Deligne I Section 2

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 12 March 2012

This is an attempt to translate Section 2 of [DeI].

The Grothendieck Theory: Poincaré duality

(2.1) To explain the relation between roots of unity and orientations, I will preliminarily retell two classic cases in a crazy language.

  1. Differentiable varieties. -- Let X be a differentiable variety of pure dimension n. The orientation sheaf on X is the sheaf locally isomorphic to the constant sheaf , with invertible sections on an open set U on X corresponding to the orientations of U. An orientation of X is an isomorphism from to the constant sheaf . The fundamental class of X is a morphism Tr: Hcn(X, ); if X is oriented, it is identified with a morphism Tr: Hcn(X, ). The Poincaré duality is expressed with the help of the fundamental class.
  2. Complex varieties. -- Let be an algebraic closure of . A smooth complex algebraic variety, or even a subjacent differentiable variety, is always orientable. To orient it, it suffices to orient itself. This amounts to a choice:
    1. to choose one of the two roots of the equation x2=-1; which is called +i;
    2. to choose an isomorphism of / with U1={z | |z|=1}; +i is the image of 1/4;
    3. a choice of one of the two isomorphisms x exp(±2πix) from / to the group of roots of unity in , which extends by continuity to an isomorphism from / to U1.

Let (1) 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 (1)= ker(exp: *). The generator y=±2πi corresponds to the isomorphism c): xexp(xy).

Let (r) be the rth tensor power of (1). If X is a complex algebraic variety of pure complex dimension r, the orientation sheaf of X is the constant sheaf of value (r).

(2.2) To "orient" an algebraic variety over an algebraically closed field k of characteristic 0, one must choose an isomorphism from / to the group of roots of unity of k. 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 p of k is different from . Let /n(1) denote the group of roots of unity of k dividing n. As n varies, the /n(1) form a projective system, with transition maps σm,n: /m(1) /n(1) : xxm-n. Put &ell(1) =limproj/m (1) and &ell(1) =&ell(1) . Let (r) be the rth tensor power of (1); for r, r negative, we also put (r) = (-r) .

As a vector space over , (1) is isomorphic to . The group of automorphisms of k always acts nontrivally on (1): it acts via the character with values in * which is given by its action on its roots of unity. In particular, if k= 𝔽q, the Frobenius substitution acts by multiplication by q.

Let X be a smooth pure algebraic variety of dimension n over k. The orientation sheaf of X in -adic cohomology is the -sheaf (n). The fundamental class is a morphism Tr:Hc2n (X, (n)) , or, alternatively, Tr:Hc2n (X, ) (-n) .

Theorem (2.3) (Poincaré duality) For X proper, smooth and pure of dimension n, the bilinear form Tr(xy): Hi(X, ) H2n-i (X,) (-n) is a perfect duality (which identifies Hi(X, ) with the dual of H2n-i (X, (n)) ).

(2.4) Let X0 be a smooth proper algebraic variety over 𝔽q, pure of dimension n, and let X over 𝔽 q be obtained from X0 by extension of scalars. The morphism (2.3) is compatible with the action of Gal( 𝔽q /𝔽q). If the (αj) are the eigenvalues of the geometric Frobenius acting on Hi (X,), the eigenvalues of F acting on H2n-i (X,) are thus the (qnαj-1).

(2.5) For simplification assume X is connected. The proof of (2.4) is translated as follows into a geometric language, then to a galoisian formulation (cf. (1.15)).

  1. The cup product puts Hi(X, ) and H2n-i (X,) in perfect duality with values in H2n (X,) , which is of dimension 1.
  2. The cup product commutes with the reciprocal image F* by the morphism of Frobenius F:XX.
  3. The morphism F is finite of degree qn:on H2n (X,) , F* is the multiplication by qn.
  4. The eigenvalues of F* thus have the property (2.4).

(2.6) Put χ(X) = i (-1)i dim Hi(X, ) . If n is odd, the form Tr(xy) on Hn(X, ) is alternating; the integer nχ(X) is always even. One easily deduces from (1.5.4) and from (2.3), (2.4) that Z(X0,t) = εq -nχ(X) 2 t-χ(X) Z(X0, q-n t-1) where ε=±1. If n is even, let N be the multiplicity of the eigenvalue qn/2 of F* acting on Hn(X, ) (i.e. the dimension of the corresponding generalised eigenspace). Then ε= { 1, ifn is odd, (-1)N, ifn is even. This is Grothendieck's formulation of the functional equation of the functions Z.

(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 X over an algebraically closed field k, we will denote the sheaf (r) by (r). This sheaf is (non canonically) isomorphic to .

Theorem (2.8) Suppose X is smooth of pure dimension n and is locally constant. Let be the dual of . The bilinear form Tr(xy) : Hi(X,) Hc2n-i (X,(n) ) Hc2n-i (X, (n) ) Hc2n (X, (n) ) is a perfect duality.

(2.9) Suppose X is connected and let x be a closed point in X. The functor x is an equivalence between the category of locally constant -sheaves with -adic representations of π1(X,x). Via this equivalence, H0 (X,) is identified with the invariants of π1(X,x) acting on x: H0 (X,) x π1(X,x) . PUT IN EQUATION NUMBER. By (2.8), for X smooth and connected of dimension n, it follows that Hc2n (X,) = H0(X, (n) ) = ((x (n) ) π1(X,x) ) The duality exchanges invariants (the largest invariant subspace) and coinvariants (the largest invariant quotient). This formula can be is rewritten as Hc2n (X,) = (x) π1(X,x) (-n), We will use this only for n=1.

Scholium (2.10) Let X be a smooth connected curve over an algebraically closed field k, x a closed point of X and a locally constant -sheaf. Then

  1. Hc0(X, )=0 if X is affine.
  2. Hc2(X, ) = (x) π1(X,x) (-1).
The assertion (i) simply indicates that does not have a section with finite support.

(2.11) Let X be a projective curve, smooth and connected over an algebraically closed field k, U an open set of X, the complement of a finite set S of closed points of X, j the inclusion UX and a locally constant -sheaf on U. Let j* be the direct image constructible -sheaf of . Its fiber at xS 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 Tr(xy): Hi(X, j*) H2-i (X,j* (1)) H2(X, j* j* (1)) H2(X, j*( )(1)) H2(X, j* (1)) = H2(X, (1)) is a perfect duality.

(2.13) It will be convenient to have at our disposal the -sheaf (r) on an arbitrary scheme X where is invertible. The point is to define the /n(1) . By definition /n(1) is the etale sheaf of (n)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 Hi: VII; definition of the Hci: 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 Z (1.14.3) is clearly treated in [1]; where, the Lefschetz formula (1.12), for a smooth projective curve X, 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 S=Spec(k), X=X, K= j*, L=) 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 X and to characteristic 0.

Notes and References

This is an attempt to translate Section 2 of [DeI].


[DeI] P. Deligne, La Conjecture de Weil I, Publ. Math. IHÉS (1974) 273-307. MR0340258.

[Gr] A. Grothendieck, Formule de Lefschetz et rationalité des fonctions L, Séminaire Bourbaki 279 décembre 1964. MR??????.

[Lf] S. Lefschetz, L'analysis situs et la géométrie algébrique (Gauthier-Villars), 1924. Reproduced in Selected papers (Chelsea Publ. Co.) MR??????.

[Rn] R.A. Rankin, Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions II, Proc. Camb. Phil. Soc., 35 (1939) 351-372. MR??????.

[Wl] A. Weil, Numbers of solutions of equations in finite fields, Bull. Am. Math. Soc. 55 (1949) 497--508. MR??????.

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