppertal.de (O. Lorscheid).1 J. López Peña’s research was supported by MCIM grant MTM2010-20940-C02-01, research group FQM-266 (Junta de Andalucía) and Max-Planck Institutefor Mathematics in Bonn.1631-073X/$ – see front matter © 2012 Published by Elsevier Masson SAS on behalf of Académie des sciences.http://dx.doi.org/10.1016/j.crma.2012.05.001456 J. López Peña, O. Lorscheid / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 455–458n-space over a ring B by +AnB = Spec(B[T1, . . . , Tn]+). Similarly, we use a superscript “+” for the usual projective space +PnBand the usual Grassmannian Gr(k,n)+B over a ring B .2. Graded blueprints and ProjLet 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 ≡ ∑ai withai ∈ 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 togetherwith additively closed subsets Bi for i ∈N such that 1 ∈ B0, such that for all i, j ∈ N and a ∈ Bi , b ∈ B j , the product ab is anelement of Bi+ j and such that for every b ∈ B , there are a unique finite subset I of N and unique non-zero elements ai ∈ Bifor every i ∈ I such that b = ∑ai . An element of ⋃i0 Bi is called homogeneous. If a ∈ Bi is non-zero, then we say, morespecifically, that a is homogeneous of degree i.We collect some immediate facts for a graded blueprint B as above. The subset B0 is multiplicatively closed, i.e. B0 canbe seen as a subblueprint of B . The subblueprint B0 equals B if and only if for all i > 0, Bi = {0}. In this case we say thatB is trivially graded. By the uniqueness of the decomposition into homogeneous elements, we have Bi ∩ B j = {0} for i = j.This means that the union⋃i0 Bi has the structure of a wedge product∨i0 Bi . Since∨i0 Bi is multiplicatively closed,it can be seen as a subblueprint of B . We define Bhom = ∨i0 Bi and call the subblueprint Bhom 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 iand s is homogeneous of degree j, then we say that b/s is a homogeneous element of degree i − j. We define S−1 B0 as thesubset of homogeneous elements of degree 0. It is multiplicatively closed, and inherits thus a subblueprint structure fromS−1 B . If S is the complement of a prime ideal p, then we write B(p) for the subblueprint (Bp)0 of homogeneous elementsof degree 0 in Bp.An ideal I of a graded blueprint B is called homogeneous if it is generated by homogeneous elements, i.e. if for everyc ∈ I , there are homogeneous elements pi,q j ∈ I and elements ai,b j ∈ B and an additive relation ∑ai pi + c = ∑b jq j in B .Let B be a graded blueprint. Then we define Proj B as the set of all homogeneous prime ideals p of B that do not containB+hom =∨i>0 Bi . The set X = Proj B comes together with the topology that is defined by the basisUh = {p ∈ X | h /∈ p}where h ranges through Bhom and with a structure sheaf OX that is the sheafification of the association Uh → B[h−1]0where B[h−1] is the localization of B at S = {hi}i0.Note that if B is a ring, the above definitions yield the usual construction of Proj B for graded rings. In complete analogyto the case of graded rings, one proves the following theorem:Theorem 1. The space X = Proj B together with OX is a blue scheme. The stalk at a point p ∈ Proj B is Ox,p = B(p) . If h ∈ B+hom , thenUh  Spec B[h−1]0 . The inclusions B0 ↪→ B[h−1]0 yield morphisms Spec B[h−1]0 → Spec B0 , which glue to a structural morphismProj B → Spec B0 . If B is a graded blueprint, then the associated semiring B+ inherits a grading. Namely, let Bhom = ∨i0 Bi the homo-geneous part of B . Then we can define B+i as the additive closure of Bi in B+ , i.e. as the set of all b ∈ B such that thereis an additive relation of the form b ≡ ∑ak in B with ak ∈ Bi . Then ∨ B+i defines a grading of B+ . Similarly, the gradingof B induces a grading on a tensor product B ⊗C D with respect to blueprint morphisms C → B and C → D under theassumption that the image of C → B is contained in B0. Consequently, a grading of B implies a grading of B inv = B ⊗F1 F12(see [9, Lemma 1.4] and [10, p. 11]) and of the ring B+Z= B+inv. Analogously, if both B and D are graded and the image of Clies in both B0 and D0, then B ⊗C D inherits a grading from the gradings of B and D .3. Projective spaceThe functor Proj allows the definition of the projective