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The structure of Deitmar schemes, I.

1. Introduction. One of the earliest realizations of [F.sub.1] was through the scheme theory developed by Deitmar [1], which is based on the observation that commutative rings over [F.sub.1] could be imagined as commutative multiplicative monoids (with an absorbing element). For these "rings", one can naturally define (prime) ideals, localization, a Spec-construction, etc., and Deitmar has proven that a natural base extension to Z of varieties in this context leads to toric varieties [2]. Also, there is a promising theory for zeta functions of Deitmar schemes [2] which agrees with Kurokawa's theory for zeta functions of schemes "defined over [F.sub.1]" [6].

More general scheme theories over [F.sub.1] have seen the light of day since Deitmar's (see for instance [7, 8] for an account), but in one way or the other, Deitmar schemes always appear to be the core of such schemes. So Deitmar schemes are one of the very basic objects in [F.sub.1]-theory, and one needs to understand them as well as possible. (We refer to the monograph [12] for background on the object [F.sub.1].)

Besides the Algebraic Geometry side of [F.sub.1]-theory, there is also the combinatorial-synthetic side (cf. [10]): in an old paper [13], Tits already described symmetric groups as Chevalley groups over [F.sub.1], seen as limit objects of projective general linear groups over finite fields, and the corresponding geometric modules (buildings), which are just projective spaces over the same fields in this case, become complete graphs with the full subgraph structure. (Tits also describes several other spherical buildings over [F.sub.1] in loc. cit.; their automorphism groups are the Weyl groups of the thick buildings of the same type; see also [10].)

In this note, we want to glue these two theories together. We will start with a loose graph, which is a more general object than a graph, and we will construct a Deitmar scheme from it of which the closed points correspond to the vertices of the loose graph. Several fundamental properties of the Deitmar scheme can then be obtained easily from the combinatorics of the loose graph, such as connectedness, the automorphism group, etc. And very interestingly, it appears that a number of combinatorial [F.sub.1]-objects (such as the combinatorial [F.sub.1]-projective space of above) are just loose graphs, and moreover, the associated Deitmar schemes will precisely be the scheme versions in Deitmar's theory of these objects. As a model example, the projective space scheme Proj([F.sub.1][[X.sub.0], ..., [X.sub.n]]) arises from the complete graph [C.sub.n + 1] on n + 1 vertices, and the latter is precisely how a projective space of dimension n over [F.sub.1] should look like (Tits). For further reference, we call a complete graph on n + 1 vertices (n any cardinal number) with the full complete subgraph structure a projective space of dimension n over [F.sub.1]. An edge with its two vertices is a combinatorial projective line over [F.sub.1], and more generally, any subset of vertices together with the induced graph structure defines a linear subspace. Then if [Aut.sub.sch](x), respectively [Aut.sub.synth](x), denotes the scheme theoretic automorphism group, respectively the combinatorial ("synthetic") automorphism group, and [GAMMA] is a loose graph while S([GAMMA]) is the corresponding Deitmar scheme, we have

(1) [Aut.sub.Sch](S([GAMMA])) [congruent to] [Aut.sub.Synth]([GAMMA]).

As a corollary, we will obtain that any group can occur as the full automorphism group of some Deitmar scheme.

So we will bring combinatorial [F.sub.1]-objects and Deitmar schemes together through loose graphs (and a more general construction).

2. Deitmar schemes and examples. Define an [F.sub.1]-ring to be a commutative monoid with an absorbing element 0.

For the definition of Deitmar scheme, we refer to [1] or [11]. Let us just mention that one defines an affine Deitmar scheme Spec(A) similarly as an affine Grothendieck scheme (by building a Zariski-type topology and structure sheaf of [F.sub.1]-rings on the set of monoidal prime ideals of the commutative monoid with zero A). A general Deitmar scheme then is a monoidal space (a topological space endowed with a sheaf of [F.sub.1]-rings), locally isomorphic to affine Deitmar schemes. (Deitmar schemes are sometimes also called [D.sub.0]-schemes or [M.sub.0]-schemes in some papers.)

2.1. Polynomial rings. Define

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

that is, the union of {0} and the (abelian) monoid generated by the [X.sub.j].

