A characterization of Lie incidence geometries of exceptional type E - - PDF document

A characterization of Lie incidence geometries of exceptional type E7,4 and E8,4 Silvia Onofrei Mathematics Department, Kansas State University 137 Cardwell Hall, Manhattan KS 66506 email: onofrei@math.ksu.edu

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2

Let D be one of the following Dynkin diagrams:

• 1
• 2
• 3

4

• 5
• 6
• 7
• 1
• 2
• 3

4

• 5
• 6
• 7
• 8

Label the node 4 by P, points, and the node 3 by L, lines. We get in this way the Lie incidence geometries (Cooperstein, [8]) of exceptional type E7,4 and E8,4. The object of this paper is the following: The Main Theorem. Let ∆ be a parapolar space with thick

lines which is locally An,3, n = 6, 7. Then ∆ is the homo- morphic image of a building of type E7 or E8.

We start with a point-line geometry, ∆ = (P, L). Our approach is based on the local assumption that the residue of a point in ∆ is a ge-

• metry of Grassmann type An,3, n = 6, 7 and on the global assumption

that ∆ is a parapolar space. Cooperstein theory provides us with two classes of maximal singular subspaces, which will be denoted ¯ A and ¯ B and with a class of symplecta S, nondegenerate polar spaces of type D4. First, we prove that there is a class of 2-convex subspaces, D, each one isomorphic to D5,5(K), for some division ring K, and such that each symplecton S ∈ S is contained in a unique member D(S) ∈ D.

• ¯

B

• L

P

• ¯

A

• ¯

B

• L

P

• ¯

A

• D
• Now, let Γ be a locally D-truncated geometry over K = (P, L, ¯

A, ¯ B, D). A residually connected sheaf over all nonempty flags of Γ is constructed. Then ∆ is the homomorphic image of a building geometry belonging to the diagram D and whose truncation to K is isomorphic to Γ.

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• 1. Preliminaries and definitions

A geometry Γ is residually connected if and only if: (i) every flag of corank one lies in a maximal flag; (ii) any flag of corank at least two has a connected residue. Let f : ¯ Γ → Γ be an epimorphism of geometries over I. Then f is said to be a full epimorphism if whenever x ∼ y in Γ, there exists some ¯ x ∈ f −1(x) and some ¯ y ∈ f −1(y) with ¯ x ∼ ¯ y in ¯ Γ. (G) A point-line geometry Γ = (P, L) is a Gamma space if for x ∈ P and L ∈ L with |x⊥ ∩ L| > 1 then x ∈ L⊥. A parapolar space is a point-line geometry Γ = (P, L) which is defined by the following axioms:

• 1. Γ is a connected gamma space.
• 2. For any L ∈ L, L⊥ is not a clique in the point-collinearity graph.
• 3. For any pair x, y ∈ P, d(x, y) = 2, x⊥ ∩ y⊥ is empty, a point or

a nondegenerate polar space of rank at least 2. If |x⊥ ∩ y⊥| = 1 then (x, y) is called a special pair, if |x⊥ ∩ y⊥| > 1 then (x, y) is a polar pair. If no special pairs exist, Γ is called a strong parapolar space. It was proved by Cooperstein [9] that whenever (x, y) is a polar pair, i.e. x⊥ ∩ y⊥ is a nondegenerate polar space of rank k, the convex clo- sure of x and y, denoted by < x, y >, is a non-degenerate polar space

• f rank k + 1, which is called a symplecton. We write S =< x, y >.

The family of all symplecta in Γ will be denoted by S. The parapolar spaces are partial linear spaces whose singular subspaces are all projec-

• tive. Every singular subspace of rank at most 2 lies in a symplecton.

Given S ∈ S, x ∈ P \ S then x⊥ ∩ S is a singular, possibly emtpy, subspace of S.