space PnB over a blueprint B . Namely, the free blueprint C =B[T0, . . . , Tn] over B comes together with a natural grading (cf. [9, Section 1.12] for the definition of free blueprints).Namely, Ci consists of all monomials bTe00 · · · T enn such that e0 + · · · + en = i where b ∈ B . Note that C0 = B and Chom = C .The projective space PnB is defined as Proj B[T0, . . . , Tn]. It comes together with a structure morphism PnB → Spec B .In case of B = F1, the projective space PnF1 is the monoidal scheme that is known from F1-geometry (see [4], [1, Sec-tion 3.1.4]) and [10, Ex. 1.6]). The topological space of PnF1is finite. Its points correspond to the homogeneous prime ideals(Si)i∈I of F1[S0, . . . , Sn] where I ranges through all proper subsets of {0, . . . ,n}.In case of a ring B , the projective space PnB does not coincide with the usual projective space since the free blueprintB[S0, . . . , Sn] is not a ring, but merely the blueprint of all monomials of the form bSe00 · · · Senn with b ∈ B . However, theassociated scheme +PnB = (PnB)+ coincides with the usual projective space over B , which equals Proj B[S0, . . . , Sn]+ .4. Closed subschemesLet X be a scheme of finite type. By an F1-model of X we mean a blue scheme X of finite type such that X+Z isisomorphic to X . Since a finitely generated Z-algebra is, by definition, generated by a finitely generated multiplicativeJ. López Peña, O. Lorscheid / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 455–458 457Fig. 1. Points of the Grassmannian Gr(2,4)F1 . Generator xij belonging to an ideal is depicted as segment i– j in .subset as a Z-module, every scheme of finite type has an F1-model. It is, on the contrary, true that a scheme of finite typepossesses a large number of F1-models.Given a scheme X with an F1-model X , we can associate to every closed subscheme Y of X the following closedsubscheme Y of X , which is an F1-model of Y . In case that X = Spec B is the spectrum of a blueprint B = A//R, and thusX  Spec B+Zis an affine scheme, we can define Y as Spec C for C = A//R(Y ) where R(Y ) is the pre-addition that contains∑ai ≡ ∑b j whenever ∑ai = ∑b j holds in the coordinate ring Y of Y . This is a process that we used already in [10,Section 3].Since localizations commute with additive closures, i.e. (S−1 B)+Z= S−1(B+Z) where S is a multiplicative subset of B , theabove process is compatible with the restriction to affine opens U ⊂ X . This means that given U = Spec(S−1 B), which is anF1-model for X ′ = U+Z , then the F1-model Y ′ that is associated to the closed subscheme Y ′ =X ′ ×X Y of X ′ by the aboveprocess is the spectrum of the blueprint S−1C . Consequently, we can associate with every closed subscheme Y of a schemeX with an F1-model X a closed subscheme Y of X , which is an F1-model of Y ; namely, we apply the above process to allaffine open subschemes of 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 Y of a projective space +PnZ, we derive the following descriptionof the associated F1-model Y in PnF1 by homogeneous coordinate rings. Let C be the homogeneous coordinate ring of Y ,which is a quotient of Z[S0, . . . , Sn]+ by a homogeneous ideal I . Let R be the pre-addition on F1[S0, . . . , Sn] that consistsof all relations∑ai ≡ ∑b j such that ∑ai = ∑b j in C . Then B = F1[S0, . . . , Sn]//R inherits a grading from F1[S0, . . . , Sn]by defining Bi as the image of F1[S0, . . . , Sn]i in B . Note that B ⊂ C and that the sets Bi equal the intersections Bi = Ci ∩ Bfor i  0 where Ci is the homogeneous part of degree i of C . Then the F1-model Y of Y equals Proj B .5. F1-models for GrassmanniansOne of the simplest examples of projective varieties that is not toric is the Grassmannian Gr(2,4). The problem of findingF1-models for Grassmannians was originally posed by Soulé in [12], and solved by the authors by obtaining a torificationfrom the Schubert cell decomposition (cf. [8,7]). In this note, we present F1-models for Grassmannians as projective varietiesdefined through (homogeneous) blueprints. The proposed construction fits within a more general framework for obtainingblueprints and totally positive blueprints from cluster data (cf. [6]).Classically, the coordinate ring