Last updates: 12 March 2012
This is an attempt to translate Section 2 of [DeI].
(2.1) To explain the relation between roots of unity and orientations, I will preliminarily retell two classic cases in a crazy language.
Let denote the free -module of rang one where the two element set of generators is in canonical correspondence with one of the two element subsets of a), b), c). The simplest is to take . The generator corresponds to the isomorphism c): .
Let be the tensor power of . If is a complex algebraic variety of pure complex dimension , the orientation sheaf of is the constant sheaf of value .
(2.2) To "orient" an algebraic variety over an algebraically closed field of characteristic 0, one must choose an isomorphism from to the group of roots of unity of . The set of such isomorphisms is a principal homogeneous space for (not only for ). Since our interest is only in the -adic cohomology, it suffices to consider the roots of unity of order a power of , and to assume that the characteristic of is different from . Let denote the group of roots of unity of dividing . As varies, the form a projective system, with transition maps Put and . Let be the tensor power of ; for , negative, we also put .
As a vector space over , is isomorphic to . The group of automorphisms of always acts nontrivally on : it acts via the character with values in which is given by its action on its roots of unity. In particular, if , the Frobenius substitution acts by multiplication by .
Let be a smooth pure algebraic variety of dimension over . The orientation sheaf of in -adic cohomology is the -sheaf . The fundamental class is a morphism or, alternatively,
Theorem (2.3) (Poincaré duality) For proper, smooth and pure of dimension , the bilinear form is a perfect duality (which identifies with the dual of ).
(2.4) Let be a smooth proper algebraic variety over , pure of dimension , and let over be obtained from by extension of scalars. The morphism (2.3) is compatible with the action of . If the are the eigenvalues of the geometric Frobenius acting on , the eigenvalues of acting on are thus the .
(2.5) For simplification assume is connected. The proof of (2.4) is translated as follows into a geometric language, then to a galoisian formulation (cf. (1.15)).
(2.6) Put . If is odd, the form on is alternating; the integer is always even. One easily deduces from (1.5.4) and from (2.3), (2.4) that where . If is even, let be the multiplicity of the eigenvalue of acting on (i.e. the dimension of the corresponding generalised eigenspace). Then This is Grothendieck's formulation of the functional equation of the functions .
(2.7) We will need other forms of the duality theorem. The case of curves will suffice for these purposes. If is a -sheaf on an algebraic variety over an algebraically closed field , we will denote the sheaf by . This sheaf is (non canonically) isomorphic to .
Theorem (2.8) Suppose is smooth of pure dimension and is locally constant. Let be the dual of . The bilinear form is a perfect duality.
(2.9) Suppose is connected and let be a closed point in . The functor is an equivalence between the category of locally constant -sheaves with -adic representations of . Via this equivalence, is identified with the invariants of acting on : PUT IN EQUATION NUMBER. By (2.8), for smooth and connected of dimension , it follows that The duality exchanges invariants (the largest invariant subspace) and coinvariants (the largest invariant quotient). This formula can be is rewritten as We will use this only for .
Scholium (2.10) Let be a smooth connected curve over
an algebraically closed field , a closed point of
and a locally constant
(2.11) Let be a projective curve, smooth and connected over an algebraically closed field , an open set of , the complement of a finite set of closed points of , the inclusion and a locally constant -sheaf on . Let be the direct image constructible -sheaf of . Its fiber at is of rank less than or equal to the rank of its fiber in a general point; this is the space of invariants under a local monodromy group.
Theorem (2.12) The bilinear form is a perfect duality.
(2.13) It will be convenient to have at our disposal the -sheaf on an arbitrary scheme where is invertible. The point is to define the . By definition is the etale sheaf of ()th roots of unity.
(2.14) Bibliographic indications on sections 1 and 2.
A) All the important results in etale cohomology are proved first for torsion sheaves. The extension to -sheaves is done by passage to formal limits. In what follows, for each theorem sited, I will not endeavour to indicate a reference where it is proved but a reference where its analogue for torsion sheaves is.
B) With the exception of the Lefschetz and (2.12), the results from etale cohomology used in this article are all proved in SGA 4. For those already introduced, the references are: definition of the : VII; definition of the : XVII 5.1; finiteness theorem: XIV 1, completed in XVII 5.3; cohomological dimension: X; Poincaré duality: XVIII.
C) The relation between the various Frobenius ((1.4), (1.11), (1.15)) is explained in detail in SGA5, XV, §§1, 2.
D) The cohomological interpretation of the functions (1.14.3) is clearly treated in ; where, the Lefschetz formula (1.12), for a smooth projective curve , is used, but not proved. For the proof, it is necessary, unfortunately, to see SGA 5.
E) The form (2.12) of the Poincaré duality follows from a general result SGA 4, XVIII (3.2.5) (for , , , ) by a local calculation which is not difficult. The statement will be given explicitly in the defnitive version of SGA 5. In the case which we will use (tame ramification of ), on may obtain it may a transcendental method, by lifting and to characteristic 0.
This is an attempt to translate Section 2 of [DeI].