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• 2. Some properties of An,3

Let V be a n + 1-dimensional vector space over some division ring K. Define a point-line geometry Γ = (P, L) as follows: the points of Γ are the 3-dimensional subspaces of V , the lines are the (2, 4)-flags of V . Then Γ is a Grassmann space of type An,3. In the sequel we restrict

• urselves to the case n = 6, 7.

There are two classes of maximal singular subspaces: (1) The class A, which are PG(3)’s; (2) The class B, which are PG(n − 2)’s. Let M = A ∪ B be the collection of all maximal singular subspaces

• f Γ. If A ∈ A, B ∈ B then A ∩ B is empty or a projective line. If

A, A′ ∈ A (B, B′ ∈ B) then A ∩ A′ (B ∩ B′) is empty or a point. The Grassmann spaces are strong parapolar spaces. Their symplecta are polar spaces of rank 3 and type D3, they are (1, 5)-flags of V . The point-collinearity graph has diameter 3. (1) If x ∈ P, S ∈ S, x ∈ S then x⊥ ∩S is empty, a point or a plane. (2) If S1, S2 ∈ S then S1 ∩ S2 can be empty, a point, a line or a common maximal singular subspace of the two symplecta. (3) If S ∈ S and M ∈ M then S ∩ M is empty, a point or a plane. In [13] Shult gave a characterization in terms of points and maximal singular subspaces, of certain parapolar spaces, including some Grass-

• manianns. We will use his results here, in order to formulate the fol-

lowing property of the Grassmann spaces A6,3 and A7,3: (D) For any maximal singular subspace M ∈ M and point p ∈ P \ M, x⊥ ∩ M is empty or a line.

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• 3. The class D of subspaces

In what follows Γ = (P, L) will be a parapolar space which is locally An,3(K) for n = 6, 7 and some division ring K (i.e. the residue of every point x ∈ P is a geometry of type An,3, n = 6, 7). (L.1) There are two classes of maximal singular subspaces ¯ A (which are PG(4)’s) and ¯ B (which are PG(n − 1)’s, n = 6, 7). Two maximal singular subspaces from the same class meet at the empty set or a line. Two maximal singular subspaces which belong to different classes meet at the empty set, a point or a plane. (L.2) Let ¯ Ai ∈ ¯ A, ¯ Bi ∈ B, i = 1, 2, S ∈ S and denote by Ai = ¯ Ai ∩ S, Bi = ¯ Bi ∩ S. Then Ai ∩ Bi can be a point or a plane and A1 ∩ A2 (or B1 ∩ B2) can be empty, a line or they are equal. In [13] Shult gave a characterization in terms of points and maximal singular subspaces, of certain parapolar spaces, including some Grass-

• manianns. We will use his results here, in order to formulate:

(L.3) Given ¯ M ∈ ¯ M and x ∈ P \ ¯ M then x⊥ ∩ ¯ M is empty, a point or a plane. (L.4) Given S ∈ S, x ∈ P \ S then x⊥ ∩ S is empty, a point, a line or a maximal singular subspace of S. Let S ∈ S. The two classes of maximal singular subspaces are: M4(S) = {A | A = ¯ A ∩ S for ¯ A ∈ ¯ A} Mn−1(S) = {B | B = ¯ B ∩ S for ¯ B ∈ ¯ B}, n = 6, 7 Then, we define: N4(S) = {x ∈ P | x⊥ ∩ S ∈ M4(S)} D(S) := S ∪ N4(S)

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• Theorem. Let Γ = (P, L) be a parapolar space which is locally An,3, n =

6, 7. There is a collection D of geodesically closed subspaces, each one isomorphic to D5,5(K), for some division ring K such that if S ∈ S then there is a unique member D(S) ∈ D containing S. In what follows, we will use the following notations: for S ∈ S and a point p ∈ N4(S) we will denote by Ap = p⊥ ∩ S ∈ M4(S) and by ¯ Ap =< p, Ap >∈ ¯ A. Proposition 1. Let S ∈ S, then D(S) is a subspace of Γ.