for Gr(k,n) is obtained by quotienting out the homogeneous coordinate ring of the pro-jective space P(nk)−1 by the homogeneous ideal generated by the Plücker relations. A similar construction can be carried outusing the framework of (graded) blueprints. We make that construction explicit for Gr(2,4).Define the blueprint OF1 (Gr(2,4)) = F1[x12, x13, x14, x23, x24, x34]//R where the congruence R is generated by thePlücker relation x12x34 + x14x23 ≡ x13x24 (the signs have been picked to ensure that the totally positive part of the Grass-mannian is preserved, cf. [6]). Since R is generated by a homogeneous relation, OF1 (Gr(2,4)) inherits a grading from thecanonical morphismπ : F1[x12, x13, x14, x23, x24, x34] −→ F1[x12, x13, x14, x23, x24, x34]//R.Let Gr(2,4)F1 := Proj(OF1 (Gr(2,4))). The base extension Gr(2,4)+Z is the usual Grassmannian, and π defines a closed em-bedding of Gr(2,4)F1 into P5F1, which extends to the classical Plücker embedding Gr(2,4)+Z↪→+ P5Z.Homogeneous prime ideals in OF1 (Gr(2,4)) are described by their generators as the proper subsets I  {x12, x13, x14, x23,x24, x25} such that I is either contained in one of the sets {x12, x34}, {x14, x23}, {x13, x24}, or otherwise I has a non-emptyintersection with all three of them. In other words, I cannot contain elements in two of the above sets without also contain-ing an element of the third one. Gr(2,4)F1 is depicted in Fig. 1. It consists of 6 + 12 + 11 + 6 + 1 = 36 prime ideals of ranks458 J. López Peña, O. Lorscheid / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 455–4580, 1, 2, 3 and 4, respectively (cf. [10, Def. 2.3] for the definition of rank), thus resulting in a model essentially different tothe one presented in [8], which had 35 points corresponding to the coefficients of NGr(2,4)(q) = 6 + 12(q − 1)+ 11(q − 1)2 +5(q − 1)3 + 1(q − 1)4. In spite of arising from different constructions, both F1-models for Gr(2,4) have 6 =(42)closed points,supporting the naive combinatorial interpretation of Gr(2,4)F1 . These six points correspond to the F1-rational Tits points ofGr(2,4)F1 , which reflect the naive notion of F1-rational points of an F1-scheme (cf. [10, Section 2.2]).As in the classical setting, the Grassmannian Gr(2,4)F1 does admit a covering by six F1-models of affine 4-space, whichcorrespond to the open subsets of Gr(2,4)F1 where one of the generators is non-zero. However, these F1-models of affine4-space are not the standard model A4F1= Spec(F1[a,b, c,d]), but the “2 × 2-matrices” M2,F1 = Spec(F1[a,b, c,d, D]//〈ad ≡ bc + D〉) in case that one of x12, x34, x14 or x23 is non-zero, and the “twisted 2 × 2-matrices” Mτ2,F1 =Spec(F1[a,b, c,d, D]//〈ad + bc ≡ D〉) in case that one of x13 or x24 is non-zero.References[1] C. Chu, O. Lorscheid, R. Santhanam, Sheaves and K -theory for F1-schemes, Adv. Math. 229 (4) (2012) 2239–2286.[2] A. Connes, C. Consani, Characteristic 1, entropy and the absolute point, preprint, arXiv:0911.3537v1, 2009.[3] A. Deitmar, Schemes over F1, in: Number Fields and Function Fields—Two Parallel Worlds, in: Progr. Math., vol. 239, Birkhäuser Boston, Boston, MA,2005, pp. 87–100.[4] A. Deitmar, F1-schemes and toric varieties, Beiträge Algebra Geom. 49 (2) (2008) 517–525.[5] K. Kato, Toric singularities, Amer. J. Math. 116 (5) (1994) 1073–1099.[6] J. López Peña, F1-models for cluster algebras and total positivity, in preparation.[7] J. López Peña, O. Lorscheid, Mapping F1-land an overview of geometries over the field with one element, in: Noncommutative Geometry, Arithmeticand Related Topics, Johns Hopkins University Press, 2011, pp. 241–265.[8] J. López Peña, O. Lorscheid, Torified varieties and their geometries over F1, Math. Z. 267 (3–4) (2011) 605–643.[9] O. Lorscheid, The geometry of blueprints. Part I: Algebraic background and scheme theory, Adv. Math. 229 (3) (2012) 1804–1846.[10] O. Lorscheid, The geometry of blueprints. Part II: Tits–Weyl models of algebraic groups, preprint, arXiv:1201.1324, 2012.[11] G. Mikhalkin, Tropical geometry, unpublished notes, 2010.[12] C. Soulé, Les variétés sur le corps à un élément, Mosc. Math. J. 4 (1) (2004) 217–244, 312.[13] K. Thas, Notes on F1, I. Combinatorics of D0-schemes and F1-geometry, in preparation.

