Deligne I Section 2

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $\mathbb{Z}\prime$ on $X$ is the sheaf locally isomorphic to the constant sheaf $\mathbb{Z}$, with invertible sections on an open set $U$ on $X$ corresponding to the orientations of $U$. An orientation of $X$ is an isomorphism from $\mathbb{Z}\prime$ to the constant sheaf $\mathbb{Z}$. The fundamental class of $X$ is a morphism $\mathrm{Tr}:H_c^n(X,\mathbb{Z}\prime )\to \mathbb{Z}$; if $X$ is oriented, it is identified with a morphism $\mathrm{Tr}:H_c^n(X,\mathbb{Z})\to \mathbb{Z}$. The Poincaré duality is expressed with the help of the fundamental class.
2. Complex varieties. -- Let $ℂ$ be an algebraic closure of $ℝ$. A smooth complex algebraic variety, or even a subjacent differentiable variety, is always orientable. To orient it, it suffices to orient $ℂ$ itself. This amounts to a choice:
   1. to choose one of the two roots of the equation $x^2=-1$; which is called $+i$;
   2. to choose an isomorphism of $ℝ/\mathbb{Z}$ with $U^1=\{z\in ℂ|\left|z\right|=1\}$; $+i$ is the image of 1/4;
   3. a choice of one of the two isomorphisms $x\mapsto \exp (\pm 2\pi ix)$ from $\mathbb{Q}/\mathbb{Z}$ to the group of roots of unity in $ℂ$, which extends by continuity to an isomorphism from $ℝ/\mathbb{Z}$ to $U^1$.

Let $\mathbb{Z}\left(1\right)$ denote the free $\mathbb{Z}$-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 $\mathbb{Z}\left(1\right)=\ker (\exp :ℂ\to {ℂ}^{*})$. The generator $y=\pm 2\pi i$ corresponds to the isomorphism c): $x\mapsto \exp \left(xy\right)$.

Let $\mathbb{Z}\left(r\right)$ be the $r^{\text{th}}$ tensor power of $\mathbb{Z}\left(1\right)$. If $X$ is a complex algebraic variety of pure complex dimension $r$, the orientation sheaf of $X$ is the constant sheaf of value $\mathbb{Z}\left(r\right)$.

(2.2) To "orient" an algebraic variety over an algebraically closed field $k$ of characteristic 0, one must choose an isomorphism from $\mathbb{Q}/\mathbb{Z}$ to the group of roots of unity of $k$. The set of such isomorphisms is a principal homogeneous space for ${\hat{\mathbb{Z}}}^{*}$ (not only for ${\mathbb{Z}}^{*}$). Since our interest is only in the $\ell$-adic cohomology, it suffices to consider the roots of unity of order a power of $\ell$, and to assume that the characteristic $p$ of $k$ is different from $\ell$. Let $\mathbb{Z}/{\ell }^n\left(1\right)$ denote the group of roots of unity of $k$ dividing ${\ell }^n$. As $n$ varies, the $\mathbb{Z}/{\ell }^n\left(1\right)$ form a projective system, with transition maps $${\sigma }_{m,n}:\mathbb{Z}/{\ell }^m\left(1\right)⟶\mathbb{Z}/{\ell }^n\left(1\right):x⟼x^{{\ell }^{m-n}}.$$ Put ${\mathbb{Z}}_{\mathrm{\&ell}}\left(1\right)=\lim \mathrm{proj}\mathbb{Z}/{\ell }^m\left(1\right)$ and ${\mathbb{Q}}_{\mathrm{\&ell}}\left(1\right)={\mathbb{Q}}_{\mathrm{\&ell}}\left(1\right){\otimes }_{{\mathbb{Z}}_{\ell }}{\mathbb{Q}}_{\ell }$. Let ${\mathbb{Q}}_{\ell }\left(r\right)$ be the $r^{\text{th}}$ tensor power of ${\mathbb{Q}}_{\ell }\left(1\right)$; for $r\in \mathbb{Z}$, $r$ negative, we also put ${\mathbb{Q}}_{\ell }\left(r\right)={{\mathbb{Q}}_{\ell }(-r)}^{\vee }$.