• Proof. Let x, y ∈ D(S) be two collinear points. If x, y ∈ S we are done

since S is a geodesically closed subspace of Γ. If x ∈ N4(S) and y ∈ S then the line L1 = xy ⊂ ¯ Ax ⊂ D(S). Thus we may assume that the line L1 = xy is such that L1 ∩ S = ∅. Ax, Ay are maximal singular subspaces of S belonging to the same class. Thus there are three possi- ble relations among them: Ax = Ay, Ax∩Ay = L a line or Ax∩Ay = ∅. a) Assume first that Ax = Ay. Let z ∈ L1 \ {x, y}. Since Ax = Ay ⊂ ¯ Ax ∩ ¯ Ay, by (L.2) it follows that ¯ Ax = ¯

• Ay. This means that L1 ⊂ ¯

Ax and thus z ∈ D(S). b) Assume now that Ax ∩ Ay = L ∈ L(S) and let z ∈ L1 \ {x, y}. Then L ⊂ z⊥. As L ⊂ z⊥ ∩ S we can have z⊥ ∩ S is a line or a maximal singular subspace of S. Let w ∈ Ay \ z⊥. Let π = w⊥ ∩ Ax. Now B =< w, π >∈ Mn−1(S). This is true since B ∩ Ax = π and B ∩ Ay =< w, L >, both planes in S. Therefore, B and Ax, Ay belong to different classes of maximal singular subspaces of S. Let R =< w, x >. Note that y ∈ R thus z ∈ R. Also B is a maximal singular subspace of R. In R, z⊥ ∩ B = πz (a plane) and therefore, by (L.4), z⊥ ∩ S = Az a maximal singular subspace of S. Thus, we have proved that if z⊥ ∩ S ⊃ L then z⊥ ∩ S must be a maximal sin- gular subspace of S. It remains to show that Az ∈ M4(S). Claim:

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x⊥ ∩ z⊥ ∩ S = L. Assume that Az ∩ Ax is a plane. Let p ∈ Ax ∩ Az \ L. Then p ∼ x, p ∼ z, x, z ∈ L1 and, by gamma space property (G), p ∼ y. Therefore p ∈ Ax∩Ay\L contradicting the fact that Ax∩Ay = L. The claim is proved. c) Assume that Ax ∩ Ay = ∅. Let u ∈ Ax, then u⊥ ∩ Ay = πu, a

• plane. Let R =< u, y >. Note that x ∈ R. Thus, in R, polar space,

x⊥ ∩πu = ∅, which contradicts the choice of x and y. Therefore, in this case x cannot be collinear to y.

• Lemma 1. Assume x ∈ N4(S) and y ∈ x⊥ is such that y⊥∩S\Ax = ∅.

Then y ∈ N4(S).

• Proof. Assume y⊥ ∩ S ⊃ {p}, p ∈ Ax. Then p⊥ ∩ Ax = πp, a plane.

Thus (x, p) is a polar pair. Let R =< x, p >. Note y ∈ x⊥ ∩ p⊥ ⊂ R. Therefore y⊥ ∩ πp = L a line and thus y⊥ ∩ S ⊃< p, L > which means that y⊥ ∩ S = M a maximal singular subspace of S. In order to prove that y ∈ N4(S) it suffices to show that M ∩ Ax = L. Clearly M ∩ Ax ⊃ L. Assume by contradiction that M ∩ Ax ⊃< q, L >= π a

• plane. Then < x, y, π > is a PG(4) inside the symplecton R. This is

a contradiction since R is a rank 4 polar space. If q ∈ L we are done. Otherwise, y ∈ ¯ Ax and p ∈ Ax, which contradicts the choice of p.

• Lemma 2. Let R, S ∈ S. If R ∩ S is a maximal singular subspace

B ∈ Mn−1(S) then D(S) = D(R).

