1-motives and geometric monodromy. (Exposé VII: 1-motifs et monodromie géométrique.) (French) Zbl 0830.14001

Fontaine, Jean-Marc (ed.), Périodes \(p\)-adiques. Séminaire de Bures- sur-Yvette, France, 1988. Paris: Société Mathématique de France, Astérisque. 223, 295-319 (1994).
For a scheme \(S\) an \(S\)-\(1\)-motive \(M = [Y \to G]\) is herein defined by
(i) an \(S\)-group scheme \(Y\) locally (in the étale topology of \(S)\) isomorphic to \(\mathbb{Z}^r\),
(ii) a commutative \(S\)-group scheme \(G\), extension of an abelian scheme \(A\) by a torus \(T\), and
(iii) an \(S\)-homomorphism \(u : Y \to G\).
If \(S\) is the spectrum of an algebraically closed field the definition reduces to that given by P. Deligne [Inst. Hautes Étud. Sci., Publ. Math. 44 (1974), 5-77 (1975; Zbl 0237.14003)]. For \(S = \text{Spec} (R)\) where \(R\) is a complete discrete valuation ring and \(K\) its function field, a \(K\)-\(1\)-motive with good reduction (resp. potentially good reduction) is defined by the property of yielding an \(R\)-\(1\)-motive (resp. after a finite extension of \(K)\). To any \(K\)-\(1\)-motive \(M = [Y \to G]\) is canonically associated a strict \(K\)-\(1\)-motive, i.e. \(M' = [Y' \to G']\) such that \(G'\) has potentially good reduction, a quasi- isomorphism \(M_{\text{rig}}' \to M_{\text{rig}}\) in the derived category of bounded complexes of \(fppf\)-sheaves on the small rigid site of \(\text{Spec} (K)\), producing a canonical isomorphism \(T_\ell (M') \cong T_\ell (M)\) between the \(\ell\)-adic realizations, for any prime \(\ell\). For a strict \(K\)-\(1\)-motive \(M = [Y \to G]\), the geometric monodromy \(\mu : Y \times Y^* \to \mathbb{Q}\) (where \(Y^*\) is the character group of the torus \(T)\) is defined by dealing with the trivialization of the Poincaré biextension, canonically associated with the Cartier dual of \(M\). The geometric monodromy is zero if and only if \(M\) has potentially good reduction. Finally, the geometric monodromy is related with the \(\ell\)-adic realization.
For the entire collection see [Zbl 0802.00019].


14A20 Generalizations (algebraic spaces, stacks)
14G20 Local ground fields in algebraic geometry
14F30 \(p\)-adic cohomology, crystalline cohomology