2.2. Affine space. Let A = [F.sub.1][[X.sub.1], ..., [X.sub.n]]. Denote Spec([F.sub.1][[X.sub.1], ..., [X.sub.n]]) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and call it the n-dimensional affine space over [F.sub.1]. The [not equal] (0) prime ideals of A are of the form

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where I is a subset of {1, ..., n} and ([X.sub.i]) = [X.sub.i]A = {[X.sub.i]a | a [member of] A}. The stalk of the structure sheaf at [p.sub.I] is the localization of A at the multiplicative set S that contains all products of elements [X.sub.j] where j [??] I.

2.3. Proj-schemes. In [9] we introduced the Proj-scheme construction for Deitmar schemes (see also [11]). We quickly repeat this procedure.

Monoid quotients. Let M be a commutative unital monoid (with 0), and I an ideal of M. We define the monoidal quotient M/I to be the set {[m] [member of] M|m [member of] M}/([m] = [0] if m [member of] I).

The Proj-construction. Consider the [F.sub.1] ring [F.sub.1][[X.sub.0], [X.sub.1], ..., [X.sub.m]], where m [member of] N. Since any polynomial is homogeneous in this ring, we have a natural grading

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [R.sub.i] consists of the elements of [F.sub.1][[X.sub.0], [X.sub.1], ..., [X.sub.m]] of total degree i, for i [member of] N. The irrelevant ideal is defined as

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(It is just the monoid minus the element 1.) Now Proj([F.sub.1][[X.sub.0],..., [X.sub.m]]) =: Proj([F.sub.1][X]) consists, as a set, of the prime ideals of [F.sub.1][[X.sub.0], [X.sub.1], ..., [X.sub.m]] which do not contain Irr (so only Irr is left out of the complete set of prime ideals). The closed sets of the (Zariski) topology on this set are defined as usual: for any ideal I of [F.sub.1][[X.sub.0], [X.sub.1], ..., [X.sub.m]], we define

(6) V(I) := {p|p [member of] Proj([F.sub.1][X]), I [[subset].bar] p}, where V(I) = [??] if I = Irr and V ({0}) = Proj([F.sub.1](X)), the open sets then being of the form

(7) D(I) := {p|p [member of] Proj([F.sub.1][X]), I [??] p}.

It is obvious that Proj([F.sub.1][X]) is a Deitmar scheme. (The structure sheafis described below in a more general setting.) Each ideal ([X.sub.i]) defines an open set D(([X.sub.i])) such that the restriction of the scheme to this set is isomorphic to Spec([F.sub.1][[X.sub.(i)]]), where [X.sub.(i)] is [X.sub.0], [X.sub.1], ..., [X.sub.m] with [X.sub.i] left out.

Remark 2.1. Note that the closed points and projective sublines of Proj([F.sub.1][[X.sub.0], ..., [X.sub.m]]) form a complete graph on m + 1 vertices, so we can easily switch between combinatorial [F.sub.1]-projective spaces and Proj([F.sub.1][X])-schemes.

In general, suppose M is a commutative unital monoid (with 0) with a grading

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the [M.sub.i] are the sets with elements of total degree i (for i [member of] N), and let, as above, their relevant ideal be Irr = {0} [upsilon] [[??].sub.i [greater than or equal to] 1] [M.sub.i]. Define the topology Proj(M) as before (noting that homogeneous (prime) ideals are the same as ordinary monoidal (prime) ideals here). For an open U, define [O.sub.M](U) as consisting of all functions

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [M.sub.(p)] is the subset of [M.sub.p] of fractions of elements with the same degree, for which f(p) [member of] [M.sub.(p)] for each p [member of] U, and such that there exists a neighborhood V of p in U, and elements u, [upsilon] [member of] M, for which [upsilon] [??] q for every q [member of] V, and f(q) = [bar.u]/[upsilon] in [M.sub.(q)].

In this way we obtain a sheaf of [F.sub.1]-rings on Proj(M) making it a Deitmar scheme.

Now recall the following results.

Theorem 2.2 ([1]). Let A be an [F.sub.1]-ring.

(i) For each p [member of] Spec(A), the stalk [O.sub.p] of the structure sheaf is isomorphic to the localization of A at p.

(ii) For global sections, we have [GAMMA](Spec(A), O) := O(Spec(A)) [congruent to] A.

Theorem 2.3 ([1]).