• Proof. Need only to show that D(S) ⊆ D(R).

Clearly S ⊆ D(R). Let w ∈ S \ R. Then w⊥ ∩ B is a plane. Thus w⊥ ∩ R contains a plane and therefore it must be a maximal singular subspace, say Aw. Since Aw ∩ B is a plane, it follows that Aw and B belong to different classes of maximal singular subspaces in R. Thus w ∈ N4(R). Suppose now that x ∈ N4(S). Let Ax = x⊥ ∩ S. Then there exists a point p ∈ Ax ∩ B = Ax ∩ R. Let q ∈ B \ {p}. Then q⊥ ∩ Ax = πq, a plane and thus Bq =< q, πq > is a maximal singular subspace such that Bq ∩ B = pq (a line) and Bq ∩ Ax = πq (a plane). Thus Bq ∈ Mn−1(S).

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Set Rq =< x, q >. In Rq, choose a point w ∈ x⊥∩q⊥∩p⊥\Bq and note that, by Lemma 1, w ∈ N4(S) and let w⊥∩S = Aw. Now, Aw∩B = πw must be a plane and by the first part of this Lemma it follows at once that w ∈ N4(R) and w⊥ ∩ R = A′

w where A′ w ∈ M4(R). Note also

that A′

w ∩ B = πw. Now ¯

Aw =< w, Aw > and ¯ A′

w =< w, A′ w > are

two maximal singular subspaces from class ¯ A which meet in a PG(3), < πw, w >. By (L.1) it follows that w ∈ R. Thus x⊥ ∩ R ⊃ {w, p}. Let y ∈ Ax \ πq. Clearly y ∈ N4(R). Apply Lemma 1 to the pair (x, y) now and get x ∈ N4(R).

• Proposition 2. D(S) is a 2-convex subspace of Γ.
• Proof. First recall that 2-convex means that if x, y ∈ D(S) are two

points at distance 2 then x⊥ ∩ y⊥ ⊂ D(S). i) If x, y ∈ S then obviously x⊥ ∩y⊥ ⊂ S since S is geodesically closed. ii) Let now x ∈ N4(S) and y ∈ S. Let z ∈ x⊥ ∩ y⊥ \ S. Apply Lemma 1 to the pair (x, z). Thus z ∈ N4(S). iii) Consider now the case when x, y ∈ N4(S), x ∼ y and let z ∈ x⊥ ∩ y⊥ \ S. a) Ax = Ay there is nothing to prove since y ∈ ¯ Ax. b) Assume Ax ∩ Ay = L. Since x⊥ ∩ ¯ Ay ⊃ L it follows that x⊥ ∩ ¯ Ay is a plane on L, πy =< L, y1 > with y1 ∈ S. So < xy1, L > is a PG(3) which meets ¯ Ax and ¯ Ay at planes and thus it lies inside a maximal singular subspace of type ¯ B. Moreover, since d(x, y) = 2, then < x, y >= R with L ⊂ R ∩ S. Note y1 ∈ R. Also z ∈ x⊥ ∩ y⊥ and so z ∈ R and therefore z⊥ ∩ L = ∅. Let p ∈ z⊥ ∩ L. Since z⊥ ∩ ¯ Ax contains a line, it follows that it must be a plane. Also note that z and < y1, L > are in R and so z⊥∩ < y1, L > is a line. Assume z ∼ y1. It follows at once that L ⊂ z⊥. Now, Bx =< xz, L > is a PG(3) inside a maximal singular subspace of ¯

• B. Similarly, By =< zy, L >

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is a PG(3) which lies inside a maximal singular subspace of ¯ B type,

• too. Note that Bx ∩ ¯

Ax =< x, L > and By ∩ ¯ Ay =< y, L > are planes. On the other side, in R the plane < z, L > lies at the intersection of two unique maximal singular subspaces of different classes. But we’ve already found that < z, L >⊂ Bx ∩ By which is a contradiction unless Bx = By. This implies that z ∈ ¯ Ax (since z ∼ y1, z cannot be in ¯ Ay),