Projective geometry for blueprints

Javier López Peña

Oliver Lorscheid

Crossref

Comptes Rendus Mathematique

López Peña, J.

Lorscheid, O. 

MPG.PuRe

PROJECTIVE GEOMETRY FOR BLUEPRINTSJAVIER LO´PEZ PEN˜A AND OLIVER LORSCHEIDABSTRACT. In this note, we generalize the Proj–construction from usual schemes toblue schemes. This yields the definition of projective space and projective varieties over ablueprint. In particular, it is possible to descend closed subvarieties of a projective spaceto a canonical F1–model. We discuss this in case of the Grassmannian Gr(2, 4).1. INTRODUCTIONBlueprints are a common generalization of commutative (semi)rings and monoids. Theassociated geometric objects, blue schemes, are therefore a common generalization ofusual scheme theory and F1–geometry (as considered by Kato [5], Deitmar [3] and Connes-Consani [2]). The possibility of forming semiring schemes allows us to talk about idem-potent schemes and tropical schemes (cf. [11]). All this is worked out in [9].It is known, though not covered in literature yet, that the Proj-construction from usualalgebraic geometry has an analogue in F1-geometry (after Kato, Deitmar and Connes-Consani). In this note we describe a generalization of this to blueprints. In private commu-nication, Koen Thas announced a treatment of Proj for monoidal schemes (see [13]).We follow the notations and conventions of [10]. Namely, all blueprints that appear inthis note are proper and with a zero. We remark that the following constructions can becarried out for the more general notion of a blueprint as considered in [9]; the reason thatwe restrict to proper blueprints with a zero is that this allows us to adopt a notation that iscommon in F1-geometry.Namely, we denote by AnB the (blue) affine n-space Spec(B[T1, . . . , Tn])over a blue-print B. In case of a ring, this does not equal the usual affine n-space since B[T1, . . . , Tn]is not closed under addition. Therefore, we denote the usual affine n-space over a ringB by +AnB = Spec(B[T1, . . . , Tn]+). Similarly, we use a superscript “+” for the usualprojective space +PnB and the usual Grassmannian Gr(k, n)+B over a ring B.2. GRADED BLUEPRINTS AND ProjLet B be a blueprint and M a subset of B. We say that M is additively closed in Bif for all additive relations b ≡ ∑ ai with ai ∈ 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 withadditively closed subsets Bi for i ∈ N such that 1 ∈ B0, such that for all i, j ∈ N anda ∈ Bi, b ∈ Bj , the product ab is an element of Bi+j and such that for every b ∈ B, thereare a unique finite subset I of N and unique non-zero elements ai ∈ Bi for every i ∈ Isuch that b ≡∑ ai. An element of⋃i≥0Bi is called homogeneous. If a ∈ Bi 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 B0 ismultiplicatively closed, i.e. B0 can be seen as a subblueprint of B. The subblueprint B0equals B if and only if for all i > 0, Bi = {0}. In this case we say that B is triviallygraded. By the uniqueness of the decomposition into homogeneous elements, we haveBi ∩Bj = {0} for i 6= j. This means that the union⋃i≥0Bi has the structure of a wedgeproduct∨i≥0Bi. Since∨i≥0Bi is multiplicatively closed, it can be seen as a subblueprintJ. Lo´pez Pen˜a’s research was supported by MCIM grant MTM2010-20940-C02-01, research group FQM-266(Junta de Andalucı´a) and Max-Planck Institute for Mathematics in Bonn.1arXiv:1203.1665v1  [math.AG]  8 Mar 20122 JAVIER LO´PEZ PEN˜A AND OLIVER LORSCHEIDof B. We define Bhom =∨i≥0Bi and call the subblueprint Bhom the homogeneous part ofB.Let S be a multiplicative subset of B. If b/s is an element of the localization S−1Bwhere f is homogeneous of degree i and s is homogeneous of degree j, then we say thatb/s is a homogeneous element of degree i− j. We define S−1B0 as the subset of