By Pierre Deligne

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Write P = �x� where x = uh − u for some u ∈ U . Observe that β(x, x) = 0 and, in the orthogonal case Q(x) = Q(uh − u) = Q(uh) + Q(u) − β(uh, u) = 2Q(u) − β(u, u) = 0. Thus x is isotropic (singular in the orthogonal case), and there is a hyperbolic plane L = �x, y�. Observe that y �∈ P ⊥ , thus it is suﬃcient to extend h to �U, y�. Now �x� is a hyperplane in L, thus L⊥ h is a hyperplane in �xh�⊥ . Thus there exists y � ∈ V \U such that �xh, y � �⊥ = L⊥ h. Now, �xh, t� � = �xh, y �� � for some y �� ∈ V \U such that (x, y � ) is a hyperbolic pair.

Any maximal totally isotropic/ totally singular subspaces in V have the same dimension. This dimension is equal to the Witt index. 2. Anisotropic formed spaces. Let (V, κ) be a formed space. Recall that (V, κ) comes in three ﬂavours. Our aim in this subsection is to reﬁne Theorem 32 in each case – the ﬁrst we can do in total generality; for the other two we restrict ourselves to vector spaces over ﬁnite ﬁelds. 1. Alternating forms. Our ﬁrst lemma is nothing more than an observation. ating germ Lemma 36.

En }, then B is the set of all upper triangular matrices. The group B and any conjugate of B is called a Borel subgroup of G. (D2) Given a basis {e1 , . . , en }, the corresponding frame is the set F = { e1 , e2 , . . , en }. Let N be the stabilizer in G of the given frame. What is N ? Answer. N is the set of all monomial matrices, that is all matrices with precisely on nonzero entry in each row and colum. (D3) Show that G = N, B . Answer. Let g ∈ GLn (k) and let j be the last row such that aj1 = 0.

### Catégories tannakiennes by Pierre Deligne

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