As a vector space over ${\mathbb{Q}}_{\ell }$, ${\mathbb{Q}}_{\ell }\left(1\right)$ is isomorphic to ${\mathbb{Q}}_{\ell }$. The group of automorphisms of $k$ always acts nontrivally on ${\mathbb{Q}}_{\ell }\left(1\right)$: it acts via the character with values in ${\mathbb{Z}}_{\ell }^{*}$ which is given by its action on its roots of unity. In particular, if $k={\overline{𝔽}}_q$, 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 $\ell$-adic cohomology is the ${\mathbb{Q}}_{\ell }$-sheaf ${\mathbb{Q}}_{\ell }\left(n\right)$. The fundamental class is a morphism $$\mathrm{Tr}:H_c^{2n}(X,{\mathbb{Q}}_{\ell }(n\left)\right)⟶{\mathbb{Q}}_{\ell },$$ or, alternatively, $$\mathrm{Tr}:H_c^{2n}(X,{\mathbb{Q}}_{\ell })⟶{\mathbb{Q}}_{\ell }(-n).$$

Theorem (2.3) (Poincaré duality) For $X$ proper, smooth and pure of dimension $n$, the bilinear form $$\mathrm{Tr}(x\cup y):H^i(X,{\mathbb{Q}}_{\ell })\otimes H^{2n-i}(X,{\mathbb{Q}}_{\ell })⟶{\mathbb{Q}}_{\ell }(-n)$$ is a perfect duality (which identifies $H^i(X,{\mathbb{Q}}_{\ell })$ with the dual of $H^{2n-i}(X,{\mathbb{Q}}_{\ell }(n\left)\right)$).

(2.4) Let $X_0$ be a smooth proper algebraic variety over ${𝔽}_q$, pure of dimension $n$, and let $X$ over ${\overline{𝔽}}_q$ be obtained from $X_0$ by extension of scalars. The morphism (2.3) is compatible with the action of $\mathrm{Gal}({\overline{𝔽}}_q/{𝔽}_q)$. If the $\left({\alpha }_j\right)$ are the eigenvalues of the geometric Frobenius acting on $H^i(X,{\mathbb{Q}}_{\ell })$, the eigenvalues of $F$ acting on $H^{2n-i}(X,{\mathbb{Q}}_{\ell })$ are thus the $\left(q^n{\alpha }_j^{-1}\right)$.

(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 $H^i(X,{\mathbb{Q}}_{\ell })$ and $H^{2n-i}(X,{\mathbb{Q}}_{\ell })$ in perfect duality with values in $H^{2n}(X,{\mathbb{Q}}_{\ell })$, which is of dimension 1.
2. The cup product commutes with the reciprocal image $F^{*}$ by the morphism of Frobenius $F:X\to X$.
3. The morphism $F$ is finite of degree $q^n$:on $H^{2n}(X,{\mathbb{Q}}_{\ell })$, $F^{*}$ is the multiplication by $q^n$.
4. The eigenvalues of $F^{*}$ thus have the property (2.4).

(2.6) Put $\chi \left(X\right)=\sum _i{(-1)}^i\dim H^i(X,{\mathbb{Q}}_{\ell })$. If $n$ is odd, the form $\mathrm{Tr}(x\cup y)$ on $H^n(X,{\mathbb{Q}}_{\ell })$ is alternating; the integer $n\chi \left(X\right)$ is always even. One easily deduces from (1.5.4) and from (2.3), (2.4) that $$Z(X_0,t)=\epsilon q^{\frac{-n\chi \left(X\right)}{2}}{t}^{-\chi \left(X\right)}Z(X_0,q^{-n}{t}^{-1})$$ where $\epsilon =\pm 1$. If $n$ is even, let $N$ be the multiplicity of the eigenvalue $q^{n/2}$ of $F^{*}$ acting on $H^n(X,{\mathbb{Q}}_{\ell })$ (i.e. the dimension of the corresponding generalised eigenspace). Then $$\epsilon =\{\begin{array}{ll}1,& \text{if}n\text{is odd},\\ {(-1)}^N,& \text{if}n\text{is even.}\end{array}$$ 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 $ℱ$ is a ${\mathbb{Q}}_{\ell }$-sheaf on an algebraic variety $X$ over an algebraically closed field $k$, we will denote the sheaf $ℱ\otimes {\mathbb{Q}}_{\ell }\left(r\right)$ by $ℱ\left(r\right)$. This sheaf is (non canonically) isomorphic to $ℱ$.