• therwise x ∼ y, which contradicts the choice of x and y. If y1 ∼ z

then z is collinear with the line yy1, which meets Ay at a point q = p. So z⊥ ∩S ⊃ {q}, q ∈ Ax. Now apply Lemma 1 and get that z ∈ N4(S). c) Assume Ax ∩ Ay = ∅. Choose v ∈ Ax. Set R =< v, y >. Note R ⊆ D(S), by Lemma 2. Thus D(R) = D(S), and it follows x ∈ D(R). Now we have z ∼ x, z⊥ ∩R\(x⊥ ∩R) = ∅ (since x ∼ y) and by Lemma 1 it follows that z ∈ D(R) = D(S). This ends the proof.

• Proposition 3. D(S) is a strong parapolar subspace of Γ.
• Proof. If x, y ∈ S or x ∈ S and y ∈ N4(S) or x, y ∈ N4(S) and

Ax ∩ Ay = ∅ the result is immediate. Thus, assume that x, y ∈ N4(S) and Ax ∩ Ay = ∅. Let p ∈ Ay and set R =< p, x >. Since R ∩ S =< p, p⊥ ∩ Ax >∈ Mn−1(S), by Lemma 2, D(R) = D(S). Now y ∈ D(S) = D(R) and therefore y⊥ ∩ R = A′

y, a maximal singular

subspace of R. But x ∈ R and x⊥ ∩A′

y is a plane contained in x⊥ ∩y⊥.

Thus, (x, y) is a polar pair.

• Lemma 3. Let S, R ∈ S be such that R ⊂ D(S) and R ∩ S = ∅. Then

R ∩ S = B ∈ Mn−1(S) and D(S) = D(R).

• Proof. Let S, R ∈ S be such that R ⊂ D(S) and R ∩ S = ∅. Let

p ∈ R ∩ S and choose x ∈ R such that x ∈ p⊥. Since R ⊂ D(S), x⊥∩S = Ax a maximal singular subspace of S. Now, in S, p⊥∩Ax = πp a plane. But R =< x, p > and πp ⊂< x, p > and therefore πp ⊂ R ∩ S. Thus, since p ∼ x, < p, πp >⊂ R ∩ S, is a maximal singular subspace. But < p, πp > ∩Ax = πp, a plane, it follows that < p, πp > and Ax belong to different classes. The second part of this lemma is immediate now, by Lemma 2, thus D(R) = D(S).

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Proposition 4. Let S ∈ S and R be some symplecton in D(S). Then D(S) = D(R).

• Proof. As mentioned above, besides S the symplecta from D(S) can

be such that: a) R ⊂ D(S) and R ∩ S = B ∈ Mn−1(S); b) T ⊂ D(S) and T ∩ S = ∅. The case a) was considered in Lemma 2. Assume now that T ⊂ D(S) and T ∩ S = ∅. Let x ∈ T, then x⊥ ∩ S = Ax. Let y ∈ S \ Ax. Then set R =< x, y >. Clearly, R ∩ S is a maximal singular subspace, thus D(S) = D(R). Now R ∩ T = ∅, since x ∈ R ∩ T and, by Lemma 4, D(S) = D(R) = D(T). This proves the proposition.

• Proposition 5. Given a symplecton S ∈ S there is a unique D(S) ∈ D

such that S ⊂ D(S). It is clear now that if x ∈ D(S) and R ⊂ D(S) a symplecton, then x⊥ ∩ R ⊂ N4(S), a maximal singular subspace of R. Now, using the characterization of D5,5(K), given by Cohen and Cooperstein in (Lemma 6, [7]), we can identify the subspace D(S) with D5,5(K), for some division ring K.