(i) For an [F.sub.1]-ring A, we have that (Spec(A), [O.sub.A]) is a monoidal space.

(ii) If [alpha] : A [right arrow] B is a morphism of monoids, then [alpha] induces a morphism of monoidal spaces

(10) (f, [f.sup.#]) : Spec(B) [right arrow] Spec(A), yielding a functorial bijection

(11) Hom(A, B) [right arrow] [Hom.sub.loc](Spec(B), Spec(A)), where on the right hand side we only consider local morphisms.

For the sake of convenience, we will make the surjectivity part of Theorem 2.3(ii) explicit.

Theorem 2.4. Any local morphism of monoidal spaces (f, [f.sup.#]): Spec(B) [right arrow] Spec(A) is induced by a monoidal morphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as in Theorem 2.3(ii).

Proof. Let (f, [f.sup.#]) be as in the statement of the theorem; then taking global sections, [f.sup.#] induces a morphism [phi] : [GAMMA](Spec(A), O) [right arrow] [gamma](Spec(B), O), which by Theorem 2.2 is a morphism [phi] : A [right arrow] B. For any p [member of] Spec(B), we have a local morphism [f.sup.#.sub.p] : [A.sub.f(p)] [right arrow] [B.sub.p] such that the following diagram commutes:

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As [f.sup.#] is a local homomorphism, we have that [[phi].sup.-1](p) = f(p), so that f coincides with the map Spec(B) [right arrow] Spec(A) induced by [phi]. It follows readily that the monoid homomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] induces (f, [f.sup.#]).

Since any automorphism is local, we have the following implication.

Corollary 2.5. If (f, [f.sup.#]) [member of] Aut(Spec(A)) is such that f = 1 implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [f.sup.#] also is trivial.

So if the topology of Spec(A) is sufficiently fine, the only element in Aut(Spec(A)) with trivial component f, is the trivial one.

3. Loose graphs. Define a loose graph ("L-graph") to be a triple (V, E, I),where V is a set of "vertices", E is a set of "edges" (V [upsilon] E = [??]), and I is a symmetric relation on (V x E) [upsilon] (E x V) (which indicates when a vertex and an edge are incident), with the additional property that each edge is incident with at most two distinct vertices. (In other words, it relaxes the definition of graphs, in that an edge can now also have one, or even no, endpoint(s). Also, since we introduce loose graphs as incidence geometries, we do not allow loops, and the geometry is undirected.)

3.1. Embedding theorem. Let [GAMMA] = (V E I) be a loose graph. We define a projective space P([GAMMA]) over [F.sub.1] as follows. Let E' [[subset].bar] E be the set of "loose edges"--edges with only a single endpoint. On each of these edges, we add a new endpoint, as such creating a point set V' which is in bijective correspondence with E'. Now P([GAMMA]) is the complete graph on the vertex set V [upsilon] V'. As such, we have an embedding of geometries

(13) [psi] : [GAMMA] [right arrow] P([GAMMA]) = P,

where P is the combinatorial projective space over [F.sub.1] of dimension [absolute value of V] + [absolute value of V'] - 1. If [GAMMA] is a graph, then E' = [??] and the dimension of P is [absolute value of V] - 1.

Theorem 3.1. The following properties clearly hold.

DIM P has minimal dimenson [absolute value of V] + [absolute value of V'] - 1 with respect to the embedding property (that is, there is no combinatorial projective space over [F.sub.1] of smaller dimension in which [GAMMA] embeds).

AUT Each automorphism of [GAMMA] is faithfully induced by an automorphism of P. (So that in particular, only the identity automorphism of P can fix any element of E [upsilon] V.)

3.2. Example. Projective completion. Note that if one starts with a combinatorial affine space A over [F.sub.1], considered as a loose graph, P(A) is precisely the projective completion of A.

4. From loose graphs to Deitmar schemes.

4.1. Patching and the functor [THETA]. Now let [GAMMA] = (V, E, I) be a not necessarily finite graph. We will give a patching" argument as follows.

