By Alexander Polishchuk

The purpose of this e-book is to offer a latest therapy of the speculation of theta services within the context of algebraic geometry. the newness of its process lies within the systematic use of the Fourier-Mukai rework. the writer starts off by means of discussing the classical idea of theta features from the perspective of the illustration thought of the Heisenberg team (in which the standard Fourier remodel performs the favourite role). He then indicates that during the algebraic method of this thought, the Fourier–Mukai remodel can usually be used to simplify the present proofs or to supply thoroughly new proofs of many vital theorems. Graduate scholars and researchers with powerful curiosity in algebraic geometry will locate a lot of curiosity during this quantity.

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Then the composition of the two maps Hc0 (Gm /k, N ) → H 0 (A1 /k, j0 ! N ) → H 0 (Gm /k, N ) is an isomorphism. Therefore the first map is injective, and this implies that G I(∞) = 0, which in turn implies that H 1 (I(∞), G) = 0 (since 3. FIBRE FUNCTORS 23 G I(∞) and H 1 (I(∞), G) have the same dimension). And the second map is surjective, which gives the vanishing of H 1 (I(0), G), and this vanishing in turn implies the vanishing of G I(0) . , it is an irreducible perverse sheaf which is not a Kummer sheaf Lχ [1].

Using the long exact cohomology sequence, we reduce immediately to the case when N is irreducible. If N is punctual, the assertion 24 3. FIBRE FUNCTORS is obvious. , Hc−1 (Gm /k, F[1]) = Hc0 (Gm /k, F) = 0; the groups Hci (Gm /k, F[1]) = Hci+1 (Gm /k, F) with i ≤ −2 or i ≥ 2 vanish trivially. CHAPTER 4 The Situation over a Finite Field Let us now turn our attention to the case of a finite field k, and a groupscheme G/k which is a form of Gm . Concretely, it is either Gm /k itself, or it is the nonsplit form, defined in terms of the unique quadratic extension k2 /k inside the chosen k as follows: for any k-algebra A, G(A) := {x ∈ A ⊗k k2 | NormA⊗k k2 /A (x) = 1}.

1. Suppose that N in Parith is ι-pure of weight zero and arithmetically semisimple. Then the following six conditions are equivalent. (1) For j : G/k ⊂ X/k the inclusion of G/k into its complete nonsingular model, the “forget supports” map is an isomorphism j! N ∼ = Rj N . (2) The natural “forget supports” and restriction maps Hc0 (G/k, N ) → ω(N ) → H 0 (G/k, N ) are both isomorphisms. (3) The cohomology group ω(N ) is ι-pure of weight zero for the action of F robk2 . (3bis) The cohomology group Hc0 (G/k, N ) is ι-pure of weight zero for the action of F robk (or equivalently, for the action of F robk2 ).

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