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• 4. Locally truncated geometries, chamber systems and

sheaves Locally truncated geometries. Let I be an index set and J ⊂ I. Set K = I \ J. Let M = (mij) be a Coxeter matrix with rows and columns indexed by I. The diagram of M is the edge labelled graph D(M) := (E, λ, I), λ : E → I, where (ij) is an edge labelled mij if and only if mij > 2. A geometry ∆ over I belongs to the diagram D(M) if and only if every residue of type {i, j} of ∆ is a generalised mij-gon. The J-truncation of ∆, J∆, is the truncation of type I \ J

• f ∆.

A geometry Γ over K is said to be locally truncated of type D

• ver I (or J-locally truncated), if and only if for every flag F of Γ,

the residue ResΓ(F) is isomorphic to the J-truncation of a geometry belonging to the diagram DI\t(F) (the restriction of D to the typeset I \ t(F)). Chamber systems. A chamber system over I is a mapping λ : C

2

• → 2I from the set of all 2-subsets of C to the set of all subsets of

the typeset I, such that for any a, b, c ∈ C: λ(a, b) ∩ λ(b, c) ⊆ λ(a, c). We denote it by C = (C, λ, I). The elements of C are called cham-

• bers. Two distinct chambers a and b are i-adjacent iff i ∈ λ(a, b).

The residue of C of type J is the set of the connected components of the graph (C, EJ, λJ), with λJ = λ |J and EJ = {e ∈ E | λJ(a, b) = ∅}. A chamber system over I is residually connected if and only if: a) For any two residues R1 and R2, of types t(R1) and t(R2) respec- tively, R1 ∩ R2 is either empty or is a residue of type t(R1) ∩ t(R2). b) Suppose R1, ...R2 is a sequence of residues with pairwise non-empty intersections, then ∩iRi is nonempty.

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A chamber system C belongs to the diagram D(M) if and only if every residue of C of type {i, j} is the chamber system of a general- ized mij-gon. A building is a chamber system of type M, with M a Coxeter matrix, in which every corank one residue is strongly gated.

• Sheaves. Let Γ be a geometry over K which is locally truncated of

type D, whose typeset is I. A sheaf over the geometry Γ is a function Σ which assigns to each non-empty flag F of Γ a geometry Σ(F) over I \ t(F) such that the following hold: (1) ResΓ(F) =J Σ(F) (2) For any two flags F1 ⊂ F2, we have: Σ(F2) ≃ ResΣ(F1)(F2 \ F1). If such a function Σ exists we say that the sheaf exists. A sheaf Σ is residually connected if and only if for each object x of the geometry Γ, the sheaf geometry Σ(x) is residually connected. Due to the functorial relation between the category of geometries and the category of the chamber systems, whenever a sheaf Σ exists, there is also a canonical defined chamber system associated to it: Lemma A. (Brouwer and Cohen [2]) Suppose Γ is a locally truncated diagram geometry of type K, over a diagram D of type I where |K| ≥ 3. Suppose that a sheaf Σ exists. There is a canonically defined chamber system C(Σ) over I. Suppose, for each object x ∈ Γ, that every rank 2 residue of Γ(x) is connected. Then the chamber system C(Σ) also belongs to diagram D.

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• 5. The main theorem

The Main Theorem. Let ∆ be a parapolar space with thick lines which is locally An,3, n = 6, 7. Then ∆ is the homomorphic image of a building of type E7 or E8. Let I = {1, ...m}, m = 7, 8 be an index set. Let K = {2, ...6}, J = I \ K. Denote by D one of the following diagrams of excep- tional Lie type:

• ¯

B

• L

P

• ¯

A

• D
• ¯

B

• L

P

• ¯

A

• D
• Let Γ be a locally D-truncated geometry over K, where Γ is the rank

5 geometry {P, L, ¯ A, ¯ B, D}.