Consider P = P([GAMMA]), and note that since [GAMMA] is a graph, P\[GAMMA]--when P is considered as a graph--is just a set S of edges. Let [mu] be arbitrary in S, and let z be one of the two (closed) points on [mu] in P = Proj([F.sub.1][[[X.sub.i]].sub.i[member of]V]) (recall Remark 2.1). Suppose that in the projective space P, z is defined by the ideal generated by the polynomials

(14) [X.sub.i], i [member of] V, i [not equal to] j = j(z).

Let P(z) be the complement in P of z; it is a hyperplane defined by [X.sub.j] = 0 (and it forms a complete graph on all the points but z). Denote the corresponding closed subset of Proj([F.sub.1][[[X.sub.i]].sub.i[member of]V]) by C(z). Let z' [not equal to] z be the other point of the edge [mu] corresponding to the index j' = j(z') [member of] V. Define the subset P(z') = P\{z'} of V, and denote the corresponding closed subset by C(z'). Finally, define

(15) C([mu]) = C(z) [upsilon] C (z').

It is also closed in Proj([F.sub.1][[[X.sub.i]].sub.i[member of]V]), and the corresponding closed subscheme is the projective space P "without the edge [mu]"; the coordinate ring is [F.sub.1][[[X.sub.i]].sub.i[member of]V]/[I.sub.[mu]] (where ([X.sub.j][X.sub.l]) =: [I.sub.[mu]]) and its scheme is the Proj-scheme defined by this ring. Now introduce the closed subset

(16) C([GAMMA]) = [[union].sub.[mu][member] of]SC([mu]).

Then C([GAMMA]) defines a closed subscheme S([GAMMA]) which corresponds to the graph [GAMMA]. We have

(17) S([GAMMA]) = Proj([F.sub.1][[[X.sub.i]].sub.i[member of]V]/[[union].sub.[mu][member] of]S [I.sub.[mu]]).

In this presentation, an edge corresponds to a relation, and we construct a coordinate ring for [THETA]([GAMMA]) = S([GAMMA]) by deleting all relations of the ambient space P([GAMMA]) which are defined by edges in the complement of [GAMMA]. We call a Deitmar scheme S([GAMMA]) constructed from a graph T a G-scheme.

A similar construction can be done for loose graphs, cf. [section][section]4.3.

4.2. Automorphism groups. The next theorem shows that the automorphism group of projective spaces from the incidence geometrical point of view, which we denote by [Aut.sub.synth](.), coincides with the automorphism group from the point of view of [F.sub.1]-schemes, denoted [Aut.sub.sch](.).

Theorem 4.1. Let P be a projective space over [F.sub.1], and let Proj([F.sub.1][[[X.sub.i]].sub.i [member of] P]) be the corresponding projective scheme. Then we have

(18) [Aut.sub.Synth](P) [congruent to] [Aut.sub.Sch](Proj([F.sub.1][[[X.sub.i]].sub.i [member of] P])).

Proof. It is clear that any element of [Aut.sub.inc](P) [congruent to] Sym(P) induces an automorphism of Proj([F.sub.1][[[X.sub.i]].sub.i[member of]P]), by permuting the set {[X.sub.i]|i [member of] P}. (This is just the action of the symmetric group on the closed points or hyperplanes.) So if [Aut.sub.sch](Proj([F.sub.1][[[X.sub.i]].sub.i [member of] P])) strictly contains [Aut.sub.synth](P), there are elements in [Aut.sub.sch](Proj([F.sub.1][[[X.sub.i]].sub.i [member of] P])) fixing all closed points. Now if some scheme automorphism of Proj([F.sub.1][[[X.sub.i]].sub.i [member of] P]) enjoys this feature, then it is easy to see that all prime ideals are also fixed (since prime ideals are of the form [[union].sub.j]R[X.sub.i], with R = [F.sub.1][[[X.sub.i]].sub.i [member of] P]), and hence the topology is fixed element-wise. By Corollary 2.5 this induces a trivial sheaf isomorphism, which is what we wanted to prove.

In fact, this is only a special case of a more general result which we will encounter later.

A similar proof (considering the action on the ideals that correspond to the "directions" instead of the closed points) leads to the same theorem for affine spaces:

Theorem 4.2. Let A be an affine space over [F.sub.1], and let Spec([F.sub.1][[[X.sub.i]].sub.i [member of] A]) be the corresponding Deitmar scheme. Then we have

(19) [Aut.sub.Synth](A) [congruent to] [Aut.sub.Sch](Spec([F.sub.1][[[X.sub.i]].sub.i [member of] A])).