homoge-neous elements of degree 0. It is multiplicatively closed, and inherits thus a subblueprintstructure from S−1B. If S is the complement of a prime ideal p, then we write B(p) forthe subblueprint (Bp)0 of homogeneous elements of degree 0 in Bp.An ideal I of a graded blueprint B is called homogeneous if it is generated by homo-geneous elements, i.e. if for every c ∈ I , there are homogeneous elements pi, qj ∈ I andelements ai, bj ∈ B and an additive relation∑aipi + c ≡∑bjqj in B.Let B be a graded blueprint. Then we define ProjB as the set of all homogeneousprime ideals p of B that do not contain B+hom =∨i>0Bi. The set X = ProjB comestogether with the topology that is defined by the basisUh = { p ∈ X | h /∈ p }where h ranges throughBhom and with a structure sheafOX that is the sheafification of theassociation Uh 7→ B[h−1]0 where B[h−1] is the localization of B at S = {hi}i≥q0.Note that if B is a ring, the above definitions yield the usual construction of ProjB forgraded rings. In complete analogy to the case of graded rings, one proves the followingtheorem.Theorem 2.1. The space X = ProjB together with OX is a blue scheme. The stalk at apoint p ∈ ProjB is Ox,p = B(p). If h ∈ B+hom, then Uh ' SpecB[h−1]0. The inclusionsB0 ↪→ B[h−1]0 yield morphisms SpecB[h−1]0 → SpecB0, which glue to a structuralmorphism ProjB → SpecB0. If B is a graded blueprint, then the associated semiring B+ inherits a grading. Namely,let Bhom =∨i≥0Bi the homogeneous part of B. Then we can define B+i as the additiveclosure of Bi in B+, i.e. as the set of all b ∈ B such that there is an additive relation ofthe form b ≡ ∑ ak in B with ak ∈ Bi. Then ∨B+i defines a grading of B+. Similarly,the grading of B induces a grading on a tensor product B ⊗C D with respect to blueprintmorphisms C → B and C → D under the assumption that the image of C → B iscontained in B0. Consequently, a grading of B implies a grading of Binv = B⊗F1 F12 andof the ring B+Z = B+inv. Along the same lines, if both B and D are graded and the imagesof C → B and C → D lie in B0 and C0 respectively, then B ⊗C D inherits a gradingobtained from the gradings of B and D.3. PROJECTIVE SPACEThe functor Proj allows the definition of the projective space PnB over a blueprint B.Namely, the free blueprint C = B[T0, . . . , Tn] over B comes together with a natural grad-ing (cf. [9, Section 1.12] for the definition of free blueprints). Namely, Ci consists of allmonomials bT e00 · · ·T enn such that e0 + · · ·+ en = i where b ∈ B. Note that C0 = B andChom = C. The projective space PnB is defined as ProjB[T0, . . . , Tn]. It comes togetherwith a structure morphism PnB → SpecB.In case of B = F1, the projective space PnF1 is the monoidal scheme that is known fromF1-geometry (see [4], [1, Section 3.1.4]) and [10, Ex. 1.6]). The topological space of PnF1is finite. Its points correspond to the homogeneous prime ideals (Si)i∈I of F1[S0, . . . , Sn]where I ranges through all proper subsets of {0, . . . , n}.In case of a ring B, the projective space PnB does not coincide with the usual pro-jective space since the free blueprint B[S0, . . . , Sn] is not a ring, but merely the blue-print of all monomials of the form bSe00 · · ·Senn with b ∈ B. However, the associatedscheme +PnB = (PnB)+ coincides with the usual projective space over B, which equalsProjB[S0, . . . , Sn]+.PROJECTIVE GEOMETRY FOR BLUEPRINTS 34. CLOSED SUBSCHEMESLet X be a scheme of finite type. By an F1-model of X we mean a blue scheme Xof finite type such that X+Z is isomorphic to X . Since a finitely generated Z-algebra is,by definition, generated by a finitely generated multiplicative subset as a Z-module, everyscheme of finite type has an F1-model. It is, on the contrary, true that a scheme of finitetype possesses a large number of F1-models.Given