• Theorem. With the notations from above: given the geometry Γ of

J-locally truncated type D, there exists a sheaf Σ of type I defined

• ver the family of all nonempty flags F of Γ with the property that

JΣ(F) = ResΓF for any nonempty flag F ∈ F.

The existence of the sheaf is proved by direct construction, using Brower- Cohen sheaf theory [2] and a result of Ellard and Shult [10] for string diagrams of type Cn. The sheaf Σ of type I defined over the set of all nonempty rank 1 and rank 2 flags of Γ and then extended to the set of all nonempty flags F

• f Γ. It assigns a geometry Σ(F) over I \ t(F) such that

JΣ(F) := ΓF

and with the property that whenever F1 ⊆ F2 is a pair of incident flags, there is an induced isomorphism Σ(F2) ≃ Σ(F1)(F2\F1)

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• f geometries over I \ t(F2). Moreover, since the sheaf values at rank

1 and 2 flags are buildings, or direct product of buildings, and hence simply 2-connected, the sheaf is residually connected. There exists a canonically defined chamber system C(Σ) over I which also belongs to the diagram D. The chamber system C(Σ) has all its rank 3 residues covered by build- ings and by Tits Local Approach Theorem, its universal 2-cover B, is the chamber system of a building of type D. Apply the functor G : Chamb → Geom: B

G

− − − → ˜ ∆

JTr

− − − → J ˜ ∆   h   h∗   h∗J C(Σ)

G

− − − → ∆

JTr

− − − → J∆ ˜ ∆ = G(B) is a building geometry belonging to the diagram D; h∗ : ˜ ∆ → ∆ is the functorially defined morphism of geometries The chamber system C(Σ) is residually connected, thus ∆ = G(C(Σ)), the geometry functorially defined by C(Σ) is also residually connected. Every flag of rank at most 2 in Γ is the image of a flag in ˜ ∆. Applying the functor JTr we get: Γ ≃J G(C(Σ)) ≃J ∆ The functorially induced map h∗J :J ˜ ∆ →J ∆ is a full epimorphism of geometries (see Lemma 12, [11]).

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Final remarks 1. We have to mention here that the existence of such a sheaf was stated without proof in Brouwer and Cohen (Theo- rem 4, [2]). They considered a finite geometry of truncated type with 4 known nodes, instead of 5. The existence of a sheaf is proved by di- rect construction. Our method uses Brouwer-Cohen sheaf theory and a technique developed by Ellard and Shult [10] to construct a sheaf over a string geometry of locally truncated type Cn. The sheaf defined over the string geometry has a global decomposition in two parts ”left” and ”right”. In the presence of the edge (3, 4) from diagram D, the global decomposability of the sheaf is lost and it is only partially restored at local level.

• 2. It doesn’t follow that ∆ is a building. There exists, for example,

a quotient of the unique building of type E7 over the field of complex numbers by a group of order 2, which is a geometry of type E7 but not a building (Example 4, [2]).

• 2. Both ˜

∆ and ∆ are parapolar spaces and there is a full epimorphism between their J-truncations. Thus we may ask ourselves in which con- ditions the above map becomes an isomorphism, which would mean that the locally truncated geometry ∆ itself will be a building. An ap- proximation to this would be to check the conditions (O), (LL), (LH) for E7 and (O), (LL), (LH), (HH) for E8, from Tits paper (Propo- sition 9, [16]). This is quite difficult in this case, since the objects involved in Tits’approach are ”square” nodes in our description, thus collections of the objects in Γ. An alternate way would be to impose certain restrictions which makes the full epirmorphism an isomorphism

• f parapolar spaces.

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References

[1] A.E. Brouwer, A.M. Cohen, Some remarks on Tits geometries: with an appen- dix by J. Tits, Nederl. Akad. Weterisch. Indag. Mat., 45(4)(1983), 393-402. [2] A. Brouwer, A. Cohen, Local recognition of Tits geometries of classical type,

• Geom. Dedicata, 20(1986), 181-199.

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