At the level of [F.sub.1] there are no translations, nor exotic scheme automorphisms (each automorphism of an affine Deitmar scheme is linear, i.e., linearly extends to an automorphism of the projective completion, as is the case for its combinatorial version). (The corresponding [F.sub.1]-ring automorphisms are the ones given by permuting indices--no proper polynomial automorphisms occur.)

The following theorem, using the notation of the introductory paragraph of this section, is easy to obtain. We will denote the category of (undirected, loopless) graphs and natural morphisms by G.

Theorem 4.3. For any element [GAMMA] [member of] G, we have that

(20) Aut[([GAMMA]).sub.synth] [congruent to] Aut[(S([GAMMA])).sub.sch].

Proof. Obviously any graph automorphism of T induces naturally a scheme automorphism of S([GAMMA]). Suppose [alpha] [member of] Aut(S([GAMMA]))\Aut([GAMMA]). As any such [alpha] induces a homeomorphism of the topology, an automorphism is naturally induced on [GAMMA], so we may as well suppose that [alpha] [equivalent to] 1 on [GAMMA], so also on the topology of S([GAMMA]). Write S([GAMMA]) = [[union].sub.i]Spec([A.sub.i]), the Spec([A.sub.i]) being affine Deitmar schemes. Then each such scheme is isomorphic to an affine space scheme (by the simple form of the G-schemes), and the theorem easily follows from Theorem 4.2.

As a corollary, we can show easily that any (finite or infinite) group can occur as the automorphism group of some Deitmar scheme.

Corollary 4.4. Each group H is the full automorphism group of some Deitmar scheme.

Proof. By combined work of Frucht [4] (for the finite case), de Groot [3] and Izbicki [5] (for the infinite case), any group H is the full automorphism group of some graph. Now apply the previous theorem.

4.3. Extension to loose graphs. Let [GAMMA] = (V, E) be a connected loose graph. We distinguish three types:

type I graphs;

type II complements of graphs [DELTA] [[subset].bar] C (where C is some complete graph in which [DELTA] is embedded);

type III loose graphs not of type I nor II.

If [GAMMA] is of type I, we have seen how to associate a closed [D.sub.0]-subscheme S([GAMMA]) of P([GAMMA]) to [GAMMA]. If [GAMMA] is of type II, then we define the Deitmar scheme S([GAMMA]) naturally on the open set of P([GAMMA]) which is the complement of the (closed) point set of the graph [[GAMMA].sup.c] (the complement of [GAMMA] in P([GAMMA])). If [GAMMA] is of type III, S([GAMMA]) is the Deitmar scheme defined by the intersection of the closed subscheme defined on its graph theoretical completion [bar.[GAMMA]] [not equal to] [GAMMA], and the open set which is the complement of the complete graph defined on the vertices of [bar.[GAMMA]]\[GAMMA]. As such we have:

Proposition 4.5. Each loose graph [GAMMA] defines a Deitmar scheme S([GAMMA]).

We call such a scheme a loose scheme.

Denote the category of loose (undirected, loop-less) graphs and natural morphisms by LG. The following theorem is obtained in a similar way as Theorem 4.3.

Theorem 4.6. For any element [GAMMA] [member of] LG, we have that

(21) Aut[([GAMMA]).sub.synth] [congruent to] Aut[(S([GAMMA])).sub.sch].

4.4. Connectedness. Elements of the category of loose schemes have many important properties which can easily be read from the corresponding loose graph--recall for instance Theorem 4.3. Another one is:

Theorem 4.7. A loose scheme S([GAMMA]) is connected if and only if the loose graph [GAMMA] is connected.

Proof. Suppose [GAMMA] = (V, E) is not connected. It is clearly sufficient to consider the case where we have two connected components, say [[GAMMA].sub.1] and [[GAMMA].sub.2]. As P([GAMMA]) = P(P([[GAMMA].sub.1]) [??]P([[GAMMA].sub.2])), the fact that there are no relations between generators in different components readily implies that S([GAMMA]) is also not connected as a Deitmar scheme.

The converse is similar.