a scheme X with an F1-model X , we can associate to every closed subschemeY of X the following closed subscheme Y of X , which is an F1-model of Y . In case thatX = SpecB is the spectrum of a blueprintB = AR, and thus X ' SpecB+Z is an affinescheme, we can define Y as SpecC for C = AR(Y ) where R(Y ) is the pre-additionthat contains∑ai ≡∑bj whenever∑ai =∑bj holds in the coordinate ring ΓY of Y .This is a process that we used already in [10, Section 3].Since localizations commute with additive closures, i.e. (S−1B)+Z = S−1(B+Z ) whereS is a multiplicative subset of B, the above process is compatible with the restriction toaffine opens U ⊂ X . This means that given U = Spec(S−1B), which is an F1-model forX ′ = U+Z , then the F1–model Y ′ that is associated to the closed subscheme Y ′ = X ′×X Yof X ′ by the above process is the spectrum of the blueprint S−1C. Consequently, we canassociate with every closed subscheme Y of a scheme X with an F1-model X a closedsubscheme Y of X , which is an F1–model of Y; namely, we apply the above process toall affine open subschemes of X and glue them together, which is possible since additiveclosures commute with localizations.In case of a projective variety, i.e. a closed subscheme Y of a projective space +PnZ ,we derive the following description of the associated F1-model Y in PnF1 by homogeneouscoordinate rings. Let C be the homogeneous coordinate ring of Y , which is a quotient ofZ[S0, . . . , Sn]+ by a homogeneous ideal I . Let R be the pre-addition on F1[S0, . . . , Sn]that consists of all relations∑ai ≡∑bj such that∑ai =∑bj in C. Then B =F1[S0, . . . , Sn]R inherits a grading from F1[S0, . . . , Sn] by defining Bi as the imageof F1[S0, . . . , Sn]i in B. Note that B ⊂ C and that the sets Bi equal the intersectionsBi = Ci ∩ B for i ≥ 0 where Ci is the homogeneous part of degree i of C. Then theF1-model Y of Y equals ProjB.5. F1–MODELS FOR GRASSMANNIANSOne of the simplest examples of projective varieties that is not a toric variety (andin particular, not a projective space) is the Grassmann variety Gr(2, 4). The problem offinding models over F1 for Grassmann varieties was originally posed by Soule` in [12], andsolved by the authors by obtaining a torification from the Schubert cell decomposition (cf.[8, 7]).In this note, we present F1-models for Grassmannians as projective varieties definedthrough (homogeneous) blueprints. The proposed construction for the Grassmannians fitswithin a more general framework for obtaining blueprints and totally positive blueprintsfrom cluster data (cf. the forthcoming preprint [6]).Classically, the homogeneous coordinate ring for the Grassmannian Gr(k, n) is ob-tained by quotienting out the homogeneous coordinate ring of the projective space P(nk)−1by the homogeneous ideal generated by the Plu¨cker relations. A similar construction canbe carried out using the framework of (graded) blueprints. In what follows, we make thatconstruction explicit for the Grassmannian Gr(2, 4).Define the blueprint OF1(Gr(2, 4)) = F1[x12, x13, x14, x23, x24, x34]R where thecongruence R is generated by the Plu¨cker relation x12x34 + x14x23 ≡ x13x24 (the signshave been picked to ensure that the totally positive part of the Grassmannian is preserved,cf. [6]). SinceR is generated by a homogeneous relation,OF1(Gr(2, 4)) inherits a grading4 JAVIER LO´PEZ PEN˜A AND OLIVER LORSCHEIDfrom the canonical morphismpi : F1[x12, x13, x14, x23, x24, x34] −→ F1[x12, x13, x14, x23, x24, x34]R.Let Gr(2, 4)F1 := Proj(OF1(Gr(2, 4))). The base extension Gr(2, 4)+Z is the usual Grass-mannian, and pi defines a closed embedding of Gr(2, 4)F1 into P5F1 , which extends to theclassical Plu¨cker embedding Gr(2, 4)+Z ↪→ +P5Z.Homogeneous prime ideals in OF1(Gr(2, 4)) are described by their generators as