Theorem (2.8) Suppose $X$ is smooth of pure dimension $n$ and $ℱ$ is locally constant. Let ${ℱ}^{\vee }$ be the dual of $ℱ$. The bilinear form $$$ Tr$(x\cup y):H^i(X,ℱ)\otimes H_c^{2n-i}(X,{ℱ}^{\vee }(n\left)\right)\to H_c^{2n-i}(X,ℱ\otimes {ℱ}^{\vee }(n\left)\right)\to H_c^{2n}(X,{\mathbb{Q}}_{\ell }(n\left)\right)\to {\mathbb{Q}}_{\ell }$$ is a perfect duality.

(2.9) Suppose $X$ is connected and let $x$ be a closed point in $X$. The functor $ℱ\mapsto {ℱ}_x$ is an equivalence between the category of locally constant ${\mathbb{Q}}_{\ell }$-sheaves with $\ell$-adic representations of ${\pi }_1(X,x)$. Via this equivalence, $H^0(X,ℱ)$ is identified with the invariants of ${\pi }_1(X,x)$ acting on ${ℱ}_x$: $$H^0(X,ℱ)\stackrel{\sim }{⟶}{ℱ}_x^{{\pi }_1(X,x)}.$$ PUT IN EQUATION NUMBER. By (2.8), for $X$ smooth and connected of dimension $n$, it follows that $$H_c^{2n}(X,ℱ)=H^0(X,{ℱ}^{\vee }(n){)}^{\vee }=(\left({ℱ}_x^{\vee }\right(n){)}^{{\pi }_1(X,x)}{)}^{\vee }$$ The duality exchanges invariants (the largest invariant subspace) and coinvariants (the largest invariant quotient). This formula can be is rewritten as $$H_c^{2n}(X,ℱ)=\left({ℱ}_x{)}_{{\pi }_1(X,x)}\right(-n),$$ 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 $ℱ$ a locally constant ${\mathbb{Q}}_{\ell }$-sheaf. Then

1. $H_c^0(X,ℱ)=0$ if $X$ is affine.
2. $H_c^2(X,ℱ)=\left({ℱ}_x{)}_{{\pi }_1(X,x)}\right(-1)$.

The assertion (i) simply indicates that $ℱ$ 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 $U\hookrightarrow X$ and $ℱ$ a locally constant ${\mathbb{Q}}_{\ell }$-sheaf on $U$. Let $j_{*}ℱ$ be the direct image constructible ${\mathbb{Q}}_{\ell }$-sheaf of $ℱ$. Its fiber at $x\in S$ 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 $$\mathrm{Tr}(x\cup y):H^i(X,j_{*}ℱ)\otimes H^{2-i}(X,j_{*}{ℱ}^{\vee }(1\left)\right)\to H^2(X,j_{*}ℱ\otimes j_{*}{ℱ}^{\vee }(1\left)\right)\to H^2(X,j_{*}(ℱ\otimes {ℱ}^{\vee }\left)\right(1\left)\right)\to H^2(X,j_{*}{\mathbb{Q}}_{\ell }(1\left)\right)=H^2(X,{\mathbb{Q}}_{\ell }(1\left)\right)\to {\mathbb{Q}}_{\ell }$$ is a perfect duality.

(2.13) It will be convenient to have at our disposal the ${\mathbb{Q}}_{\ell }$-sheaf ${\mathbb{Q}}_{\ell }\left(r\right)$ on an arbitrary scheme $X$ where $\ell$ is invertible. The point is to define the $\mathbb{Z}/{\ell }^n\left(1\right)$. By definition $\mathbb{Z}/{\ell }^n\left(1\right)$ is the etale sheaf of (${\ell }^n$)^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 ${\mathbb{Q}}_{\ell }$-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 $H^i$: VII; definition of the $H_c^i$: 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 $S=\mathrm{Spec}\left(k\right)$, $X=X$, $K=j_{*}ℱ$, $L={\mathbb{Q}}_{\ell }$) 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 $ℱ$), on may obtain it may a transcendental method, by lifting $X$ and $ℱ$ to characteristic 0.

Notes and References

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

References

[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 $\tau \left(n\right)$ 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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