5. Weighted incidence geometries and Deitmar schemes. It is clear that the functor

(22) L : LG [right arrow] D :[GAMMA] [right arrow] S([GAMMA])

from the category of loose graphs to the category of Deitmar schemes is not surjective at all. Still, it is possible to adapt the ideas of above to make the functor surjective. We propose to associate a Deitmar scheme to a weighted incidence geometry (that is, an incidence geometry coming with a weight function on the point set) in a related way as one does for loose graphs. As such, all Deitmar schemes could be constructed from a combinatorial geometry, and they could be studied through these geometries.

Let [GAMMA] = (P, B, I) be a rank 2 incidence geometry, that is, P is a set of which the elements are "points", B consists of "lines" (P [union] B = [??]), and I is a symmetric relation on (P x B) [union] (B x V) called "incidence". Let [omega] : P [right arrow] [N.sup.x] be a weight function which assigns a strictly positive integer to each point. Assume now that

(#)1 any line has only a finite number of points;

(#)2 this number is at least two.

Define a Deitmar scheme as follows. For a line L, let [P.sub.L] be the points incident with L. Now define the Deitmar scheme S([GAMMA]) as

(23) S([GAMMA]) := Spec([F.sub.1][[[X.sub.u]].sub.u[member of]P]/[I.sub.[GAMMA]]),

where [I.sub.[GAMMA]] is the ideal

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and define S[??] as

(25) S[??] := Proj([F.sub.1][[[X.sub.u]].sub.u[member of]P]/[I.sub.[GAMMA]]).

So each line of T defines an ideal in [F.sub.1][[[X.sub.u]].sub.u[member of]P]. (It is now clear to the reader why we need (#)1 and (#)2.)

Remark 5.1. Note that the constructions of [section]4 and [section]5 can be adapted to Z-schemes.

In a future paper, we will study the functors S(x) and S[??] in this setting, and explain the connection with the loose graph functor. If everything works out well, we will then be able to describe and study (e.g.) toric varieties through rank 2 incidence geometries.

doi: 10.3792/pjaa.90.21

References

[1] A. Deitmar, Schemes over [F.sub.1], in Number fields and function fields--two parallel worlds, Progr. Math., 239, Birkhauser Boston, Boston, MA, 2005, pp. 87-100.

[2] A. Deitmar, Remarks on zeta functions and K-theory over [F.sub.1], Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 8, 141-146.

[3] J. de Groot, Groups represented by homeomorphism groups, Math. Ann. 138 (1959), 80-102.

[4] R. Frucht, Herstellung von Graphen mit vorge-gebener abstrakter Gruppe, Compositio Math. 6 (1939), 239-250.

[5] H. Izbicki, Unendliche Graphen endlichen Grades mit vorgegebenen Eigenschaften, Monatsh. Math. 63 (1959), 298-301.

[6] N. Kurokawa, Zeta functions over [F.sub.1], Proc. Japan Acad. Ser. A Math. Sci.81 (2005), no. 10, 180-184.

[7] J. Lopez Pena and O. Lorscheid, Mapping [F.sub.1]-land: an overview of geometries over the field with one element, in Noncommutative geometry, arithmetic, and related topics, Johns Hopkins Univ. Press, Baltimore, MD, 2011, pp. 241-265.

[8] O. Lorscheid, A blueprinted view on [F.sub.1]-geometry, in Absolute Arithmetic and [F.sub.1]-Geometry. (Submitted).

[9] K. Thas, Notes on [F.sub.1], I, Unpublished notes, 2012.

[10] K. Thas, The Weyl functor--Introduction to Absolute Arithmetic, in Absolute Arithmetic and [F.sub.1]-Geometry. (Submitted).

[11] K. Thas, The combinatorial-motivic nature of [F.sub.1]-schemes, in Absolute Arithmetic and [F.sub.1]-Geometry. (Submitted).

[12] K. Thas (ed.), Absolute Arithmetic and [F.sub.1]-Geometry. (Submitted).

[13] J. Tits, Sur les analogues algebriques des groupes semi-simples complexes, in Colloque d'algebre superieure, tenu a Bruxelles du 19 au 22 decembre 1956, Centre Belge de Recherches Mathematiques, Etablissements Ceuterick, Louvain, 1957, pp. 261-289.

Koen Thas

Ghent University, Department of Mathematics, Krijgslaan 281, S25, B-9000 Ghent, Belgium
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