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On the semi-simplicity of Galois actions. (English) Zbl 1167.14308

The author gives a short proof of the “well-known” (but never published) result in arithmetic geometry stating that on a smooth, projective algebraic variety \(X\) over a finite field \(K\), the property that the absolute Galois group \(G_K\) acts partially semi-simply on the étale cohomology of \(X\) (Property \(S(X)\)) implies the statement of a semi-simple action (Property \(SS(X)\)). The first proof of this statement appeared in manuscript notes informally distributed at the time of the 1991 Seattle conference on motives by U. Jannsen, N. Katz and W. Messing. A precise description of the properties \(S(X)\) and \(SS(X)\) is the following.
Let \(K\) be a finitely generated field and let \(l\) be a prime number different from \(\text{char}(K)\).
Property \(SS(X)\). The action of \(G_K\) on \(H^\ast(\overline X,\mathbb{Q}_l)\) is semi-simple.
Property \(S(X)\). For all \(i\geq 0\), the action of \(G_K\) on \(H^{2i}(\overline X,\mathbb{Q}_ l(i))\) is “semi-simple at the eigenvalue \(1\)”, i.e. the map induced by the identity \[ H^{2i}(\overline X,\mathbb{Q}_ l(i))^{G_K} \to H^{2i}(\overline X,\mathbb{Q}_ l(i))_{G_K} \] is bijective (equivalently said, Property \(S^i(X)\) holds \(\forall i\geq 0\)).
In the paper the author shows that if \(X\) is of dimension \(d\) over a field \(K\) contained in the algebraic closure of a finite field of positive characteristic different from \( l\), then \(S^d(X\times X)\Longleftrightarrow SS(X)\) (Theorem 6).
While the implication \(SS(X) \Rightarrow S^d(X\times X)\) is shown to hold for any field \(K\) (of characteristic different from \( l\)), by using classical results in the theory of Lie algebras associated to finite-dimensional semi-simple algebras over fields of characteristic zero, the proof of the implication \(S^d(X\times X) \Rightarrow SS(X)\) relies on the further hypothesis on \(K\).
The main result of the paper is the following (Theorem 7).
Theorem: Let \(F\) be a finitely generated field over \(F_p\) and let \(X\) be a smooth, projective variety of dimension \(d\) over \(F\). Let \(\mathcal O\) be a valuation ring of \(F\) with finite residue field, such that \(X\) has good reduction at \(\mathcal O\). Let \(Y\) be the special fibre of a smooth projective model \(\tilde X\) of \(X\) over \(\mathcal O\). Then, \(S^d(Y\times Y)\Rightarrow SS(X).\)
The proof relies on P. Deligne’s geometric semi-simplicity theorem [Publ. Math., Inst. Hautes Étud. Sci. 52, 137–252 (1980; Zbl 0456.14014) (Cor. 3.4.13)].
In the final part of the paper the author shows, using a similar technique, a related result when \(F\) is finitely generated over \(Q\).

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
12F10 Separable extensions, Galois theory

Citations:

Zbl 0456.14014
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References:

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