theproper subsets I ( {x12, x13, x14, x23, x24, x25} such that I is either contained in one ofthe sets {x12, x34}, {x14, x23}, {x13, x24}, or otherwise I has a nonempty intersectionwith all three of them. In other words, I cannot contain elements in two of the above setswithout also containing an element of the third one.FIGURE 1. Points of the Grassmannian Gr(2, 4)F1 .Generator xij belonging to an ideal is depicted as segment i–j in 1 234The structure of the set of (homogeneous) prime ideals of OF1(Gr(2, 4)) is depicted inFigure 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 resultingin a model essentially different to the one presented in [8] by means of torifications, whichhad 6+12+11+5+1 = 35 points, in correspondence with the coefficients of the countingpolynomial NGr(2,4)(q) = 6 + 12(q− 1) + 11(q− 1)2 + 5(q− 1)3 + 1(q− 1)4. It is worthnoting that despite arising from different constructions, both F1-models for Gr(2, 4) have6 =(42)closed points, corresponding to the combinatorial interpretation of Gr(2, 4)F1 asthe set of all subsets with two elements inside a set with four elements. These six pointscorrespond to the F1-rational Tits points of Gr(2, 4)F1 , which reflect the naive notion ofF1-rational points of an F1-scheme (cf. [10, Section 2.2]).Like in the classical geometrical setting, the Grassmannian Gr(2, 4)F1 does admit acovering by six F1-models of affine 4-space, which correspond to the open subsets ofGr(2, 4)F1 where one of x12, x34, x14, x23, x13 or x24 is non-zero. However, these F1-models of affine 4-space are not the standard model A4F1 = Spec(F1[a, b, c, d]), but the“2 × 2-matrices” M2,F1 = Spec(F1[a, b, c, d]〈ad ≡ bc + D〉) in case that one of x12,x34, x14 or x23 is non-zero, and the “twisted 2×2-matrices”Mτ2,F1 = Spec(F1[a, b, c, d]〈ad+ bc ≡ D〉) in case that one of x13 or x24 is non-zero.PROJECTIVE GEOMETRY FOR BLUEPRINTS 5REFERENCES[1] Chenghao Chu, Oliver Lorscheid, and Rekha Santhanam. Sheaves and K-theory for F1-schemes. Adv.Math., 229(4):2239–2286, 2012.[2] Alain Connes and Caterina Consani. Characteristic 1, entropy and the absolute point. Preprint,arXiv:0911.3537v1, 2009.[3] Anton Deitmar. Schemes over F1. In Number fields and function fields—two parallel worlds, volume 239of Progr. Math., pages 87–100. Birkha¨user Boston, Boston, MA, 2005.[4] Anton Deitmar. F1-schemes and toric varieties. Beitra¨ge Algebra Geom., 49(2):517–525, 2008.[5] Kazuya Kato. Toric singularities. Amer. J. Math., 116(5):1073–1099, 1994.[6] Javier Lo´pez Pen˜a. F1-models for cluster algebras and total positivity. In preparation.[7] Javier Lo´pez Pen˜a and Oliver Lorscheid. Mapping F1-land. an overview of geometries over the field withone element. In Noncommutative geometry, arithmetic and related topics, pages 241–265. Johns HopkinsUniversity Press, 2011.[8] Javier Lo´pez Pen˜a and Oliver Lorscheid. Torified varieties and their geometries over F1. Math. Z., 267(3-4):605–643, 2011.[9] Oliver Lorscheid. The geometry of blueprints. Part I: algebraic background and scheme theory. Adv. Math.,229(3):1804–1846, 2012.[10] Oliver Lorscheid. The geometry of blueprints. Part II: Tits-Weyl models of algebraic groups. Preprint,arXiv:1201.1324, 2012.[11] Grigory Mikhalkin. Tropical geometry. Unpublished notes, 2010.[12] Christophe Soule´. Les varie´te´s sur le corps a` un e´le´ment. Mosc. Math. J., 4(1):217–244, 312, 2004.[13] Koen Thas. Notes on F1, I. Combinatorics of D0-schemes and F1-geometry. In preparation.DEPARTMENT OF MATHEMATICS, UNIVERSITY COLLEGE LONDON, 25 GOWER STREET, LONDON WC1E6BT, UNITED KINGDOME-mail address: jlp@math.ucl.ac.ukDEPARTMENT OF MATHEMATICS, UNIVERSITY OF WUPPERTAL, GAUSSSTR. 20, 42097 WUPPERTAL,GERMANY.E-mail address: lorscheid@math.uni-wuppertal.de

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